we will discuss a classification of all simple weight modules over witt algebras with finite multiplicities. the result generalizes a well known result of o. mathieu for the virasoro algebra. this is a joint result with y. billig (carleton, canada).

i will present recent development about the structure of four dimensional shrinking ricci solitons. i will show how some basic information about scalar curvature allows us to better understand such solitons. for example, assuming the scalar curvature is bounded, these manifolds must have their curvature operator non-nengative at
infinity. furthermore, if the scalar curvature converges to zero, then they are asymptotically conical. some generalizations in higher dimension will also be discussed. this talk is based on joint work with jiaping wang.

the question of global regularity vs. finite time blow-up remains open for many fluid equations. even in the cases where global regularity is known, solutions may develop small scales as time progresses. in this talk, i will first discuss an active scalar equation which is an interpolation between the 2d euler equation and the surface quasi-geostrophic equation. we study the patch dynamics for this equation in the half-plane, and prove that the solutions can develop a finite-time singularity. i will also discuss a passive transport equation whose solutions are known to have global regularity, and our goal is to study how well a given initial density can be mixed if the incompressible flow satisfies some physically relevant quantitative constraints. this talk is based on joint works with a. kiselev, l. ryzhik and a. zlatos.

the problem of asymptotically efficient estimation of different kind of functionals for various statistical models has been extensively discussed in the literature. this talk is concerned with the estimation of spectral functionals for continuous-time gaussian stationary models.

suppose we observe a finite realization $\{x(t), \, 0\le t\le t\}$ of a centered real-valued stationary gaussian process $\{x(t), \, t\in\mathbb{r}\}$ with an {\it unknown\/} spectral density $\theta(\lambda).$ assume that $\theta(\lambda)$ belongs to a given (infinite-dimensional) class ${\mbox{\boldmath$\theta$}}$ of spectral densities possessing some smoothness properties. let $\phi(\cdot)$ be some {\it known\/} functional, the domain of definition of which contains ${\mbox{\boldmath$\theta$}}$. the problem is to estimate the value
$\phi (\theta)$ of the functional $\phi(\cdot)$ at an unknown point
$\theta \in {\mbox{\boldmath$\theta$}}$, and investigate the asymptotic properties of the suggested estimators, depending on the dependence structure of the model and smoothness structure of the "parametric" set ${\mbox{\boldmath$\theta$}}$. the main objective is construction of asymptotically efficient estimators for $\phi(\theta).$

we define the concepts of $h$- and ik-efficiency of estimators, based on the variants of h\'ajek convolution theorem and h\'ajek-le cam local asymptotic minimax theorem, respectively, and show that the simple "plug-in" statistic
$\phi (i_t)$, where $i_t=i_t(\lambda)$ is the periodogram of the underlying process $x(t)$, is $h$- and ik-asymptotically efficient estimator for a linear functional $\phi (\theta),$ while for a nonlinear smooth functional $\phi (\theta)$,
an $h$- and ik-asymptotically efficient estimator is the statistic
$\phi (\widehat \theta_t)$, where $\widehat \theta_t$ is a suitable sequence of the so-called "undersmoothed" kernel estimators of the unknown spectral density $\theta(\lambda).$

exact asymptotic bounds for minimax mean square risks of estimators of linear functionals will also be presented.

the hales--jewett theorem is one of the pillars of ramsey theory, from which many other results follow.

a celebrated result of shelah from 1988 gives a significantly improved bound for this theorem. a key tool used in his proof, now known as the cube lemma, has become famous in its own right. hoping to further improve shelah's result, more than twenty years ago, graham, rothschild and spencer asked whether there exists a polynoimal bound for this lemma. in this talk, we present the answer to their question and discuss numerous connections of the cube lemma with other problems in ramsey theory.

joint work with david conlon (oxford), jacob fox (mit), and benny sudakov (eth zurich)

let c be a smooth, absolutely irreducible genus-3 curve over a number field m. suppose that the jacobian of c has complex multiplication by a sextic cm-field k. suppose further that k contains no imaginary quadratic subfield. we give a bound on the primes p of m such that the stable reduction of c at p contains three irreducible components of genus 1.

joint work with bouw, cooley, lauter, manes, newton, ozman.

i will discuss various geometric gluing constructions. first i will discuss constructions for constant mean curvature hypersurfaces in euclidean spaces including my earlier work for two-surfaces in three-space which settled the hopf conjecture for surfaces of genus two and higher, and recent generalizations in
collaboration with christine breiner in all dimensions. i will then briefly mention gluing constructions in collaboration with mark haskins for special lagrangian cones in $c^n$. a large part of my talk will concentrate on doubling and
desingularization constructions for minimal surfaces and on applications on closed minimal surfaces in the round spheres, free boundary minimal surfaces in the unit ball, and self-shrinkers for the mean curvature flow. finally i will discuss my collaboration with simon brendle on constructions for einstein metrics on four-manifolds and related geometric objects.

low-rank structures are common in modern data analysis and signal processing, and they usually play essential roles in various estimation and detection problems. it is challenging to recover the underlying low-rank structures reliably from corrupted or undersampled measurements. in this talk, we will introduce convex and nonconvex optimization methods for low-rank recovery by two examples. the first example is community detection in network data analysis. in the literature, it has been formulated as a low-rank recovery problem, and then sdp relaxation methods can be naturally applied. however, the statistical advantages of convex optimization approaches over other competitive methods, such as spectral clustering, were not clear. we show in this talk that the methodology of sdp is robust against arbitrary outlier nodes with strong theoretical guarantees, while standard spectral clustering may fail due to a small fraction of outliers. we also demonstrate that a degree-corrected version of sdp works well for a real-world network dataset with a heterogeneous distribution of degrees. although sdp methods are provably effective and robust, the computational complexity is usually high and there is an issue of storage. for the problem of phase retrieval, which has various applications and can be formulated as a low-rank matrix recovery problem, we introduce an iterative algorithm induced by nonconvex optimization. we prove that our method converges reliably to the original signal. it requires far less storage and has much higher rate of convergence compared to convex methods.

in many areas of imaging science, it is difficult to determine the phase of linear measurements. for example, in some applications, one only has access to the pointwise absolute value of various masked fourier transforms of the desired signal. in this setting, the goal is to reconstruct the signal from the intensity measurements, that is, perform phase retrieval. this talk describes how to leverage ideas from spectral graph theory to design fourier masks in a way that allows for efficient signal reconstruction.

the well-known analytic class number formula, linking the special value at s=0 of the dedekind zeta function of a number field to its class number and regulator, has been the foundation and prototype for the highly conjectural theory of special values of l-functions for close to two centuries. we will discuss generalizations of the class
number formula to the context of equivariant artin l-functions which capture refinements of the brumer-stark and coates-sinnott conjectures. these generalizations relate various algebraic-geometric invariants associated to a global field, e.g. its quillen k-groups and etale cohomology groups, to various special values of its galois-equivariant l-functions. they illustrate the subtle interactions of number theory with complex and p-adic analysis, algebraic geometry, topology and homological algebra.

the density of primes $p\equiv 1\pmod{8}$ (resp. $p\equiv
7\pmod{8}$) such that the class number of $\mathbb{q}(\sqrt{-p})$ (resp.
$\mathbb{q}(\sqrt{-2p})$) is divisible by $2^{k+2}$ is conjectured to be
$2^{-k}$ for all positive integers $k$. the conjecture has been resolved
for $k = 1$ by the chebotarev density theorem. for the family of quadratic
fields $\mathbb{q}(\sqrt{-2p})$, we use methods of friedlander and iwaniec
to prove the conjecture for $k = 2$. moreover, we show that there are
infinitely many primes $p$ for which the class number of
$\mathbb{q}(\sqrt{-p})$ is divisible by $16$.

with jon weare (chicago), we study the continuum limit of a family of kinetic monte carlo models of crystal surface relaxation that includes both the solid-on-solid and discrete gaussian models. with computational experiments and theoretical arguments we are able to derive several partial differential equation limits identified (or nearly identified) in previous studies and to clarify the correct choice of surface tension appearing in the pde and the correct scaling regime giving rise to each pde. we also provide preliminary computational and analytic investigations of a number of interesting qualitative features of the large scale behavior of the models. the pde models involved are fully non-linear fourth order diffusion type equations with many interesting geometric features. we will also discuss recent progress analyzing existence and regularity properties of solutions to such pde.

due to the advance of single-molecule techniques, stochastic phenomena in chemistry and biology have been widely observed, which promotes the rapid development of stochastic modeling. i will discuss several stochastic processes in single-molecule enzyme kinetics, transcriptional burst and toggle switch. the stochastic modeling not only can explain some unexpected law from the trajectory perspective, but also can help uncover certain molecular mechanism and carefully analyze the effect of noise within gene regulation.

i will discuss the influence of the seed in models of randomly growing trees; in particular, i will focus on the preferential attachment and uniform attachment models. in both of these models, different seeds lead to different distributions of limiting trees from a total variation point of view. i will discuss the differences and similarities in proving this for the two models. this is based on joint work with sebastien bubeck, ronen eldan, and elchanan mossel.

i will explain part of my journey in this beautiful subject:

* super-strong approximation and its applications to other areas of math.

* counting lattices.

* ratner's theorems and its applications to number theory.

in each one of these topics, some open problems will be mentioned.

(disclaimer: most likely one of these topics will be discussed only briefly because of time restriction.)

angelakis proved in his thesis (2015) that there are many different imaginary quadratic fields that all have the "same" absolute abelian galois group, i.e., their absolute abelian galois group is isomorphic to some universal profinite group.

in this talk, we show that his techniques can be used to prove similar results for the adelic point groups associated to elliptic curves.

in random matrix theory, we study the histogram of eigenvalues of matrices sampled from given distributions on their entries, asymptotically as the dimension grows. the (computationally) nicest case -- independent gaussian entries -- was initiated by wigner in the 1950s, and by the 1980s this case was pretty completely understood.

in the last decade, the subject has exploded. most attention has thus far been focused on two natural broad generalizations of the gaussian case: either to matrices with independent entries sampled from some other distribution, or matrices whose joint density of entries has a log-concave potential. in both scenarios, we now know the asymptotic behavior is "universal": the limit law of eigenvalues, their fluctuations around this limit, and finer statistics like asymptotic spacing and edge behavior, are all the same essentially independent of the specific model used.

in this talk i will discuss the main story of random matrices as we know it today, and also discuss a very new direction: a third broad generalization of "gaussian" matrices. thinking of a gaussian as the solution to the heat equation on a linear space of matrices, i have been studying the eigenvalues of heat kernel distributed random matrices on lie groups. as we will discuss, some of the "universal" behavior from the studied models persists (like the size and form of the fluctuations), but in most cases, these models yield completely new asymptotics that we are only beginning to understand.

i will discuss divisibility and wild kernels in algebraic k-theory of number fields $f$ and present basic results concerning the divisible elements in k-groups. without appealing to the quillen-lichtenbaum conjecture one can prove that the group of divisible elements is isomorphic to the corresponding group of \' etale divisible elements.

one can apply this result for the proof of the $lim^1$ analogue of the quillen-lichtenbaum conjecture. one can also apply it to investigate: the imbedding obstructions in homology of $gl,$ the splitting obstructions for the quillen localization sequence, the order of the group of divisible elements via special values of $\zeta_{f}(s).$ i will discuss the relation of divisible elements with kummer-vandiver and iwasawa conjectures.

i will also present recent results, joint with cristian popescu, concerning brummer-stark conjecture and galois equivariant stickelberger splitting map in quillen localization sequence. the stickelberger splitting map is a basic tool to investigate the structure of the group of divisible elements.

the notion of universal enveloping algebra is introduced concerning a commutative poisson algebra in order to study the category of poisson modules over the poisson algebra. the main purpose of this talk is to propose the study of universal enveloping algebra for its own sake. some basic ring-theoretic and homological properties of the universal enveloping algebra are proved. we also ask some open questions, which suggests us to use universal enveloping algebra to understand the poisson structure of any commutative algebra.

compressed sensing is a new field that arose as a response to inefficient traditional signal acquisition schemes. under the assumption that the signal of interest is sparse, one wishes to take a small number of linear samples and later utilize a reconstruction algorithm to accurately recover the compressed signal. typically, one assumes the signal is sparse itself or with respect to some fixed orthonormal basis. however, in applications one instead more often encounters signals sparse with respect to a tight frame which may be far from orthonormal. in the first part of this lecture, we will introduce the compressed sensing problem as well as recent results extending the theory to the case of sparsity in tight frames. the second part of the lecture focuses on dictionary learning which is also a new field, but closely related to compressive sensing. briefly speaking, a dictionary is an overcomplete and redundant system consisting of prototype signals that are used to express other signals. due to the redundancy, for any given signal, there are many ways to represent it, but normally the sparsest representation is preferred for simplicity and easy interpretability. a good analog is the english language where the dictionary is the collection of all words (prototype signals) and sentences (signals) are short and concise combinations of words. in this lecture, we will introduce the problem of dictionary learning, its origin and applications, and existing solutions.

this talk gives an introduction to the most recent advances in optimization and numerical linear algebra.

