from demaratus and histiaeus to ceasar, from
mary queen of scots to hitler, cryptography has played
an important role in world history.
substitution, transposition and polyalphabetic ciphers,
and the famous enigma machine will be discussed.
we will finish with personal experiences working outside of academia.

we incorporate the poisson--boltzmann (pb) theory of electrostatics into the variational implicit-solvent model (vism) for the solvation of charged molecules in an aqueous solvent. the principle of vism is to determine equilibrium solute-solvent interfaces and estimate the molecular solvation free energies by minimizing a mean-field free-energy functional of all possible solute-solvent interfaces. the functional consists mainly of solute-solvent interfacial energy, solute-solvent van der waals interaction energy, and electrostatic energy. we develop highly accurate numerical methods for solving the dielectric pb equation and for computing the dielectric boundary force. these methods are integrated into a robust level-set method for numerically minimizing the vism functional. we test and apply our level-set vism with pb theory to the solvation of some single ions, two charged particles, and two charged plates, and to the solvation of the host-guest system cucurbit[7]uril and bicyclo[2.2.2]octane. our computational results show that vism with pb theory can capture well the sensitive response of capillary evaporation to the charge in hydrophobic confinement and the polymodal hydration behavior, and can provide accurate estimates of binding affinity of the host-guest system. we also discuss several issues for further improvement of vism. this is a joint work with li-tien cheng, joachim dzubiella, bo li, and j. andrew mccammon.

algebraic k-theory is a deep and subtle invariant of rings and schemes, carrying information about arithmetic and geometry. when applied to the group ring of the loop space of a manifold, it captures information about the diffeomorphism group. over the past 25 years, the study of algebraic k-theory has been revolutionized by the introduction of trace methods, which use trace (or chern character) maps to the simpler but related theories of (topological) cyclic and hochschild homology.

a unifying perspective on the properties of algebraic k-theory and these related theories is afforded by viewing the input as a category of compact modules (i.e., a piece of an enhanced triangulated category). this talk will survey recent work describing the structural properties of these theories using various models of the homotopical category of module categories.

natural images tend to be compressible, i.e., the amount of information needed to encode an image is small. this conciseness of information -- in other words, low dimensionality of the signal -- is found throughout a plethora of applications ranging from mri to quantum state tomography. it is natural to ask: can the number of measurements needed to determine a signal be comparable with the information content? we explore this question under modern models of low-dimensionality and measurement acquisition.

a conjecture of debarre predicts that the only subvarieties of ppav with minimal cohomology class are brill-noether loci in jacobians and the fano surface of line in the intermediate jacobian of a cubic three fold. we present a work in progress with luigi lombardi in which we made some progress toward the conjecture. in particular we give a generalization to higher dimension of matsusaka-ran criterion.

we study some problems relating to polynomials evaluated
either at random gaussian or random bernoulli inputs. we present a
structure theorem for degree-d polynomials with gaussian
inputs. in particular, if p is a given degree-d polynomial, then p
can be written in terms of some bounded number of other polynomials
$q_1,...,q_m$ so that the joint probability density function of
$q_1(g),...,q_m(g)$ is close to being bounded. this says essentially
that any abnormalities in the distribution of $p(g)$ can be explained by
the way in which p decomposes into the $q_i$. we then present some
applications of this result.

i will present some new results on classical and quantum integrable systems, emphasizing the interactions between symplectic geometry and spectral theory. i will also briefly describe some recent advances in the study of invariants in symplectic geometry.

in this talk i will present results on a stochastic model of spatial evolution on a lattice, motivated by the process of carcinogenesis (or cancer initiation) from healthy epithelial tissue. cancer often arises through a sequence of genetic alterations or mutations. each of these alterations may confer a fitness advantage to the cell, resulting in a clonal expansion. to model this we will consider a generalization of the biased voter process which incorporates successive mutations modulating fitness. under this model we will investigate a possible mechanism for the phenomenon of ``field cancerization," which refers to the clinical observation that multiple independent primary tumors often arise in the same region of tissue. (joint work w/ k. leder, m. ryser, and r. durrett)

in this talk we utilize a support vector machine feature selection
procedure via concave minimization to solve the well-known disputed
federalist papers classification problem. first we find a separating
plane that classifies correctly all the training set consisting of
papers of known authorship, based on the relative frequencies of three
words only. then, using this three-dimensional separating plane, all
of the 12 disputed papers ended up on one side of the separating
plane. our result coincides with previous statistical and
combinatorial method results.