in classical optimization theory, much has been known about local optimizers. there are standard conditions for local optimality, like the karush-kuhn-tucker (kkt) condition, second order sufficiency condition (sosc), strict complementarity condition (scc). there are no such conditions for global optimality in nonlinear programming theory. however, when all the functions are polynomial, there is a global optimality condition, which are closely related to kkt, sosc and scc.

in numerical linear algebra, much has been known about matrix computations, like ranks, eigenvalues, singular value decompositions, low rank approximations and matrix completions. tensors are natural generalizations of matrices, arising from wide applications. similar questions like for matrices need to be solved for tensors. however, such questions are mostly open for tensors.

this talk will give an overview of the new research results in the area, as well as the remaining challenges.

following a question of serre, the sylow subgroups of the absolute galois group of a p-adic field were studied and completely understood by labute. however, the structure of the p-sylow subgroups of the absolute galois group of the field of rational numbers is much more subtle and mysterious. we make progress towards its determination via a surprisingly simple decomposition. joint work with lior bary-soroker and moshe jarden.

a fundamental invariant associated to every complex rational function f is its monodromy group, that is, the galois group of the covering $f:p^1->p^1$ on the projective line $p^1$. we shall discuss the accumulating work towards determining the degree n complex rational functions with a specified monodromy group that is not $a_n$ or $s_n$, and its applications to problems in number theory, complex analysis, and complex dynamics. joint work with michael zieve.

we will describe uc's calculus online, now available to all uc 2022年亚洲世界杯预选赛 through our new cross campus enrollment system as well as to all non matriculated 2022年亚洲世界杯预选赛 including foreign nationals. calculus i for science and engineering 2022年亚洲世界杯预选赛 has been successfully running for over a year and calculus ii since last spring. calculus iii and iv are currently in development.

the courses have many components, from introductory welcome lectures, historical enrichment video lectures, online lecture videos (all synchronized with an online interactive e-text originally developed for print by ucla professor jon rogawski), to an online discussion forum platform all accessible via uc's canvas learning management system. we would very much welcome questions and suggestions.

perfectoid spaces were very recently introduced by scholze and kedlaya-liu, independently. these objects provide geometric context to the field-of-norms isomorphism from p-adic hodge theory, which relates absolute galois groups in characteristic 0 and characteristic p. in this talk, i will discuss what perfectoid spaces are, why they are soo hawt right now, and then present some natural questions about their geometry which i am interested in answering.

much of the structure of a division algebra can be garnered from the structure of its subspaces. for example, a division algebra of prime degree is cyclic if and only if it has a subspace each of whose elements have $p$-power in the center. we consider this and other conditions on the subspace, and describe the growth of $\{ \dim_k (kak)^i : i \geq 1 \}$, for a maximal separable subfield $k$ of a central simple algebra $a$ and $a \in a \setminus k.$ we tie this to brauer factor sets, the trace form, and the commutator question. this is joint work with matzri, saltman, and vishne, and in part with chapman.

a generic poisson algebra is a commutative associative algebra with an
anticommutative product (a bracket), which satisfies the leibnitz identity but, in general, does not satisfy the jacobi identity. we prove that the freiheitssatz holds in the variety of generic poisson algebras. in other words, every non-trivial equation over the free generic poisson algebra p has a solution in some extension of p. earlier this result was proved for ordinary poisson algebras by l.makar limanov and u.umirbaev. this is a joint work with p.kolesnikov and l.makar limanov.

the resolution of all microscopes is limited by diffraction. the observed signal is a convolution of the emitted signal with a low-pass kernel, the point-spread function (psf) of the microscope. the frequency cut-off of the psf is inversely proportional to the wavelength of light. hence, the features of the object that are smaller than the wavelength of light are difficult to observe. in single-molecule microscopy the emitted signal is a collection of point sources, produced by blinking molecules. the goal is to recover the location of these sources with precision that is much higher than the wavelength of light. this leads to the problem of super-resolution of positive sources in the presence of noise. we show that the problem can be solved using convex optimization in a stable fashion. the stability of reconstruction depends on rayleigh-regularity of the support of the signal, i.e., on how many point sources can occur within an interval of one wavelength. the stability estimate is complemented by a converse result: the performance of the convex algorithm is nearly optimal. i will also give a brief summary on the ongoing project, developed in collaboration with the group of prof. w.e. moerner, where we use the theoretical ideas to improve microscopes.

we study on-diagonal heat kernel estimates and exit time estimates for continuous time random walks (ctrws) among i.i.d. random conductances with a power-law tail near zero. for two types of natural ctrws, we give optimal exponents of the tail such that the behaviors are standard (i.e. similar to the random walk on the euclidean space) above the exponents. we then establish the local clt for the ctrws. we will also compare our results to the recent results by andres-deuschel-slowik.
this talk is a joint work with o. boukhadra (constantine) and p. mathieu (marseille).

the card game set is a game of pattern recognition. in this talk, we discuss the interesting mathematics behind a deck of set cards. we will discuss how set can be used to visualize the finite affine geometries of order 3. we also present recent results and discuss what terence tao has described as "perhaps [his] favourite open question."

this talk should be accessible to undergraduates and graduate 2022年亚洲世界杯预选赛 of all ages. the only background required is algebra on the level of math 100 and combinatorics on the level of an algebraist.

(joint work with m. follett, k. kalail, e. mcmahon, and c. pelland.)

mathematical signal processing (aka applied and computational harmonic analysis) deals with the theory of data acquisition, representation, reconstruction (and more). viewing a signal as a vector in an appropriate space, one seeks sampling theorems that prescribe how to measure the signal, and subsequently recover it from the measurements. a classical example is the standard sampling theorem for band-limited functions. on the other hand, compressing (transform coding) a class of signals entails representing its members sparsely in an appropriate system. so one is interested both in finding such a system and obtaining theoretical guarantees on the error resulting from representing functions using a few its elements. also of interest are practical methods of digitizing the measurements, and here we seek theoretical guarantees to quantify tradeoffs between the number of measurements and bits, and the reconstruction accuracy.

in recent years, this area has seen many exciting developments based on the following simple observation. many real-world signals can be modeled as vectors having only a few degrees of freedom, and more specifically as linear combinations of relatively few basis elements. such vectors are said to be sparse. great strides in sparse approximation theory and its application have been made, spurred by the rapidly growing area of compressed sensing. this is a sampling paradigm that entails efficiently recovering estimates of sparse n-dimensional vectors from m linear measurements where $m<<n$.

we will provide an introduction to some of the essentials of compressed sensing, discuss its extensions, draw connections to some of the areas of mathematics and computer science that it interacts strongly with, and give an overview of some interesting research problems.

there is still no analogue of hironaka's type resolution for algebraic varieties over fields of positive characteristics. however a theorem of dejong states that, up to taking a purely inseparable cover, there exists a birational model whose singularities are at worst quotient. in this talk i will discuss a resolution algorithm of tame quotient singularities.

we consider some recent applications of techniques of p-adic analysis to algebra and geometry. specifically, we consider three applications. first, we show that it gives a solution to a problem of keeler, rogalski, and stafford asking to show that if the orbit of a point under an automorphism of a complex projective variety has the property that it intersects some subvariety infinitely often then the orbit cannot be zariski dense. next, we show that one can give a new proof of a result of bass and lubotzky showing that the burnside problem has an affirmative solution for automorphism groups of quasiprojective varieties. finally, we consider an application that gives a result of bogomolov and tschinkel: a k3 surface defined over a number field $f$ with an infinite automorphism group has a dense set of $k$-points for some finite extension of $f$. this includes joint work with dragos ghioca, zinovy reichstein, daniel rogalski, sue sierra, and tom tucker.

in this talk we explain how this question has many different types of answers, depending on what you mean by "many" and what you mean by "symmetry". we also explore how answering this question leads from very classical algebraic geometry to the modern theory of classification of varieties.

the theory of intra-surface viscous flow on lipid bilayers is developed by combining the equations for flow on a curved surface with those that describe the elastic resistance of the bilayer to flexure. the model is derived directly from balance laws and augments an alternative formulation based on a variational principle. conditions holding along an edge of the membrane are emphasized and the coupling between flow and membrane shape is simulated numerically. particular applications that will be discussed in this talk include lipid flow on minimal surfaces, osmotic pressure drive membrane deformation, and flow due to protein adsorption.

operator splitting schemes break a complicated and possible nonsmooth optimization problem into simple matrix-vector multiplication, gradient, projection, and proximal steps. the resulting algorithms are often short, easy to code, and have (nearly) state-of-the-art performance for large-scale optimization problems that arise in machine learning, compressed sensing, medical imaging, geophysics, and bioengineering. the importance of operator splitting, a technique that dates back to the 1950’ and since then been widely used in numerical linear algebra and numerical pde, has significantly increased in the past decade.

this talk will review the basic operator splitting schemes and introduce a new splitting scheme. their special cases cover a large number of existing algorithms such as von neumann's alternating projection, iterative soft-thresholding algorithm, admm, and various primal-dual algorithms. their convergence results are presented. through examples, we also demonstrate that they lead to high-performance low-cost methods for large-scale optimization problems.

this talk includes joint work with damek davis, wei deng, ming-jun lai, zhimin peng, and ming yan.

voting has been an important topic for human activities and a central theme in the social choice theory in economics, which is featured with the celebrated impossibility theorems by nobel laureates ken arrow and amartya sen. despite of the intrinsic conflicts between the faithful representation of individuals and the desire for consistent social orders, in reality we are still looking for possible preference aggregation rules out of impossibilities. hodge theory, as a bridge between the algebraic topology and geometry, is surprisingly enabling us a tool of preference aggregation as a generalization of the classical borda count, arguably the most consistent and tractable social choice rule. it not only enables us to find aggregated preference with nearly linear complexity algorithms to deal with the rapid growth of crowdsourcing data, but also provides us ways to characterize the conflicts of interests arising locally or globally. we shall discuss these from a revisit of those impossibility theorems to see what is possible that hodge decomposition provides us.

relative gromov-witten theory gives virtual counts of curves with specified tangency conditions. the most natural setting for this theory is in the category of log schemes. in this talk, i will describe the basics of log geometry, discuss their relationship with relative gromov-witten theory, and provide a concrete sample calculation.

the arithmetic regularity lemma due to green [gafa 2005] is an analogue of the famous szemeredi regularity lemma in graph theory. it shows that for any abelian group g and any bounded function $f : g \rightarrow [0, 1]$, there exists a subgroup $h \leq g$ of bounded index such that, when restricted to most cosets of $h$, the function f is pseudorandom in the sense that all its nontrivial fourier coefficients are small. quantitatively, if one wishes to obtain that for $1 - \epsilon$ fraction of the cosets, the nontrivial fourier coefficients are bounded by $\epsilon$, then green shows that $|g/h|$ is bounded by a tower of twos of height $1/\epsilon^3$. he also gives an example showing that a tower of height $\omega(\log 1/\epsilon)$ is necessary. here, we give an improved example, showing that a tower of height $\omega(1/\epsilon)$ is necessary.

joint work with shachar lovett, guy moshkovitz, and asaf shapira.

consider two straightline planar drawings g and h of the same planar triangulation, in which the outer face is fixed. a morph between g and h is a continuous family of drawings of the triangulation, beginning with g and ending with h. we say a morph between g and h is planar if each intermediate drawing is a straightline planar drawing of the triangulation. a morph is called linear if each vertex moves from its initial position in g to its final position in h along a line segment at constant speed. it is easy to see that in general the linear morph from g to h will not be planar.

here we consider the algorithmic problem of finding a planar morph between two given drawings g and h with fixed outer face. for various reasons it is desirable to find morphs in which each vertex trajectory is fairly simple. thus we focus on the problem of constructing a planar morph consisting of a polynomial number of steps, in which each step is a planar linear morph.

(joint work with fidel barrera-cruz and anna lubiw)

recently, quantitative stochastic homogenization of operators in divergence form has witnessed important progress, starting from the work of gloria and otto. our goal is to go beyond the error bound and further analyze the statistical fluctuations around the homogenized limit. using a probabilistic representation, the kipnis-varadhan method applied to diffusion in random environment, and a quantitative martingale central limit theorem, we prove a pointwise two-scale expansion by a stationary corrector. this is joint work with jean-christophe mourrat.

ricci shrinkers are shrinking self-similar solutions to the ricci flow. these objects arise as limits in the study of singular solutions of the ricci flow. we highlight some results on the geometry of ricci shrinkers.

cell locomotion is essential for early development, angiogenesis, tissue regeneration, the immune response, and wound healing in multicellular organisms, and plays a very deleterious role in cancer metastasis in humans. locomotion involves the detection and transduction of extracellular chemical and mechanical signals, integration of the signals into an intracellular signal, and the spatio-temporal control of the intracellular biochemical and mechanical responses that lead to force generation, morphological changes and directed movement. we will discuss some of the mathematical and computational challenges that the integration of these processes poses and describe recent progress on some component processes.

random walks on graphs are well-studied, and numerous results exist relating properties of a random walk on a graph to the spectra of various matrices associated with the graph. particularly, the eigenvalues of the adjacency matrix of a graph give us information about the rate of convergence of a random walk to its stationary distribution. less well-understood are non-backtracking random walks, which are random walks with the extra condition that we cannot return to the immediately previous state. these are much harder to analyze in general. a paper of alon, benjamini, lubetzky, and sodin from 2007 proves that for regular graphs, in most cases a non-backtracking random walk will converge to its stationary distribution more quickly than an ordinary random walk. we will discuss some known results on random walks ways to approach the generalization of these results to non-backtracking random walks.

the cyclic sieving phenomenon is a decade-old object in enumerative combinatorics introduced by reiner, stanton, and white related to counting the fixed-point set sizes associated with a finite cyclic group acting on a finite set. we will give examples of the csp for combinatorial actions on young tableaux, polygon dissections, and set partitions. we will see that, while the statement of the csp is purely enumerative, the ``best" proofs of csps are algebraic and instances of the csp can predict results in algebra and geometry.

let x be an algebraic symplectic variety, and assume that its structure sheaf admits a deformation quantization. a line bundle on a lagrangian subvariety of x may be viewed as a module over the structure sheaf of x and we could ask weather the module structure admits a deformation quantization as well. we discuss some old and new results in this direction.

linear response eigenvalue problems (lreps) arise from the study of collective motion of many particle systems, such as excited states of electronic structures. large scale lreps are challenging due to its non-symmetric structure. in this talk,
we will present recent theoretical results on variational principles of lreps and discuss conjugate gradient-like algorithms. numerical results of very large lreps for computing multiple low-lying excitation energies of molecules by the time-dependent density functional theory will be presented. this is a joint work with ren-cang li, dario rocca and giulia galli.

if $k[g]$ denotes the group algebra of g over the field k, then the trace map $tr : k[g] \to k$ picks off the identity coefficient of each element of the ring. in this talk, i will discuss what is known about traces of algebraic elements of $k[g]$. of particular interest are the traces of idempotents and nilpotent elements. if time permits, i will also consider analogous results for twisted group algebras.

stochastic mechanical behaviors of random media is relevant to various of
engineering fields, one example of random media is, in order to simulate the fault formation of earthquake, stochastic treatment on ground surface is applied which consists of several layers of not fully known properties and structures.

in this presentation, we examine a general linearized elastostatic problem in random media. a complete analysis in solution space is provided including existence and uniqueness. the single integral formulation of weak form and stochastic collocation method are applied to solve this problem. moreover, the prior error estimators of stochastic collocation method are derived which imply the rate of convergence is exponential along with the order of polynomial increasing in the space of random
variables. as expected, the numerical experiments admit the exponential rate of convergence verified by a posterior error analysis. at last, a adaptive strategy derived by the posterior error analysis is designed.

this will be a panel discussion on the academic job search given by current 5th year graduate 2022年亚洲世界杯预选赛 . the panel will be interactive, so come prepared with questions!

the einstein constraints equations are of fundamental interest in the the study of einstein's theory of general relativity. this coupled nonlinear elliptic system must also be solved reliably and efficiently for gravitational wave simulation. the equations have been studied intensively for half a century; they are a particular example of a "critical exponent" problem, often arising in geometric analysis.

in this lecture, we begin with an overview of the most useful mathematical formulation of the constraint equations, and then summarize the known existence, uniqueness, and multiplicity results through 2008. we then present a number of new existence and multiplicity results developed since 2008 that substantially extend the solution theory for the constraint equations. the techniques needed for developing new results are wide-ranging, and include fixed-point arguments, maximum principles, a priori estimates, bifurcation theory, and other techniques in nonlinear analysis and partial differential equations.

we then shift gears a bit and consider galerkin and petrov-galerkin type
approximation methods for developing "good" numerical methods for solving this system. we examine how one proves rigorous error estimates for particular classes of numerical methods, including both classical finite element methods and newer methods from the finite element exterior calculus.

this project is joint work with a number of collaborators over several years.

let m be the moduli scheme of stable sheaves on a complex elliptic
surface. we want to know its birational structure, especially its kodaira
dimension. to this end, it is important to understand its singularities.
what is known about such problems now?

this is a joint work with alexander olshanskii. we study the exponents of the growth and cogrowth of subalgebras and subideals in free lie algebras. in the case of ideals it is known that the growth of a nonzero ideal of a free lie algebra is equivalent to the growth of the whole algebra. in the case of finitely generated subalgebras the same can be said about the cogrowth of proper finitely generated subalgebras. the most intriguing case is that of subideals.

this is joint work with jiri lebl and asif shakeel. we consider orthonormal bases in n qubits that consist of product states (unentangled orthonormal basis--uob) and a device for each qubit that can do an arbitrary unitary transformation and a measurement on the qubit. given a uob and an element in the basis an locc protocol for determining which element we have chosen involves the following steps:

1. set up and order for doing the local operations (unitary operation and measurement) by the devices.

2. after each local operation in this order the outcome of all of the earlier operations is communicated to the next instrument in the order and that instrument chooses it's local operation using the outcomes of all the previous operations.

it has been shown that in 3 or more qubits there are uob such that there is no protocol that can pick out a specific basis element with certainty. we show that in n qubits the generic uob does not have this non-locality and completely classify these uob's. thus the ones not on our list have this non-locality.

in this talk we discuss some recent results on wenger graphs and their modification, including spectra, expansion and cycles lengths. most of these results represent joint work with s. cioaba, w. li, a. thomason and y. wang.

the algebraic k theory space $k(r)$ is defined as a topological group completion, which on $\pi_0$ is just the usual algebraic group completion of a monoid which yields $k_0(r)$. amazingly, it turns out that this space not only has a multiplication on it which is associative and commutative up to
homotopy, but it is an infinite loop space. this means that it represents
a spectrum (the stable analogue of a space), and therefore a cohomology
theory. we construct equivariant algebraic k-theory for g-rings. however,
spectra with g-action (called naive g-spectra) are not robust enough for
stable homotopy theory, and the objects of study in equivariant stable
homotopy theory are genuine g-spectra, which correspond to cohomology
theories graded on representations.

our construction of ``genuine" equivariant algebraic k-theory recovers as
its fixed points the k-theory of twisted group rings, and as particular
cases equivariant topological real and complex k-theory, atiyah's real
k-theory and statements previously formulated in terms of naive g-spectra
for galois extensions. for example, we can reinterpret the map from the
quillen-lichtenbaum conjecture and the assembly map from carlsson's
conjecture in terms of genuine g-spectra and their fixed points.

we will not assume background in topology and will explain all the concepts from homotopy theory that arise in the talk.

motivated by perturbative computations of feynman integrals, kontsevich has conjectured that a certain group known as the grothendieck-teichmuller group acts on the moduli space of field theories. moreover he has explicitly described this action in certain examples. we first describe a stable context for this question, which can be interpreted as inverting free bosonic field theories. we then show
that a quotient of the gt group acts on the stable object in agreement with computations made by kontsevich. we will try to make this talk as self contained as possible.

by the work of griffiths, the cohomology of a family of complex projective manifolds determines a period map from the base of the family to the quotient of a flag domain d. in the case where d is hermitian symmetric, these quotients admit a number of partial compactifications including the baily-borel and toroidal amrt compactifications. i describe recent work with matt kerr on computing the mumford-tate group of the analogs of the armt boundary components of a degeneration of hodge structure arbitrary weight.

in the past decade computer science literature has studied the effect of introducing random choices to classical processes. for example, sequentially place n balls into n bins. for each ball, two bins are sampled uniformly and the ball is placed in the emptier of the two. this process does a much better job of evenly distributing the balls than the "choiceless" version where one places each ball uniformly.

consider the continuous version: form a random sequence in the unit interval by having the n*th* term be whichever of two uniformly placed points falls in the larger gap between the previous n-1 points. i'll confirm the intuition that this sequence is a.s. equidistributed, solving an open problem from itai benjamini, pascal maillard and elliot paquette. the history goes back a century to weyl and more recently to kakutani. several open problems will be discussed.

the mathematics of juggling began with the task of characterizing all possible physical juggling patterns. over the past few decades this study has expanded, inspiring related questions from many areas of math. in this talk we will discuss several new probabilistic, algebraic and combinatorial results and state some unsolved problems. attendees are invited to bring chainsaws and/or lit torches for the interactive portion of the talk.

light refreshments will be provided thanks to the sponsorship of the gsa.

i will discuss recent joint work with conley, gaboriau, and marks in which we show that all free pmp actions of groups with planar cayley graphs are treeable. this provides the first examples of nonamenable groups with one end which are strongly treeable.

for a projective variety x and an ample divisor h on it, an h-polar cylinder in x is an open ruled affine subset whose complement is a support of an effective q-divisor q-rationally equivalent to h. this notion links together affine, birational and kahler geometries. i will show how to prove existence and non-existence of h-polar cylinders in smooth and mildly singular del pezzo surfaces (for different polarizations). the obstructions comes from log canonical thresholds and fujita numbers. as an application, i will answer an old question of zaidenberg and flenner about additive group actions on the cubic fermat affine threefold cone. this is a joint work with jihun park (postech) and joonyeong won (kaist).

we will discuss our on-going investigation of adaptive strategies for finite element equations. we show first that under modest assumptions, any robust and efficient a posteriori error indicator must behave as simple interpolation error for the exact finite element solution. we then use interpolation error to study several popular h and hp adaptive algorithms.

this will be the first in a series of talks on the theory of evaluating constant term identities and their applications to combinatorics and algebra.

we consider a model of a population of fixed size $n$ in which each individual acquires beneficial mutations at rate $\mu$. each individual dies at rate one, and when a death occurs, an individual is chosen with probability proportional to the individual's fitness to give birth. we obtain rigorous results for the rate at which mutations accumulate in the population, the distribution of the fitnesses of individuals in the population at a given time, and the genealogy of the population. our results confirm predictions of desai and fisher (2007), desai, walczak, and fisher (2013), and neher and hallatschek (2013).

this will be a panel discussion on the process of finding a thesis advisor. the panel will be interactive, so come prepared with questions!

sponsored by the gsa.

i will first describe certain conjectural tate classes in the middle dimensional etale cohomology of many shimura varieties over finite fields (e.g. hilbert and picard modular surfaces at inert primes). according to the tate conjecture, there should exist corresponding algebraic cycles. surprisingly, we find that these cycles are provided by the supersingular (or more precisely basic) loci of these shimura varieties. this is based on a joint work with liang xiao.

manin's conjecture predicts that the behavior of rational points on a variety x is controlled by certain geometric constants associated to x. i will discuss how the minimal model program can be used to analyze the behavior of these constants. this is joint work with sho tanimoto and yuri tschinkel.

given a vector space $v$ over a field $k$, let $\mathrm{end}(v)$ denote the algebra of linear endomorphisms of $v$. if $v$ is finite-dimensional, then it is well-known that the diagonalizable subalgebras of $\mathrm{end}(v)$ are characterized by their internal algebraic structure: they are the subalgebras isomorphic to $k^n$ for some natural number $n$.

in case $v$ is infinite dimensional, the diagonalizable subalgebras of $\mathrm{end}(v)$ cannot be characterized purely by their internal algebraic structure: one can find diagonalizable and non-diagonalizable subalgebras that are isomorphic. i will explain how to characterize the diagonalizable subalgebras of $\mathrm{end}(v)$ as \emph{topological} algebras, using a natural topology inherited from $\mathrm{end}(v)$. i also hope to show how this characterization relates to an infinite-dimensional wedderburn-artin theorem that characterizes ``topologically semisimple'' algebras.

consider a solution to an ordinary differential equation driven along smooth vector fields $v=\left( v^{1},...,v^{d}\right) $ of linear growth.
\begin{equation}
dy_{t}=v\left( y_{t}\right) dx_{t},\text{ started from }y_{0}.\label{ode}%
\end{equation}
if $x$ has finite $1-$variation then gronwall's inequality gives a bound on
$y$ of the type%
\[
\left\vert y\right\vert _{1-var}\leq c\exp\left( c\left\vert x\right\vert
_{1-var}\right) .
\]
if $x$ has finite $p-$variation for $p>2$ then rough path theory needs to be
used to understand (\ref{ode}), and a corresponding growth estimate of the
form
\[
\left\vert y\right\vert _{p-var}\leq c\exp\left( c\left\vert \mathbf{x}%
\right\vert _{p-var}^{p}\right)
\]
can be derived in some cases. for a large class of random rough paths
$\mathbf{x=x}\left( \omega\right) $, e.g. the brownian rough path, the right
hand side of this inequality is not integrable (fernique's theorem). this has
implications for some applications of interest, such as showing the existence
and smoothness of densities of rdes via malliavin calculus.