cells are highly sophisticated information processing devices, sensing diverse inputs abstract:to make complex decisions. yet despite increasingly elegant ways to measure pathway outputs, we largely lack control over the inputs delivered to the cell in space and time. i will describe how optogenetic techniques can overcome this challenge to control the exact combinations, dynamics, and spatial locations of pathway activity in live cells. applied to mammalian growth factor signaling, they revealed the ras/erk module accurately transmits a huge range of steady-state and dynamic signals to downstream response programs. a light-based screen uncovered many of these dynamics-sensitive responses, including a cell-cell communication circuit acting through il-6 family cytokines.

invariant theory is a beautiful field. the area dates back
over 100 years to the work of hilbert, klein, gauss, and many others.
it is a very active area of research today, particularly from the
viewpoint of algebraic geometry and combinatorics. it also has far
reaching applications in representation theory, coding theory,
mathematical modeling, and even air target recognition. (i just
happened to run across this last application on google and it will
*not* be explained.)

in this talk, i aim to illustrate the beauty of noncommutative
invariant theory. all basic notions will defined. namely, i will
explain the noncommutative analogues of each of the following terms:
"groups", "acting on", "polynomial rings". i will also provide an
overview of recentwork pertaining to quantum group actions on
(noncommutative) regular algebras. the results discussed here are
from joint works with kenneth chan, pavel etingof, ellen kirkman,
yanhua wang, and james zhang: see arxiv:math/1210.6432, 1211.6513,
1301.4161, 1303.7203.

we study hodge theory for symplectic laplacians on compact
symplectic manifolds with boundary. these laplacians are novel as they
can be associated with symplectic cohomologies and be of fourth-order.
we prove various hodge decompositions and use them to obtain the
isomorphisms between the cohomologies and the spaces of harmonic fields
with certain prescribed boundary conditions. in order to establish
hodge theory in the non-vanishing boundary case, we are required to
introduce new concepts such as the j!dirichlet boundary condition and
the j!neumann boundary condition. when the boundary is of contact
type, these conditions are closely related to the reeb vector field.
another application of our results is to solve boundary value problems
of differential forms.

an isogeny graph is a graph whose vertices are principally polarized
abelian varieties and whose edges are isogenies between these varieties. in
his thesis, kohel described the structure of isogeny graphs for elliptic
curves and showed that one may compute the endomorphism ring of an elliptic
curve defined over a finite field by using a depth first search algorithm
in the graph. in dimension 2, the structure of isogeny graphs is less understood and existing algorithms for computing endomorphism rings are very expensive.
we fully describe the isogeny graphs between genus 2 jacobians with complex multiplication,
with the assumptions that the real multiplication subring is maximal and
has class number one. we derive a depth first search algorithm for computing endomorphism rings locally at prime numbers,
if the real multiplication is maximal. to the best of our knowledge, this is the first dfs-based algorithm in genus 2. (joint work with emmanuel thomé).

in biomolecular and colloidal systems, the dielectric environment often depends on the local ionic concentrations. recent experiments and molecular dynamics simulations have revealed quantitatively such dependence, and indicated that the electrostatic interaction in a system with such a dependence can be significantly different from that predicted by theories assuming a uniform dielectric constant. in this work, we develop a mean-field model for the electrostatic interactions with ionic concentration-dependent dielectrics. we minimize the electrostatic free energy of ionic concentrations using poisson's equation with a concentration-dependent dielectric coefficient to determine the electrostatic potential. our analysis leads to the the corresponding generalized boltzmann distributions of the equilibrium ionic concentrations that is quite different from the usual ones with a uniform dielectric constant we show by constructing an example that the free-energy functional is in fact nonconvex. this implies the existence of multiple local minimizers, a property that can be of physical significance. by numerical computations using our continuum model, we find many interesting phenomena such as the non-monotone ionic concentration profile near a charged surface, and the unusual shift of concentration peak due to the increase of surface charge density. it is clear that the effect of local dielectrics has a significant impact on the system and our models and results have potential applications to large biological systems. this is joint work with bo li and shenggao zhou.

i'll present an assortment of quantum phenomena that are actually quite easy to understand mathematically. i will provide the necessary background.

we prove asymptotic stability of shear flows close to the planar, periodic couette flow in the 2d incompressible euler equations. that is, given an initial perturbation of the couette flow small in a suitable regularity class, specifically gevrey space of class smaller than 2, the velocity converges strongly in l2 to a shear flow which is also close to the couette flow. the vorticity is asymptotically mixed to small scales by an almost linear evolution and in general enstrophy is lost in the weak limit. joint work with nader masmoudi. the strong convergence of the velocity field is sometimes referred to as inviscid damping, due to the relationship with landau damping in the vlasov equations. recent work with nader masmoudi and clement mouhot on landau damping may also be discussed.