in this talk we show how this obstacle can be bypassed by consideration of the
so-called \textit{accumulated p-variation }$m\left( \mathbf{x}\right) $ of a
$p$-rough path $\mathbf{x}$ over $\left[ 0,t\right] $ which is given by%
\[
m\left( \mathbf{x}\right) =\sup_{\overset{d=\left\{ 0=t_{0}<t_{1}%
<...t_{r}=t\right\} }{\left\vert \left\vert \mathbf{x}\right\vert \right\vert
_{p\text{-}var,\left[ t_{i},t_{i+1}\right] }\leq1}}\sum_{i=1}^{r}\left\vert
\mathbf{x}\right\vert _{p\text{-}var,\left[ t_{i},t_{i+1}\right] }^{p}.
\]
we prove precise tail estimates for $m\left( \mathbf{x}\left( \omega\right)
\right) $ in two important cases: (i) general gaussian rough paths and (ii)
the class of markovian rough paths associated to subelliptic dirichlet form.
this is based on joint works with, respectively, christian litterer and terry
lyons for case (i) and marcel ogrodnik for case (ii).

in this talk, we discuss a very efficient algorithm for projecting a point onto a polyhedron. this algorithm solves the projeciton problem through its dual and fully exploits the sparsity. the sparsa (sparse reconstruction by separable approximation) is used to approximately identify active constraints in the polyhedron, and the dual active set algorithm (dasa) is used to compute a high precision solution. some interesting convergence properties and very promising numerical results compared with the state-of-the-art software ipopt and cplex will be discussed in this talk.

this will be the second in a series of talks on the theory of evaluating constant term identities and their applications to combinatorics and algebra.

this will be a laid back introduction to affine schemes, one of the basic objects of study of modern algebraic geometry. primarily, we will explore some simple and concrete examples, which will motivate the otherwise abstract theory. finally, we will give a description of the arithmetic surface, and showcase david mumford's intimate and revealing portrait of $\text{spec}(\mathbb{z}[x])$. no knowledge is assumed beyond a vague recollection of what a ring is.

the jacobian $j_0(23)$ of the modular curve $x_0(23)$ is a semi-stable abelian variety over $\bbb q$ with good reduction outside $23$. it is simple. we prove that every simple semi-stable abelian variety over $\bbb q$ with good reduction outside $23$ is isogenous over $\bbb q$ to $j_0(23)$.

we investigate the curvature-dependence of water dynamics in the vicinity of hydrophobic spherical solutes using molecular dynamics simulations. for both, the lateral and perpendicular diffusivity as well as for h-bond kinetics of water in the first hydration shell, we find a non-monotonic solute-size dependence, exhibiting extrema close to the well-known structural crossover length scale for hydrophobic hydration. additionally, we find an apparently anomalous diffusion for water moving parallel to the surface of small solutes, which, however, can be explained by topology effects. the intimate connection between solute curvature, water structure and dynamics has implications for our understanding of hydration dynamics at heterogeneous biomolecular surfaces.

in many areas of imaging science, such as diffraction imaging, astronomical imaging, microscopy, etc., optical detectors can often only record the squared modulus of the fraunhofer or fresnel diff raction pattern of the radiation that is scattered from an object. in such setting, it is not possible to measure the phase of the optical wave reaching the detector. so, it is needed to reconstruct a signal from intensity measurements only. this problem is called phase retrieval.
we are going to consider the case when the measurement frame is a gabor frame, that is, the case of time-frequency structured measurements. the main motivation is that in this case, the frame coefficients are of the form of masked fourier coefficients, where the masks are time shifts of the gabor window. this makes measurements meaningful for applications, but at the same time preserves the flexibility of the frame-theoretic approach. the most efficient existing algorithms, such as phaselift, work with randomly generated gaussian frames. i am going to present the recovery algorithm with a sufficiently small number of measurements required, which is working with time-frequency structured measurements. the algorithm is based on the idea of polarization, first proposed by alexeev, bandeira, fickus and mixon.

the eigenvalues of the laplacian encode fundamental geometric information
about a riemannian metric. as an example of their importance, i will
discuss how they arose in work of cao, hamilton and illmanan, together
with joint work with stuart hall, concerning stability of einstein
manifolds and ricci solitons. i will outline progress on these problems
for einstein metrics with large symmetry groups. we calculate bounds on
the first non-zero eigenvalue for certain hermitian-einstein four
manifolds. similar ideas allow us estimate to the spectral gap (the
distance between the first and second non-zero eigenvalues) for any toric
kaehler-einstein manifold m in terms of the polytope associated to m. i
will finish by discussing a numerical proof of the instability of the
chen-lebrun-weber metric.

genetically encoded biosensors based on fluorescence resonance energy transfer (fret) have been widely applied to visualize the molecular activity in live cells with high spatiotemporal resolution. however, the rapid diffusion of biosensor proteins hinders a precise reconstruction of the actual molecular activation map. based on fluorescence recovery after photobleaching (frap) experiments, we have developed a finite element (fe) method to analyze, simulate, and subtract the diffusion effect of mobile biosensors. this method has been applied to analyze the mobility of src fret biosensors engineered to reside at different sub-compartments in live cells. the results indicate that the src biosensor located in the cytoplasm moves 4-8 folds faster than those anchored on different compartments in plasma membrane. the mobility of biosensor at lipid rafts is slower than that outside of lipid rafts and is dominated by two-dimensional diffusion. when this diffusion effect was subtracted from the fret ratio images, high src activity at lipid rafts was observed at clustered regions proximal to the cell periphery, which remained relatively stationary upon epidermal growth factor (egf) stimulation. this result suggests that egf induced a src activation at lipid rafts with well-coordinated spatiotemporal patterns. our fe-based method also provides an integrated platform of image analysis for studying molecular motility and reconstructing the spatiotemporal activation maps of signaling molecules in live cells. furthermore, we developed a correlative fret imaging microscopy (cfim) approach to quantitatively analyze the subcellular coordination between the enzymatic activity and the structural focal adhesion (fa) dynamics. by cfim, we found that different fa subpopulations have distinctive regulation mechanisms controlled by local kinase activity. therefore, our work highlights the importance of dynamic single live-cell imaging and its integration in-depth mathematical analysis.

by studying variation of motivic path torsors associated to a variety, we show how certain nondensity assertions in diophantine geometry can be translated to problems concerning k-groups. concrete results then follow based on known (and conjectural) vanishing theorems.

a tracial von neumann algebra $m$ is called prime if it cannot be decomposed as the tensor product of two nontrivial (not type ${\rm i}$) subalgebras. naturally, if $m$ is not prime, one asks if $m$ can be uniquely factored as a tensor product of prime subalgebras. the first result in this direction is due to ozawa and popa in 2003, who gave a large class of groups $\mathcal{c}$ such that for any $\gamma_1, \dots, \gamma_n \in \mathcal{c}$, the associated von neumann algebra $l(\gamma_1) \otimes \cdots \otimes l(\gamma_n)$ is uniquely factored. this talk will focus on von neumann algebras arising from a class of measured equivalence relations, and how the techniques of ozawa and popa can be adapted to this setting.

a brief introduction to discrete variational mechanics, motivated by the goal of constructing symplectic one-step methods, will be presented. followed by the presentation of a new type of variational integrator that utilizes taylor's method in it's construction. time permitting, we will make some remarks on the pros/cons and stability of constructing the variational integrator from the lagrangian variational framework versus the hamiltonian variational framework.

asymptotic behavior of random walks on a group, such as decay of return probability and growth of entropy, reflects large scale geometry of the underlying group. the notion of isoperimetric profiles, motivated by classical isoperimetric inequalities and faber-krahn inequalities, captures useful geometric information that affects behavior of random walks. in this talk i will explain some interesting examples of groups acting on trees, where random walks exhibit rich exotic behavior. a connection between return probability lower bound and entropy growth will also be discussed.

greedy algorithms are a popular choice for solving problems because they are often simple, intuitive, and efficient. however, they can also be misleading, sometimes yielding the worst possible solution to a problem. in this talk we'll go through some examples of greedy algorithms (we'll look at the best example first, and go from there) and discuss when they perform terribly, optimally, and when we can bound how close the greedy solution is to the optimal solution. finally, we'll review what matroids are and prove that a greedy algorithm yields an optimal solution to a particular class of problems if and only if we can rephrase the problem as a statement about a matroids.

sponsored by the gsa.

let $k$ be a number field and let $a$ be an abelian variety over $k$. in an effort of proper setting of the sato-tate conjecture concerning the equidistribution of frobenius elements in the representation of the galois group $g_k$ on the tate module of $a$, one of attempts is the introduction of the algebraic sato-tate group
$ast_{k}(a)$. maximal compact subgroups of $ast_{k}(a)(\mathbb{c})$ are
expected to be the key tool for the statement of the sato-tate conjecture for $a$. at the lecture, following an idea of j-p. serre, an explicit construction of $ast_{k}(a)$ will be presented based on p. deligne's motivic category for absolute hodge cycles. i will discuss the arithmetic properties of $ast_{k}(a)$ along with explicit computations of $ast_{k}(a)$ for some families of abelian varieties. i will also explain how this construction extends to absolute hodge cycles motives in the deligne's motivic category for absolute hodge cycles. this is joint work with kiran kedlaya.

in 1972 j.-p. serre proved an important theorem concerning the galois action on the torsion points of elliptic curves over number fields. we describe a conjectural uniform version of this theorem and its relation to rational points on modular curves.

we study the holomorphic embedding problem from a compact real algebraic hypersurface into a sphere. by our theorem, for any integer $n$, there is a family of compact real algebraic strongly pseudoconvex hypersurfaces in $c^2$ , none of which can be locally holomorphically embedded into the unit sphere in $c^n$. this shows that the whitney (or remmert) type embedding theorem in differential topology(or in the stein space theory, respectively) does not hold in the setting above. this is a joint work with xiaojun huang and xiaoshan li.

obliquely reflected diffusions in smooth domains are classical objects that have been well understood for half a century. on the other hand, many fundamental questions remain in the study of reflected diffusions in non-smooth domains, which arise in a variety of fields ranging from mathematical physics to stochastic networks. we first describe some recent results on reflected diffusions in piecewise smooth domains. we then introduce a new approach to the construction and characterization of obliquely reflected brown motions in bounded, simply connected planar domains. the class of processes we construct also includes certain processes with jumps that have arisen in the study of sle. this talk is partially based on joint works with chris burdzy, zhenqing chen, weining kang and donald marshall.

one of the main challenges of computational biology is predicting the rate constants of association and dissociation of large molecules. in order to make the computations feasible, simplifying assumptions are made that still allow essential features of the physical process to be captured. discussion will include current and upcoming features of the software as well as the physics and algorithms involved.

let $w = \prod x^{a_i} y^{b_i}$ be an element of a free group on two generators $x, y$. for every group $g$ the corresponding word map $w : g^2 \to g$ is defined as $w(g_1, g_2) = \prod g_1^{a_1} g_2^{a_2}$. i will speak about surjectivity and equidistribution of such word maps on groups $\operatorname{psl}(2)$ and $\operatorname{sl}(2)$. this is a report on the joint works with sh. garion, f. grunewald, b.kunyavskii, eu. plotkin, and n. gordeev.

we discuss nontrivial multiplicative relations among eigenvalues of frobenius endomorphisms of abelian varieties over finite fields. (the trivial relations are provided by the riemann hypothesis that was proven by a. weil.) we classify all abelian varieties over finite fields of a certain dimension which admit the nontrivial relations and give an explicit construction of corresponding exotic tate classes.

a classical theorem of jordan asserts that every finite subgroup $b$ of the complex general linear (matrix) group $gl(n)$ contains an abelian (normal) subgroup $a$ such that the index of $a$ in $b$ does not exceed a universal constant that depends only on $n$. we discuss analogues of jordan's theorem where the matrix group is replaced by the group of biregular (or birational) automorphisms of a complex algebraic variety, or by the diffeomorphism group of a smooth real manifold.

we explore the (noncommutative) geometry of representations of locally finite lie algebras. let l be one of these lie algebras, and let i in u(l) be the annihilator of a locally simple l-module. we show that for each such i, there is a quiver q so that locally simple l-modules with annihilator i are parameterized by "points" in the "noncommutative space" corresponding to the path algebra of q. we classify the quivers that occur and along the way discover a beautiful connection to characters of the symmetric groups $s_n$.

certain imaging applications such as x-ray crystallography require the recovery
of an underlying signal from intensity (or magnitude) measurements - a problem
commonly referred to as phase retrieval. in this talk, we discuss a framework for
solving the discrete phase retrieval problem using block circulant measurement
constructions and angular synchronization. we develop an algorithm which is
near-linear time, making it computationally feasible for large dimensional
signals. theoretical and experimental results demonstrating the method's
speed, accuracy and robustness will be presented. we also present an extension
of the framework to sparse phase retrieval, including the first known
sublinear-time compressive phase retrieval algorithm.

i will give an overview of kodaira's vanishing theorem and its applications in higher-dimensional algebraic geometry. i will then proceed to give an introduction to formal schemes and their applications. then i will state a version of the vanishing theorem for formal schemes, along with some possible further results to be investigated.

random matrix theory is filled with laws of large numbers and central limit theorems, all specialized to different statistics and scaling regimes. the two most famous laws of large numbers are for gaussian wigner (hermitian) matrices: the bulk (empirical distribution) of eigenvalues converge a.s. to the semicircle law, and the largest eigenvalue converges to twice the common variance of the entries. the corresponding central limit theorems give gaussian fluctuations in the bulk, but a completely different distribution (the tracy-widom law) for the largest eigenvalue.

one can view a gaussian wigner matrix as (a marginal of) brownian motion on the lie algebra of hermitian matrices. it is then natural to study the analogous questions for the eigenvalues of the brownian motion on the associated lie group: the unitary group. for the bulk, the a.s. limit empirical eigenvalue distribution was discovered by biane and rains independently in the late 1990s; the corresponding bulk central limit result was largely found by l\'evy and ma\"ida in 2010

elliptic curves appear prominently in wiles's resolution of fermat's last theorem, the birch and swinnerton-dyer conjecture (a millennium problem), and lenstra's factoring algorithm, and they are also important in modern cryptography. in this talk, we will give a gentle introduction to elliptic curves and then explain some of these applications without assuming any background in algebraic geometry or number theory.

sponsored by the gsa.

we investigate timelike and null vector flows on closed lorentzian manifolds and their relationship to ricci curvature. the guiding observation, first observed for closed riemannian 3-manifolds by harris & paternain '13, is that positive ricci curvature tends to yield contact forms, namely, 1-forms metrically equivalent to unit vector fields with geodesic flow. we carry this line of thought over to the lorentzian setting. first, we observe that the same is true on a closed lorentzian 3-manifold: if x is a global timelike unit vector field with geodesic flow satisfying $ric(x,x) > 0$, then $g(x,•)$ is a contact form with reeb vector field x, at least one of whose integral curves is closed. second, we show that on a closed lorentzian 4-manifold, if x is a global null vector field satisfying $\nabla_xx = x$ and $ric(x) > divx - 1$, then $dg(x,•)$ is a symplectic form and x is a liouville vector field.

electrostatic interactions play an important role in many complex charged systems, such as biological molecules, soft matter material, nanofluids, and electrochemical devices. in this work, we develop mathematical theories and computational methods to understand such interactions, particularly in charged biological molecular systems. our main contributions include: 1. theoretical studies of mean-field variational models of ionic solution and that of molecular surfaces with the poisson-boltzmann electrostatics; 2. design and implementation of the corresponding computational algorithms, and conduct extensive monte carlo simulations and numerical solutions of partial differential equations for charge-charge interactions; 3. discovery of various interesting properties of charged molecules, validate some experimental results, and clarify some confusion in literature. a common theme of this work is the variational approach. many physical effects such as ionic size effects, solvent entropy, concentration dependent dielectric response can be incorporated into a mean-field free-energy functional of ionic concentrations coupled with the poisson equation for electrostatics. the techniques of analysis developed in this work may help improve the understanding of the underlying physical properties of charged systems and provide new ways of studying analytically and numerically other problems in the calculus of variations.

a celebrated 1942 result of schoenberg characterizes all entry-wise functions which preserve positivity of matrices of any size. i will present a characterization of polynomials which preserve positivity when applied entry-wise on matrices of a fixed dimension. all put in historical context and motivated by recent demands of statistics of large data and optimization theory. a sketch of the proof will take a detour through the representation theory of the symmetric group.

an algebraic cycle homologous to zero on a variety leads to an
extension of hodge-theoretic data, and in a variational context to a
family of extensions called a normal function. these may be viewed as
"horizontal" sections of a bundle of complex tori, and are used to detect
cycles modulo algebraic (or rational) equivalence. conversely, the
existence of normal functions can be used to predict that interesting
cycles are present... or absent: a famous theorem of green and voisin
states that for projective hypersurfaces of large enough degree, there are
no normal functions (into the intermediate jacobian bundle associated to
these hypersurfaces) over any etale neighborhood of the coarse moduli
space. inspired by recent work of friedman-laza on hermitian variations of
hodge structure and oort's conjecture on special (i.e. shimura)
subvarieties in the torelli locus, r. keast and i wondered about the
existence of normal functions over etale neighborhoods of shimura
varieties. here the function is supposed to take values in a family of
intermediate jacobians associated to a representation of a reductive
group. in this talk i will explain our classification of the cases where a
green-voisin analogue does *not* hold and where one therefore expects
interesting cycles to occur, and give some evidence that these predictions
might be "sharp".

this is joint work with alan noell and sivaguru ravisankar. we prove a hartogs-bochner type theorem for a bounded domain $u$ with smooth boundary in ${\mathbb c}^n \times {\mathbb r}$ with nodegenerate, flat, and elliptic cr singularities (all natural conditions for the problem). that is, a cr function on the boundary $\partial u$ extends to a cr function on $u$, smooth up to the boundary.

self-avoiding walk of length n on the integer lattice $z^d$ is the uniform measure on nearest-neighbour walks in $z^d$ that begin at the origin and are of length $n$. if such a walk closes, which is to say that the walk's endpoint neighbours the origin, it is natural to complete the missing edge connecting this endpoint and the origin. the result of doing so is a self-avoiding polygon. we investigate the numbers of self-avoiding walks, polygons, and in particular the "closing" probability that a length n self-avoiding walk is closing. developing a method (the "snake method") employed in joint work with hugo duminil-copin, alexander glazman and ioan manolescu that provides closing probability upper bounds by constructing sequences of laws on self-avoiding walks conditioned on increasing severe avoidance constraints, we show that the closing probability is at most $n^{-1/2 + o(1)}$ in any dimension at least two. developing a quite different method of polygon joining employed by madras in 1995 to show a lower bound on the deviation exponent for polygon number, we also provide new bounds on this exponent. we further make use of the snake method and polygon joining technique at once to prove upper bounds on the closing probability below $n^{-1/2}$ in the two-dimensional setting.

we will describe the problem of detecting linear dependence of points in mordell-weil groups a(f) of abelian varieties. this is done via reduction maps. we determine the sufficient conditions for the reduction maps to detect linear dependence in a(f).
we also show that our conditons are very close to be or perhaps are the best possible. in particular we try to determine the conditions for detecting linear dependence in mordell-weil groups via finite number of reductions. the methods combine applications of transcedental, l-adic and mod v techniques. this is joint work with g. banaszak.

fundamental biological molecular processes, such as protein folding, molecular recognition, and molecular assemblies, are mediated by surrounding aqueous solvent (water or salted water). continuum description of solvent is an efficient approach to understanding such processes. in this work, we develop a solvent fluid model and computational methods for solvent dynamics and solute-solvent interface motion. the key components in our model include the stokes equation for the incompressible solvent fluid which governs the motion of the solute-solvent interface, the ideal-gas law for solutes, and the balance on the interface of viscous force, surface tension, van der waals type dispersive force, and electrostatic force. we use the ghost fluid method to discretize the flow equations that are reformulated into a set of poisson equations, and design special numerical boundary conditions to solve such equations to allow the change of solute volume. we move the interface with the level-set method. to stabilize our schemes, we use the schur complement and least-squres techniques. numerical tests in both two and three-dimensional spaces will be shown to demonstrate the convergence of our method, and to demonstrate that this new approach can capture dry and wet hydration states as observed in experiment and molecular dynamics simulations.

two of the most important examples in $ii_1$ factors are the amenable $ii_1$ factor and the free group factors. the first one is well-understood, thanks to connes' fundamental work, while the structure of free group factors is still under intense study. regarding amenable subalgebras inside free group factors, jesse peterson conjectured that any diffuse amenable subalgebra of a free group factor has a unique maximal amenable extension. in this talk i will show that any diffuse subalgebra of the radial masa in a free group factor with finitely many generators, has a unique maximal amenable extension.

let $g/k$ be a riemannian symmetric space. its complexification $g^c / k^c$ is a stein manifold, and left-translations by $g$ are holomorphic transformations of $g^c / k^c$. in this setting, invariant domains and their envelopes of holomorphy are natural objects of study. if $g/k$ is compact, then every invariant domain $d$ in $g^c / k^c$ intersects a complex torus orbit in a lower dimensional reinhardt domain $\omega_d$. in this case, complex analytic properties of $d$ can be expressed in terms of those of $\omega_d$. if $g/k$ is a non-compact, then the situation is fully understood only in the rank-one case. in this talk we present some univalence results for the envelope of holomorphy of a $g$-invariant domain in $g^c / k^c$, when the space $g/k$ is a non-compact hermitian symmetric space (joint work with a. iannuzzi).

we conjecture two combinatorial interpretations for a symmetric function arising from an eigenoperator on the macdonald polynomials called the delta operator. both interpretations generalize the famous shuffle conjecture, which connects these eigenoperators to parking functions. we prove several cases of these new conjectures using objects such as ordered set partitions, rook placements, tesler matrices, and llt polynomials. in particular, we obtain an extension of macmahon's classical equidistribution theorem from permutations to ordered set partitions, which was described in the speaker's advancement to candidacy.

for an $r$-uniform hypergraph $h$, let $f(h)$ be the minimum number of complete $r$-partite $r$-uniform subhypergraphs of $h$ whose edge sets partition the edge set of $h$.
in this talk, i will discuss the value of $f(h)$ for the random hypergraph $h$. namely, i will prove that if $(\log n)^{2.001}/n \leq p \leq 1/2$ and $h \in h^{(r)}(n,p)$, then with high probability $f(h)=(1-\pi(k^{(r-1)}_r)+o(1))\binom{n}{r-1}$, where $\pi(k_{r}^{(r-1)})$ is the tur\'an density of $k_{r}^{(r-1)}$.

consider the following system of sdes $d\lambda_i = \sigma_i(\lambda_i)db_i+\left(b_i(\lambda_i)+\sum_{j\neq i}\frac{h_{ij}(\lambda_i,\lambda_j)}{\lambda_i-\lambda_j}\right)dt\/,\quad i=1,\ldots,p\/$, (1)
describing ordered particles $\lambda_1(t)\leq \ldots\leq \lambda_p(t)$, $t\geq0$ on $r$. here $b_i$ denotes a collection of one-dimensional independent brownian motions.

let $sym(p\times p)$ be the vector space of symmetric real $p\times p$ matrices. the sdes systems (1) contain the systems describing, for the starting point having no collisions, the eigenvalues of the $sym(p\times p)$-valued process $x_t$ satisfying the following matrix valued stochastic differential equation $dx_t = g(x_t)dw_th(x_t)+h(x_t)dw_t^tg(x_t)+b(x_t)dt\/$, where the functions $g,h,b$ act spectrally on $sym(p\times p)$, and $w_t$ is a brownian matrix of dimension $p\times p$. thus the systems (1) contain dyson brownian motions, squared bessel particle systems, their $\beta$-versions and other particle systems crucial in mathematical physics and physical statistics.

note that the functions $\displaystyle{\frac{h_{ij}(lambda_i, \lambda_j)}{\lambda_i-\lambda_j}}$ describe the repulsive forces with which the particle $\lambda_i$ acts on the particle $\lambda_j$. on the other hand the singularities $\displaystyle{\frac{1}{\lambda_i-\lambda_j}}$ make the sdes system (1) difficult to solve, especially when the starting point $\lambda(0)$ has a collision $\lambda_i(0)= \lambda_j(0)$. the most degenerate case $\lambda_1(0)= \ldots= \lambda_p(0)$ is of great importance in applications.

in some particular cases (dyson brownian motions, some squared bessel particle systems), the existence of strong solutions of (1) has been established by c\'epa and l\'epingle, using the technique of multivalued sdes.

we prove the existence of strong and pathwise unique non-colliding solutions of (1), with a degenerate colliding initial point $\lambda(0)$ in the whole generality, under natural assumptions on the coefficients of the equations in (1). our approach is based on the classical it\^o calculus, applied to elementary symmetric polynomials in $p$ variables $x=(x_1, \ldots,x_p)$

$e_n(x) = \sum_{i_1<\ldots<i_n}x_{i_1}x_{i_2}\ldots x_{i_n}\/$,

as well as to symmetric polynomials of squares of differences between particles $v_n = e_n(a)\/,\quad \textrm{where } a = \{a_{ij}=(\lambda_i-\lambda_j)^2:1\leq i<j\leq p\}\/$.

in the case of squared bessel particle systems $d\lambda_i = 2\sqrt{\lambda_i}db_i+ \left(\alpha+\sum_{j\neq i}\frac{\lambda_i+\lambda_j}{\lambda_i-\lambda_j}\right)dt\/$, describing the eigenvalues of the matrix squared bessel process $dx_t = \sqrt{x_t}dw_t+dw_t^t\sqrt{x_t+}\alpha i dt\/$, we use our stochastic approach in order to determine the so-called wallach set of permissible parameters $\alpha$, known before only by harmonic analysis methods. we also determine the admissible starting points of such processes for $\alpha=1,\ldots, p-2$.

geometry has been, in recent times, a great inspiration for mathematical problems. it is therefore useful to consider numerical methods in effort to visualize and explore properties of the solutions to the partial differential equations arising from these problems. we examine a modern framework, hilbert complexes, which abstracts many of the essential details relevant for partial differential equations, such as exterior derivatives and coderivatives, laplacians, and poincaré inequalities. this viewpoint also proves useful for approximation via the finite element method. we prove some abstract error estimates and apply these results to the case of numerically computing parabolic equations on riemannian hypersurfaces.

in this talk, we shall describe some recent new results about analyticity of cr maps of positive codimension, generalizing a number of earlier results in the field.