can we describe complex behaviors of living systems, arising from millions of intracellular and intercellular interactions, in simple mathematical models without dizzying number of parameters? for two examples, i combine simple models and experiments to show that this is possible. first, i reduce the complexity of yeast cell's growth from food (glucose) consumption to a phenomenological model with two parameters - cell's perception of and uptake rate of glucose. second, i show that a cell tunes how much it "talks" to itself versus to its neighbors by secreting and sensing just one signaling molecule. encoding who talks to whom, a population of "secrete-and-sense cell" realizes a rich repertoire of complex behaviors.

compressed sensing is a signal acquisition paradigm that utilizes the sparsity of a signal (a vector in $r^n$ with $s << n$ non-zero entries ) to efficiently reconstruct it from very few (say n, $s < n << n$) generalized linear measurements. these measurements often take the form of inner products with random vectors drawn from appropriate distributions, and the reconstruction is typically done using convex optimization algorithms or computationally efficient greedy algorithms.
in this talk we cover some recent results in compressed sensing and related areas where the signal acquisition is also done via such generalized linear measurements. we discuss signal recovery from compressed sensing measurements using weighted $\ell_1$ minimization, when erroneous support information is available. we provide reconstruction guarantees that hold with high probability and improve on the standard results, provided the support information is accurate enough. time permitting, we also discuss some recent results on the digitization of generalized linear measurements, including compressed sensing measurements.

during embryonic development, an entire organism is generated from a single cell. genetics and biochemistry have identified developmental signaling pathways, however, how embryonic patterns emerge in space and time remains more obscure. i will discuss our work using a combination of live‐cell imaging, microfluidics, micropatterning, and mathematical modeling to investigate spatial and temporal aspects of embryonic patterning by the tgf‐β signaling pathway, a key morphogen during all phases of development. our work reveals novel dynamic properties of the pathway and their implications for embryonic development and establishes human embryonic stem cells as a promising system for studying self‐organized patterning quantitatively.

we consider the einstein constraint equations on an n-dimensional, asymptotically euclidean manifold m with boundary. by leveraging both our own recent recent work as well as the work of some of our collaborators, we show that far-fromcmc and near-cmc solutions exist to the conformal formulation of the einstein constraints when robin-type marginally trapped surface boundary conditions are imposed to ensure that expansion scalars along null geodesics perpendicular to the boundary region are non-positive. therefore, assuming a suitable form of weak cosmic censorship, the results we develop in this article provide a method to construct initial data that will evolve into a space-time containing an arbitrary number of black holes. a particularly important feature of our results are the minimal restrictions we place on the mean curvature, giving both near- and far-from-cmc results that are new.

let $e/\mathbb{q}$ be an elliptic curve with full 2-torsion over
$\mathbb{q}$. we
wish to study the distribution of the ranks of the 2-selmer groups of
twists of $e$ as we vary the twist parameter. a recent result of
swinnerton-dyer shows that if $e$ has no cyclic 4-isogeny defined over
$\mathbb{q}$, then the density of twists with given rank approaches a
particular
distribution. unfortunately swinnerton-dyer used an unusual notion of
density essentially given as the number of primes dividing the twist
parameter goes to infinity. we extend this result to cover density in
the natural sense.

spectral graph theory involves the study of graph properties
that are related to the eigenvalues of various matrices, such as the
adjacency matrix, the combinatorial laplacian, and the normalized
laplacian. we will also discuss connection graphs, which are weighted
simple graphs for which each edge is associated with a rotation matrix.
connection graphs have been studied recently in a variety of areas that
involve high dimensional data sets. we will describe higher dimensional
versions of the adjacency, laplacian, and normalized laplacian matrices,
and what their eigenvalues tell us about the connection graph.

the vertebrate segmentation clock is a gene expression oscillator controlling segmentation of the vertebral column. oscillatory hes/her family genes create a transcriptional negative feedback loop. we developed a stochastic two-dimensional computational model that successfully reproduces how the period, amplitude and synchronization of the segmentation clock are regulated in different genetic backgrounds. we validated the major predictions of our model by quantifying rapid degradation of her7 protein and spatially-increasing gradient of translational time-delay of her1 protein in zebrafish embryos. we further demonstrated that expression of oscillating genes are activated by notch and wnt but repressed by ra signaling.