solvent fluctuations play a fundamental role in many water-mediated molecular interactions. the significance of an efficient implicit-solvent model that can capture solvent fluctuations cannot be overemphasized. in this talk, solvent fluctuations are incorporated into a variational implicit-solvent model in two different approaches. in the first approach, solute-solvent interface fluctuations are taken into account by using a stochastic level set method with noises in the normal velocity. numerical simulations show that the method can capture dewetting transitions of hydrophobic confinements and can estimate the activation energy barrier of such transitions. the other approach employs a binary level-set representation of the solute and solvent regions. the solvent fluctuation is incorporated through ising-type monte carlo flips of the binary level-set values. coupled with solute fluctuations, this approach is able to study the folding and unfolding processes of a hydrophobic polymer. some preliminary results are presented.

we present new murnaghan-nakayama rules for certain refinements of the schur functions, the quasisymmetric schur functions and the generalized demazure atoms. we also briefly explore the enumeration of the combinatorial objects called permuted basement fillings that generate the generalized demazure atoms.

while stable sheaves on surfaces with trivial canonical bundle, i.e. k3 or abelian surfaces, have been extensively studied, the enriques case is much less understand. we will first discuss the past work of kim, hauzer, and yoshioka on some existence and irreducibility results, and then present our new results which give necessary and sufficient conditions for the existence of stable sheaves with prescribed chern classes. furthermore, we will discuss some new irreducibility results as well as some of the geometry of these moduli spaces. time permitting, we will describe a near complete on-going project describing the mmp of these moduli spaces using bridgeland stability techniques.

i will give an overview of recent results obtained in joint work with yoshimichi ueda on the structure and the rigidity of arbitrary free product von neumann algebras. first, i will explain that in any free product von neumann algebra, any amenable von neumann subalgebra that has a diffuse intersection with one of the free components is necessarily contained in this free component. this result completely settles the problem of maximal amenability inside free product von neumann algebras. then i will present new kurosh-type rigidity results for free product von neumann algebras. namely, i will explain that for any family of nonamenable factors belonging to a large class of (possibly type iii) factors including nonprime factors, nonfull factors and factors with a cartan subalgebra, the corresponding free product von neumann algebra with respect to arbitrary states retains the cardinality of the family as well as each factor up to unitary conjugacy, after permutation of the indices.

\def\z{\mathbb{z}}
\def\q{\mathbb{q}}
we give an essentially self-contained proof of the fact that a certain
$p$-adic power series
$$
\psi= \psi_p(t) \in t + t^{2}\z[[t]]\;,
$$
which trivializes the addition law of the formal group of witt
$p$-covectors $\widehat{\rm cw}_{\z}$, is $p$-adically entire and
assumes values in $\z_p$ all over $\q_p$. we also carefully examine its
valuation and newton polygons. we will recall and use the isomorphism
between the witt and hyperexponential groups over $\z_p$, and the
properties of $\psi_p$, to show that, for any
perfectoid field extension $(k,|\,|)$ of $(\q_p,|\,|_p)$, and to a
choice of a pseudo-uniformizer $\varpi = (\varpi^{(i)})_{i \geq 0}$ of
$k^\flat$, we can associate a continuous additive character
$\psi_{\varpi}: \q_p \to 1+k^{\circ \circ}$, and we will give a formula
to calculate it. the character $\psi_{\varpi}$ extends the map $x
\mapsto \exp \pi x$, where
$$\pi := \sum_{i\geq 0} \varpi^{(i)} p^i + \sum_{i<0}
(\varpi^{(0)})^{p^{-i}} p^i \in k\;.
$$
i will also present numerical computation of the first coefficients of
$\psi_p$, for small $p$, due to m. candilera.

we study variations of git quotients of log pairs (x,d) where x is a hypersurface of some fixed degree and d is a hyperplane section. git is known to provide a finite number of possible compactifications of such pairs, depending on one parameter. any two such compactifications are related by birational transformations. we describe an algorithm to study the stability of the hilbert scheme of these pairs, and apply our algorithm to the case of cubic surfaces. finally, we relate this compactifications with the (conjectural) moduli space of log k-semistable pairs.

this is work in progress with patricio gallardo (university of georgia).

appearances not withstanding this is a talk about rational curves because they're the cause of the failure. similarly, since homotopy groups are constant on the fibres of topological fibrations, a counterexample has to be in positive or mixed characteristic, and the specific one which i'll discuss is bi-disc quotients over spec
z. the example also has considerable logical implications for studying boundedness of rational curves on surfaces of general type, i.e. it cannot be implied by any theorem in $acf_0$. conversely, and more substantially, this boundedness can be proved uniformly in sufficiently large primes $p$ in $acf_0$ provided the surface enjoys $c_1^2> c_2$.

causal set theory is an approach to quantum gravity based on the idea that spacetime is fundamentally discrete and lorentz invariant. structurally, a causal set is a locally finite partial order, or a transitive dag (directed acyclic graph). after a brief introduction to the idea, i will give an overview of some of the recent research activity in causal sets. some topics i hope to touch on are: measures of discrete geometry (e.g. distance, curvature,…), dynamics on causal sets (e.g. classical and quantum fields), and phenomenology from causal sets (e.g. dark energy).

in causal set theory the ordering of the spacetime "atoms" and their number are meant to give us all of the geometrical information. thus, we should be able to construct causal set expressions that give us back particular geometrical quantities. in this talk i will describe a recent result that allows one to determine the gibbons-hawking-york (ghy) boundary term for a causal set. the ghy term appears in the action for a spacetime with a boundary, and is simply the integral over the extrinsic curvature of that boundary. a side effect of the work behind this result was the emergence of a new way to calculate the dimension of a causal set. finally, i would like to discuss the boundary contributions that are "hidden" in the bulk action for a causal set.

modifications to quantum mechanics have been proposed as potential solutions to the black hole information paradox. in particular, maldecena and horowitz have proposed a final-state projection model in which the black hole singularity constitutes a boundary to spacetime with an associated boundary condition for the wavefunction. this proposal has gained renewed interest in light of the amps "firewalls" argument. we examine the computational and information-theoretic implications of small deviations from unitarity that can arise in this model. we find that any nonunitarity allows signalling over arbitrary spacelike intervals, with channel capacity determined by the condition number of the black hole s-matrix. furthermore, grover search can be sped up using the nonunitary dynamics, but polynomial-time solution for exponential search problems implies a 1/polynomial channel capacity for instantaneous signaling. thus, within this context, we find that the no-signaling principle implies the grover search lower bound, and allowing exponential small deviations from no-signaling allows only exponentially small improvements to grover search.

our main contribution here is the discovery of a new family of standard
tableaux
$
{\cal t}^k_n
$ which are in bijection with the family ${\cal d}_{m,n}$
of rational dyck paths for $m=k\times n\pm1$ (the so called ``fuss'' case). using this family we
give a new proof of the invertibility of the sweep map in the fuss case by
means of a very simple explicit algorithm.

this is joint work with guoce xin.

the full group of a probability measure preserving action (of a group) is the group of the measure preserving transformations of the space whose graph is contained in the orbit equivalence relation of the action. for countable groups, these full groups were defined by dye in '59. he showed that such groups admit a polish topology and are complete invariant of orbit equivalence. in a joint work with f. le maître, we extend the notion of full group to probability measure preserving actions of locally compact second countable groups. these full groups also have a polish topology and they are also complete invariants of orbit equivalence. in this talk, i will define full groups and their topology and i will discuss about some of their topological properties, such as the topological rank.

in their seminal work, s.s. chern and j. moser constructed normal forms for real-analytic hypersurfaces with non-degenerate levi form. more recently, considerable work has been done to construct normal forms in degenerate cases at the level of formal power series. however, apart from that work of chern-moser, no other cases have been known where the normal form converges. the fundamental obstacle to the convergence being the non-uniformity of the underlying cr geometry.

in a joint work with ilya kossovsky, we have constructed such convergent normal forms by constructing canonical connections along natural stratifications of the cr structure.

great progress has recently been made in understanding thermodynamics beyond macroscopic limits by developing quantum thermodynamics as a resource theory, describing what state transitions are possible given specified thermodynamic resources. thermodynamics has been cited as one of the most robust aspects of physical theory; for example, it has made the transition from classical to quantum. i report preliminary results in an investigation of simple, physically meaningful properties of a physical theory that underly the possibility of such a
thermodynamic resource theory, by studying quantum thermodynamics within the broader realm of "general probabilistic theories". four such physical properties were shown by barnum, mueller, and ududec to give rise to the finite-dimensional quantum framework of density matrices and positive operator valued measures: they are (1) abstract spectrality (2) strong symmetry (3) no higher-order interference and (4) energy observability. i will explain this result and discuss whether or not all four are needed for a reasonable thermodynamics. with a slight strengthening of (1), to unique spectrality, and a significant weakening of (2), to the conjunction of (a) projectivity (an abstraction of certain aspects of the quantum projection postulate (l¸ders' version)) and (b) symmetry of transition probabilities under exchange of states with the unique finegrained effects they make certain, we have the property, important in quantum thermodynamics, that the outcome probabilities for any fine-grained measurement are majorized by the spectrum of a state, and hence that measurement-probability-based generalizations of classical entropy-like functions are given by the classical function applied to the spectrum.

this is joint work with markus mueller and cozmin ududec (characterization of quantum formalism) and with jonathan barrett, marius krumm, and markus mueller (thermodynamics).

jointly with rafael andrist we have recently shown that an affine variety $x$ is determined, up to base field automorphisms, by the abstract semigroup of endomorphisms, provided $x$ contains a copy of the affine line.

a more interesting question is how much information about $x$ can be retrieved from the group $aut(x)$ of automorphisms of $x$. this group has the structure of an ind-group, i.e. an infinite dimensional algebraic group, a concept introduced by shafarevich in 1966. it was recently studied by several authors, in particular in the case of affine $n$-space $a^n$. however, not much is known about this group in general, but there are a number of very interesting examples and conjectures.

in connection with the question above, we can prove the following.

theorem. if $x$ is a connected affine variety such that $aut(x)$ is isomorphic to $aut(a^n)$ as an ind-group, then $x$ is isomorphic to $a^n$ as a variety.

we will explain these concepts and results, and describe some recent development.

in 1966 shafarevich introduced the notion of “infinite dimensional algebraic group”, shortly “ind-group”. his main application was the automorphism group of affine $n-space \ a^n$ for which he claimed some interesting properties. recently, jointly with j.-ph. furter we showed that the automorphism group of any finitely generated (general) algebra has a natural structure of an ind-group, and we further developed the theory.

it turned out that some properties well-know for algebraic groups carry over to ind-groups, but others do not. e.g. every ind-group has a lie algebra, but the relation between the group and its lie algebra still remains unclear. as another by-product of this theory we get new interpretations and a better understanding of some classical results, together with short and transparent proofs.

an interesting “test case” is $\ aut(\ a^2)$, the automorphism group of affine 2-space, because this group is the amalgamated product of two closed subgroups which implies a number of remarkable properties. e.g. a conjugacy class of an element $g \in \ aut(\ a^2)$ is closed if and only if $g$ is semi-simple, a result well-known for algebraic groups. a generalization of this to higher dimensions would have very strong and deep consequences, e.g. for the linearization problem.

note: there will be a $pre-talk$ for graduate 2022年亚洲世界杯预选赛 from 2:30-3:00. the speaker has kindly accepted to tell our graduate 2022年亚洲世界杯预选赛 what an ind-group is. the regular talk will begin at 3:00.

the name schwarz iterative methods has been coined in the late 1980ies as a theoretical framework for investigating domain decomposition and multilevel methods for variational problems. they are based on the notion of stable space splittings of a hilbert space $h$ into a family of auxiliary hilbert spaces $h_i$, each equipped with its own variational subproblem, and in essence represent a more constructive version of the alternating directions method (adm) which in turn is a special instance of a pocs algorithm. we consider the multiplicative schwarz iteration \[ u^{j+1} = u^j + \omega_j w^j_i, \] where finding the update direction $w^j_i$ involves solving the $i$-th subproblem, and $\omega_j$ is a relaxation parameter. which $i=i_j$ is chosen in the $j$-th update step matters, until recently only deterministic orderings have been considered.

the renewed interest in investigating random (and greedy) orderings was triggered by results by strohmer and vershynin on the convergence of a randomized kaczmarz method for solving $ax=b$ (= gauss-seidel with random ordering for $aa^\ast y=b$) which received much attention, both due to the simplicity of the error analysis and the sometimes improved performance. we generalized their result to the setting of schwarz iterative methods with finitely many subproblems. even though the a priori bounds for greedy and random orderings are identical, the numerical tests show that the greedy version leads to faster convergence in practice, and that combinations of randomization and greedy approaches can remedy slow convergence of the cheaper randomized iteration. extensions to infinitely many subproblems are considered. for the case of solving linear systems (sor-method), some further improvements have recently been made for both deterministic and random orderings.

in the talk, i will briefly review the setup of schwarz iterative methods and then outline the recently obtained convergence estimates for sor-type methods with deterministic and randomized orderings for linear systems.

the ginzburg-landau lattice model is a system of stochastic differential equations that is known to converge, if properly rescaled, to a linear spde. when introducing an asymmetry in the model that vanishes in the limit, we prove convergence to the (semilinear) stochastic burgers equation. the theory for this stochastic pde is non-trivial, and we apply the theory of an energy solution. the latter was introduced by goncalves-jara and under a slight reformulation gubinelli-perkowski were recently able to show well-posedness for this equation.

this talk is about recent work with luc lapointe of the universidad de talca in chile. i will present what it means to be a schur function in superspace and what properties they have. one of the first steps to building a theory of symmetric function theory in superspace is to describe the pieri rules for these superspace versions of schur functions. in our research, we have
described them and proved that they are correct.

furthermore, we have just started to study the pieri rules for the jack polynomials in superspace as a stepping stone to understand the pieri rules for macdonald polynomials in superspace. we uncovered a surprise connection to the set of alternating sign matrices and the partition functions of square ice!!!

we consider a gibbs distribution over random walk paths on the square lattice, proportional to the cardinality of the path's boundary. we show that in the zero temperature limit, the paths condensate around an asymptotic shape. this limit shape is characterized as the minimizer of the functional, mapping open connected subsets of the plane to the sum of their principle eigenvalue and perimeter (with respect to some norm). a prime novel feature of this limit shape is that it is not in the class of wulff shapes.

this term i'll give some talks about topological quantum field theory and related topics in topology, algebra, geometry, physics or whatever. my main plan is not to plan... it's much more fun to digress freely and "do requests" from the audience.

kazhdan's property (t) is a representation-theoretic property of groups, which was introduced by kazhdan in 1967, but has found numerous applications in an amazingly large variety of subjects from representation theory and ergodic theory to combinatorics (expanders) and the theory of networks. i will start with a gentle introduction to the emerging subject of "noncommutative real algebraic geometry," a subject which deals with equations and inequalities in noncommutative
algebra over the reals, with the help of analytic tools such as representation theory and operator algebras. i will then present a surprisingly simple proof that a group $g$ has property (t) if and
only if a certain inequality in the group algebra ${\bf r}[g]$ is satisfied. this inequality is rather amenable to computer-assisted analysis and is useful in finding new examples of property (t) groups or better bounds of the kazhdan constants of known property (t) groups.

the classic linear preserver problem asks to determine, for a polynomial function f on a vector space v, the linear transformations g of v such that fg = f. in case f is invariant under a simple algebraic group g acting irreducibly on v , we note that the subgroup of gl(v) stabilizing f often has identity component g and we give applications realizing various groups, including the largest exceptional group e8, as automorphism groups of polynomials and algebras. we show that starting with a simple group g and an irreducible representation v, one can almost always find an f whose stabilizer has identity component g and that no such f exists in the short list of excluded cases. the main results are new even in the special case where the field is the complex numbers. this talk is about joint work with bob guralnick.
pre-talk for graduate 2022年亚洲世界杯预选赛 2:30-3:00.

the level set topology optimization method has been drawing attention as a method to integrate conceptual design to manufacturing. the use of level set method has a distinct advantage in naturally handling splitting and merging of boundaries. topology optimization has been identified as the ideal design method for additive manufacturing and many case studies quote revolutionary weight saving or performance improvement in the order of 20% - 80%. my research has been developing the level set topology optimization method for engineering design problems. this seminar will present the existing challenges that will potentially benefit from collaboration with mathematics.

the study of the spectrum of non-hermitian random matrices with independent, identically distributed entries was introduced by ginibre and girko. i will present two generalizations of the iid model where the independence and identical distribution assumptions are relaxed.

in typical applications of model theory to various areas of mathematics, deciding which properties are axiomatizable in the sense of first-order logic is usually key. thus, when it was first realized that many important properties of c* algebras were not axiomatizable (e.g. nuclearity), it appeared that all hope waslost for an interesting development of the model theory of c* algebras. that pessimism was soon reversed when it was realized that many of these aforementioned properties were so-called “omitting types” properties, which is a specific type of infinitary axiomatizability. using the omitting types theorem, this has allowed for some interesting results to be proven about c* algebras. i will give a survey of some of the applications of omitting types in c* algebras, including the kirchberg embedding problem, the failure of a finitary version of the arveson extension theorem, and a question of bankston on the pseudo-arc. i will also mention some open questions where the omitting types machinery might prove useful, e.g. a problem of ozawa asking for an example of a non-nuclear c* algebra with both wep and llp. i will assume no knowledge of c* algebras or model theory (although some prior knowledge will help in some parts of the talk). some of the talk will represent joint work with thomas sinclair, martino lupini, alessandro vignati, and christopher eagle.

given a representation $v$ of a group $g$, it is a classical problem to determine the centralizer of its action on the n-th tensor power of $v$. if $v$ is the natural module of a classical lie group, this led to the famous schur-weyl and brauer-weyl dualities.

in this talk, we solve this problem for the spinor representation $s$. for even-dimensional spin groups, the centralizer on the n-th tensor power of $s$ is given by a representation of $so(n)$, with a similar result also for the odd-dimensional spin groups. time permitting, we discuss generalizations of this result to quantum groups and to classification of tensor categories.

please note: there will be a pre-talk for graduate 2022年亚洲世界杯预选赛 from 2:30 - 3:00. the regular talk will begin at 3:00.

many important problems in computational science and engineering involve partial differential equations posed on moving domains. this talk will present numerical methods for the solution of such problems, as well as theoretical tools for analyzing their accuracy. i will first introduce a family of high-order finite element methods for moving-boundary problems that can handle large domain deformations with ease while representing the geometry of the moving domain exactly. at the core of our approach is the use of a universal mesh: a background mesh that contains the moving domain and conforms to its geometry at all times by perturbing a small number of nodes in a neighborhood of the moving boundary. i will then introduce a unified analytical framework for establishing the convergence properties of a wide class of numerical methods for moving-boundary problems. this class includes, as special cases, the technique described above as well as conventional deforming-mesh methods (commonly known as arbitrary lagrangian-eulerian, or ale, schemes).

the question of global regularity remains open for many fundamental models of fluid dynamics. in two dimensions, solutions to the incompressible euler equations have been known to be globally regular since the 1930s, although their derivatives can grow double-exponentially with time. on the other hand, this question has not yet been resolved for the more singular surface quasi-geostrophic (sqg) equation, which is used in atmospheric models. the latter state of affairs is also true for the modified sqg equations, a natural family of pde which interpolate between these two models.

i will present two results about the patch dynamics version of these equations on the half-plane. the first is global-in-time regularity for the euler patch model, even if the patches initially touch the boundary of the half-plane. the second is local-in-time regularity for those modified sqg patch equations which are only slightly more singular than euler, but also existence of their solutions which blow up in finite time. the latter appears to be the first rigorous proof of finite time blow-up in this type of fluid dynamics models.

we explain limit theorems associated with a family of random ordinary differential equations on manifolds, driven by randomly perturbed vector fields. after rescaling, the differentiable random curves converge to a markov process whose markov generator can be written explicitly in hormander form. we also give rates of convergence in the wasserstein distance.

example 1. a unit speed geodesic, which chooses a direction randomly and uniformly at every instant of order $1\epsilon$, converges to a brownian motion as epsilon tends to 0. furthermore their horizontal lifts converge to the horizontal brownian motion.

examples 2. inspired by the problem of the convergence of berger's spheres to a $s ^ {1/2}$, we introduce a family of interpolation equations on a lie group $g$. these are stochastic differential equations on a lie group driven by diffusion vector fields in the direction of a subgroup $h$ rescaled by $1\epsilon$, and a drift vector field in a transversal direction. if there is a reductive structure, we identify a family of slow variables which, after rescaling, converges to a markov process on $g$. furthermore, the projection of the limiting markov process to the orbit manifolds $g/h$ is markov. the limits can be identified in terms of the eigenvalue of a second order differential operator on the subgroup and the $ad(h)$ invariant decomposition of the lie algebra.

we consider a finite element discretizations of the biot's model in poroelasticity with lowest order (mini and stabilized p1-p1) elements. we show convergence of discrete schemes which are implicit in time and use these types of elements in space. we also address the issue related to the presence of non-physical oscillations in the pressure approximations for low permeabilities and/or small time steps and present numerical results confirming the monotone behavior of two stabilized schemes. this is a joint work with carmen rodrigo, francisco gaspar (univ. zaragoza) and xiaozhe hu (tufts).

some physical and mathematical theories have the unfortunate feature that if one takes them at face value, many quantities of interest appear to be infinite! various techniques, usually going under the common name of “renormalisation” have been developed over the years to address this, allowing mathematicians and physicists to tame these infinities. we will tip our toes into some of the mathematical aspects of these techniques and we will see how they have recently been used to make precise analytical statements about the solutions of some equations whose meaning was not even clear until now.

our approach for fitting dynamic nonstationary factor models to multivariate time series is based
on the principal components of the estimated time-varying spectral-density matrix. this approach
allows the spectral matrix to be smoothly time-varying, which imposes very little structure on the
moments of the underlying process. however, the estimation delivers time-varying filters that are
high-dimensional and two-sided. moreover, the estimation of the spectral matrix strongly depends
on the chosen bandwidths for smoothing over frequency and time. as an alternative, we propose a
new semi-parametric approach in which only part of the model is allowed to be time-varying. more
precisely, the latent factors admit a dynamic representation with time-varying autoregressive
coefficients while the loadings are constant over time. estimation of the model parameters is
accomplished by application of the em algorithm and the kalman filter. the time-varying parameters are modeled locally by polynomials and estimated by maximizing the likelihood locally. compared to estimation of the factors by principal components, our new approach produces superior results in particular for small cross-sectional dimension.

let $\omega\subseteq {\rm gl}_n(f_p(t))$ be a finite symmetric set containing the identity. let $\gamma$ be the group generated by $\omega$ and let $\mathbb{g}$ be the zariski-closure of $\gamma$. we discuss conditions on which the family of cayley graphs $\{{\rm cay}(\gamma ({\rm mod} q), \omega)\}$ is a family of cayley graphs as $q$ ranges through a certain subset $\sigma$ of $f_p[t]$. this problem is a positive characteristic variation of the work of bourgain-gamburd, varju, salehi golsefidy-varju, and others. we focus on the difficulties that arise in positive characteristic.

please note: there will be pre-talk for graduate 2022年亚洲世界杯预选赛 from 2:30 - 3:00. the regular talk will begin at 3:00.

numerically solving elliptic partial differential equations for a large number of degrees of freedom requires the parallel use of many computer processors. this in turn requires algorithms to partition domains into subdomains in order to distribute the work.
in this talk, we present five novel algorithms for partitioning domains that utilize information from the underlying pde. when a pde has strong convection or anisotropic diffusion, directional dependence exists and a partition that favors this direction is desirable. our schemes fall into two classes; one class creates rectangular shaped subdomains aligned in this direction and one class creates subdomains that increase in size as you move in this direction.

these schemes are mathematically described and analyzed in detail. then they are tested on a variety of experiments which include solving the convection-diffusion equation for 1/4 billion unknowns on 512 processors using over 1 teraflop of computing power.

theory and experiments demonstrate that these schemes improve the domain decomposition convergence rate when the underlying pde has directional dependence. in our hundreds of experiments, the number of dd iterations required for convergence reduces by a factor between 0.25 and 0.75. and these methods maintain or improve the final finite element solution's accuracy also.

this talk presents joint work with nicholas loehr, jeffrey remmel, and bruce sagan. the m-level rook placements are a generalization of ordinary rook placements. by factoring the m-level rook polynomials of ferrers boards, it is possible to partition them into equivalence classes.

there is a p,q-analogue of the m-level rook numbers. we have a bijective proof that two boards with the same m-level rook numbers have the same q-analogues of their m-level rook numbers. surprisingly, they may not have the same p-analogues.

we investigate the asymptotic structure of (possibly type iii) crossed product von neumann algebras arising from arbitrary actions of bi-exact discrete groups (e.g. free groups) on amenable von neumann algebras. we particularly prove a spectral gap rigidity result for the crossed products and, using recent results of boutonnet-ioana-salehi golsefidy, we provide the first example of group measure space type iii factors with no central sequences. this is joint work with c. houdayer.

the noise stability of a euclidean set is a well-studied quantity. this quantity uses the ornstein-uhlenbeck semigroup to generalize the gaussian perimeter of a set. the noise stability of a set is large if two correlated gaussian random vectors have a large probability of both being in the set. we will first survey old and new results for maximizing the noise stability of a set of fixed gaussian measure. we will then discuss some recent results for maximizing the low-correlation noise stability of three sets of fixed gaussian measures which partition euclidean space. finally, we discuss more recent results for maximizing the low-correlation noise stability of symmetric subsets of euclidean space of fixed gaussian measure. all of these problems are motivated by applications to theoretical computer science.

i will report some recent works on structure-preserving and stable discretization of magnetohydrodynamics and robust preconditioning methods for the resulting algebraic systems. in particular, judging from theoretical and/or numerical analysis of several mathematical models for magnetohydrodynamics (mhd) which involve the coupling of navier-stokes with maxwell equations, i will argue that some more complicated models may be easier to simulate than some simplified models that have been often used in practice.

the first weyl algebra $a = k\langle x, y \rangle/(xy - yx - 1)$ is $\mathbb{z}$-graded with deg x = 1 and deg y = -1. susan sierra and paul smith studied the category of graded modules over a, showing that this category was equivalent to coherent sheaves on a certain quotient stack. in this talk, we investigate the graded module categories over $\mathbb{z}$-graded rings called generalized weyl algebras. we construct commutative rings with equivalent graded module categories. in the pre-talk, we will discuss some preliminaries on categories and graded rings before giving an overview of noncommutative projective geometry.

please note: there will be a pre-talk for graduate 2022年亚洲世界杯预选赛 from 2:30 - 3:00. the regular talk will begin at 3:00.

recently, anton mellit and i gave a proof of the famous shuffle conjecture of haglund, haiman, loehr, ulyanov, and remmel, which predicts a combinatorial formula for the character of the diagonal coinvariant algebra, and other quantities in algebraic geometry. i'll explain what this conjecture is about, and explain the algebraic structures that go into this recent proof. hopefully if there's time, i'll explain some of remarkable unsolved generalizations, and their role in algebraic geometry.

graeme segal once conjectured that there should be a proof of the kac character formula for affine kac-moody algebras using equivariant localization on the affine grassmannian, analogous to the famous example done by atiyah and bott for the weyl-character formula using borel-weil-bott. he noticed that the idea formally gives the desired answer, modulo several technical considerations such as the infinite dimensionality of the space, singularities, and vanishing of higher cech cohomology groups. i'll explain how these formulas in type a follow provided one is able to switch two limits in the equivariant k-theory of the infinite dimensional grassmannian variety into which the affine grassmannian imbeds. i'll explain a theorem of mine that says gives sufficient (and possibly necessary) conditions in the form of some subtle inequalities for when the limit switching holds, and explain some other examples from macdonald theory that follow such as the famous constant term formula in type a. i'll then explain the value in studying this question for other varieties, perhaps associated to other root systems.

a sandpile on a graph is an integer-valued function on the vertices. it evolves according to local moves called topplings. some sandpiles stabilize after a finite number of topplings, while others topple forever. for any sandpile $s_0$ if we repeatedly add a grain of sand at an independent random vertex, we eventually reach a "threshold state'' $s_t$ that topples forever. poghosyan, poghosyan, priezzhev and ruelle conjectured a precise value for the expected amount of sand in $s_t$ in the limit as $s_0$ tends to negative infinity. i will outline how this conjecture was proved by means of a markov renewal theorem.

subword complexes are simplicial complexes introduced by knutson and miller to illustrate the combinatorics of schubert polynomials and determinantal ideals. they proved that any subword complex is homeomorphic to a ball or a sphere and asked about their geometric realizations. we show that a family of subword complexes can be realized geometrically via triangulations of root polytopes. this implies that a family of $beta$-grothendieck polynomials are special cases of reduced forms in the subdivision algebra of root polytopes. based on joint work with laura escobar.

a remarkable phenomenon discovered by g.a.margulis in the 1970s, now called super-rigidity,
concerns linear representations of discrete subgroups groups in such groups as sl(3,r).
as one of the applications of super-rigidity margulis proved arithmeticity of these groups. in the 1980s r.j.zimmer extended margulis' work to the context of cocycles, that found many applications in ergodic theory and geometry of actions on manifolds. in the talk i plan to give a broad overview of some of these topics and discuss some recent developments.

popa's deformation/rigidity theory led to a remarkable cocycle superrigidity theorem for bernoulli actions of groups with property (t) (2005) and of products of non-amenable groups (2006). more precisely, popa obtained that every cocycle for such an action with values in a countable (and more generally, in a $\mathcal u_{fin}$) group is cohomologous with a group homomorphism. in this talk, we will present a generalization of this theorem to coinduced actions.

the advent of novel engineered or smart materials, whose properties can be significantly altered in a controlled fashion by external stimuli, has stimulated the design and fabrication of smaller, faster, and more energy-efficient devices. as the need for even more powerful technologies grows, networks have become popular alternatives to advance the fundamental limits of performance of individual devices. thus, in the first part of this talk we provide an overview of fifteen years of work aimed at combining ideas and methods from equivariant bifurcation theory to model, analyze and fabricate novel technologies such as: ultra-sensitive magnetic and electric field sensors; networks of nano oscillators; and multi-frequency converters.

in the second part of the talk, we discuss more recent work on networks of vibratory gyroscopes systems. under normal conditions of operation, the model equations can be reformulated in a hamiltonian structure and the corresponding normal forms are then derived. through a normal form analysis, we investigate the effects of various coupling topologies and unravel the nature of the bifurcations that lead a ring of gyroscopes of any size into and out of synchronization. the synchronization state is particularly important because it can lead to a significant reduction in phase drift, thus enhancing performance. the hamiltonian approach can, in principle, be readily extended to other symmetry related systems.

i will present recent results on the consequences of semi-definite holomorphic sectional curvature for the geometric structure of projective kaehler manifolds. in the semi-positive case, the focus is on the existence of rational curves. in the semi-negative case, the focus is on semi-positivity of the canonical line bundle and splitting theorems. this talk covers various joint works with s. lu, b. wong and f. zheng.

the uniformization theorem in one complex variable states that a simply connected riemann surface is either isomorphic to the riemann sphere, the poincare disk, or the complex plane. the uniformization conjecture, proposed by yau in 1974, looks for possible generalizations in higher dimensions. the conjecture states that a complete noncompact kahler manifold with positive bisectional curvature is biholomorphic to the complex euclidean space. in this talk, i will discuss some recent progress on this conjecture and related problems.

we present an algebraic method to study four-dimensional toric varieties by lifting matrix equations from the special linear group $sl(2,\mathbb{z})$ to its preimage in the universal cover of $sl(2,\mathbb{r})$. with this method we recover the classification of two dimensional toric fans, and obtain a description of their semitoric analogue. as an application to symplectic geometry of hamiltonian systems, we give a concise proof of the connectivity of the moduli space of toric integrable systems in dimension four, recovering a known result, and extend it to the case of semitoric integrable systems with a fixed number of focus-focus points. (joint work with daniel kane and alvaro pelayo)

please note: there will be a pre-talk for graduate 2022年亚洲世界杯预选赛 from 2:30 - 3:00. the regular talk will begin at 3:00.

the finite element method in its most simple form considers piecewise polynomials on a triangulated space $\omega$. while the general theory does not require structure on $\omega$ beyond admitting a finite triangulation, it is quickly realized that in order to solve partial differential equations, $\omega$ must be endowed with a differentiable structure. we introduce a new framework which has the potential to allow the finite element method access to a broad class of differentiable manifolds of non-trivial topology.

optimization problems with second order cone constraints have wide range of applications in engineering, control, and management science. in this talk, we present an efficient method based on krylov subspace approximation for solving the second order cone linear complementarity problem (soclcp). here, we first show that soclcp can be solved by finding a positive zero $s_*\in \mathbb{r}$ of a particular rational function $h(s)$, and then propose a krylov subspace method to reduce $h(s)$ to $h_{\ell}(s)$ as in the model reduction. the zero $s_*$ of $h(s)$ can be accurately approximated by that of $h_{\ell}(s)=0$ which itself can be casted as a small eigenvalue problem. the new method is made possible by our complete description of the curve $h(s)$, and it is suitable for large scale problems. the new method is tested and compared against the bisection method recently proposed and two other state-of-the-art packages: sdpt3 and sedumi. our numerical results show that the method is very efficient both for small-to-medium dense problems as well as for large scale problems.

this is a joint work with lei-hong zhang (shanghai university of finance and economics),
wei hong yang (fudan university), and chungen shen (shanghai finance university).

with larger data sizes and often unobservable variables, statistical models have become progressively larger and structurally complex and even computationally intractable. in applications, work-arounds have arisen that add say log-likelihoods and adjust for breakdown in bartlett relations, but usually with the loss of first order accuracy. meanwhile, likelihood asymptotics has sought progressively higher accuracy for p-value functions and now provides definitive third-order p-values and related likelihood functions. we apply the likelihood asymptotic approach to this combining problem and are able to convert composite likelihood to a fully first-order accurate procedure. the methods can then be extended to the combining of statistically dependent p-values and scores.

let g be a group acting (faithfully) on a set x. a base for this action is a subset y of x so that no element of g fixes every element of y. the question of what is the minimal size of a base is a classical subject going back to the early days of finite permutation group theory.

in this talk i will mostly focus on the case that g is simple algebraic group and x is an irreducible variety. a closely related problem is to determine a generic stabilizer (if it exists). note that base size 1 is the same as saying some stabilizer is trivial (and indeed base size b on x is the same as saying base size 1 on b copies of x).

we will consider the case where x = g/h for some maximal closed subgroup h and for the case that x is an irreducible rational g-module. even if one is only interested in the case of finite groups, these cases are relevant.

some of this is joint work with burness and saxl, some with lawther and some with garibaldi.

please note: there will be a pre-talk for graduate 2022年亚洲世界杯预选赛 from 2:30 - 3:00. the regular talk will begin at 3:00.

iwasawa theory concerns the growth of arithmetic objects in towers of number fields. the unramified iwasawa modules that are the inverse limits of p-parts of class groups up such a tower are central objects in the study of iwasawa theory. i will discuss a variety of conjectures and results on the structure of unramified iwasawa modules, along with applications. in particular, i intend to explain iwasawa-theoretic refinements of leopoldt's reflection principle which form part of joint work with bleher, chinburg, greenberg, kakde, pappas, and taylor.

the classical liouville property asserts that bounded harmonic functions on euclidean space are necessarily constant. this property has been extended to $\mu$-harmonic functions related to a random walk $s$ in a locally compact group $g$ with defining measure $\mu$. in the present talk the dependence on $g$ and $\mu$, of the asymptotic entropy $h(g,\mu)$ of $s$, will be studied. the case $h(g,\mu)=0$ characterizes the liouville property, and $h(g,\mu)>0$ leads to the well-known boundary theory of h. furstenberg.

in 1844, kummer showed that cyclotomic integer rings can fail to be principal ideal domains. in 1976 and 1984, ribet and mazur-wiles used galois representations attached to modular forms to partially describe class groups that measure the extent of this failure. the exact structure of these class groups remains a mystery to this day. i will explain how to attach ideal classes to geodesics in the complex upper half-plane. a conjecture of mine states these two constructions are inverse to each other in an appropriate sense. i hope to motivate a broader philosophy, developed jointly with takako fukaya and kazuya kato, that certain arithmetic objects attached to galois representations of global fields can be described using higher-dimensional modular symbols.

the first characterization of groups by an asymptotic description of random walks on their cayley graphs dates back to kesten’s criterion of amenability. i will first review some connections between the random walk parameters and the geometry of the underlying groups. i will then discuss a flexible construction that gives solution to the inverse problem (given a function, find a corresponding group) for large classes of speed, entropy and return probability of simple random walks on groups of exponential volume growth. based on joint work with jeremie brieussel.

the kardar-parisi-zhang (kpz) universality class is a broad collection of probabilistic models including stochastic pdes, random growth models and directed polymers. models in this class show unusual, non-gaussian fluctuations, which for some special classes of initial data are described by objects coming from random matrix theory (rmt). while the connection between kpz and rmt is well understood in the case of curved initial data, the case of flat initial data has remained a bit of a mystery.

after introducing these topics, i will present a result about a system of $n$ brownian motions conditioned not to intersect on a finite time interval. the result shows that the distribution of the squared maximal height of the top path in this system coincides with that of the largest eigenvalue of a certain (finite) random matrix, known as a real wishart or loe matrix. i will describe how this result provides an explanation for the connection between kpz and rmt in the flat case, and how it generalizes some older results concerning its $n \to \infty$ scaling limit. based on joint work with gia bao nguyen.

we define fibonacci analogues of the stirling numbers of the first and second kind and the lah numbers. we will describe a new rook theory model which allows us to give combinatorial
interpretations the fibonacci analogues the stirling numbers of the first and second kind and the lah numbers as well as certain $q$-analogues and $p,q$-analogues of such numbers.

i'll discuss two problems, which on the surface seem quite different. the first, which comes up in signal processing and in algorithm design, is the problem of coming up with linear, geometry-preserving maps which are efficient to store and manipulate. the second, which comes up in coding theory and theoretical computer science, is the problem of establishing the list-decodability -- a combinatorial property -- of error correcting codes. i'll establish a connection between these two problems, and discuss how techniques from high-dimensional probability can be used to handle both. punchlines include improved fast johnson-lindenstrauss transforms and structured rip matrices, and the answer to some longstanding combinatorial open questions in coding theory.

using techniques introduced by h. thomas and n. williams in "cyclic symmetry of the scaled simplex," we prove that modular sweep maps are bijective. we conclude that the general sweep maps defined by d. armstrong, n. loehr, and g. warrington in "sweep maps: a continuous family of sorting algorithms" are bijective. as a special case, this proves that the zeta map on rational dyck paths is a bijection.