we propose a variable metric framework for minimizing the sum of a self-concordant function and a possibly non-smooth convex function endowed with a computable proximal operator. we theoretically establish the convergence of our framework without relying on the usual lipschitz gradient assumption on the smooth part. an important highlight of our work is a new set of analytic step-size selection and correction procedures based on the structure of the problem. we describe concrete algorithmic instances of our framework for several interesting large-scale applications, such as graph learning, poisson regression with total variation regularization, and heteroscedastic lasso.
here is a link to the document that contains technical parts of the
presentation: http://arxiv.org/abs/1308.2867

chemical reaction networks (crns) are used to model a variety of chemical and biological processes. a crn may be viewed as a directed graph, whose nodes are chemical complexes and whose edges are chemical reactions. wegscheider proposed, in 1901, the principle of detailed balance for chemical kinetics resulting from microscopic reversibility. in the deterministic setting of a dynamical system, the principle of detailed balance manifests as a set of conditions on the rate constants arising from the graphical structure of the crn. when modeled as a markov chain, a different graphical structure is associated with a crn, one whose nodes are population numbers of the chemical species, and whose edges are transitions with non-zero probabilities. we show that the two notions -- deterministic detailed balance (ddb) and stochastic detailed balance (sdb) -- are intimately related, but not equivalent. in particular, ddb implies sdb, but the converse is not true. however, achieving sdb without ddb requires stringent conditions on the rate constants, that are rarely realized in practice. an important exception to this is a birth and death process, for which sdb always holds but ddb does not hold in general.

for each real quadratic order o, there is a weierstrass curve w
in the hilbert modular surface parametrizing jacobians with real
multiplication by o. the curve w emerges from the study of billiards in
polygons and is important in teichmuller theory because its natural
immersion into the moduli space of curves is isometric. such an immersion
is called a teichmuller curve.

we will present explicit algebraic models for weierstrass curves obtained
by studying hilbert modular forms. we will also present evidence from our
examples that suggest a rich arithmetic associated to teichmuller curves.
this work is joint with a. kumar.

this talk will be geared towards a general audience. assuming only familiarity with p-adic numbers, we
will first discuss the local langlands correspondence for gl(n), and explain why it's a wide-ranging generalization of local class field theory. then we will follow emerton and helm in trying to extrapolate it to a correspondence between representations over more general local rings -- whereupon we reformulate the conjectural ihara lemma
in this language. the latter is a big open problem for n>2, which occurred in work of clozel, harris, and taylor, in their attempt to mimic the proof of fermat's last theorem in the context of gl(n).

finite element exterior calculus (feec) is a framework that allows for results proved on general differential complexes to be applied to a large class of mixed finite element problems. in earlier work, using this framework, we introduced a convergence and optimality result for a class of adaptive mixed finite element problems posed on polygonal domains. in this talk we discuss the extension of these results to problems on euclidean hypersurfaces. more specifically, we introduce a method and prove rates of convergence for problems posed on surfaces implicitly represented by level-sets of smooth functions.

speed, resolution, photo damage, and penetration depth are crucial for almost every application of microscopy in biological research. exciting new techniques like localization microscopy, structured illumination microscopy, and light-sheet microscopy enable faster, gentler, higher-resolution imaging that was once thought impossible. however, these techniques typically involve tradeoffs; for example, structured illumination microscopy sacrifices speed and penetration depth to achieve high resolution. i'll describe improved optical and computational approaches to localization microscopy, structured illumination microscopy, and light-sheet microscopy which eliminate these drawbacks, creating viable replacements for the current generation of conventional microscopes.

we study the global pointwise decay properties of solutions to the linear wave equation in 3+1 dimensions on a family time-dependent, non-trapping, radiating space-times. assuming a local energy decay estimate we prove that sufficiently regular solutions to this equation satisfy a conformal energy estimate and higher order conformal energy estimate with one vector field. as an application we establish a global pointwise decay estimate in terms of a weighted norm on initial data.

i will discuss the problem of stability of pluricanonical maps of varieties of positive kodaira dimension. even though it is a very natural question, few results are known. i will show how some particular cases of this problem are related to an old conjecture of fujita, usually called log spectrum conjecture. roughly speaking, it says that the set of pseudo-effective thresholds of polarized pairs has only accumulation points from below. i will explain a proof of this conjecture using recent results in the minimal model program due to hacon, mckernan and xu.

the problem of retrieving phase information from amplitude measurements alone has appeared in many scientific disciplines over the last century. phaselift is a recently introduced algorithm for phase recovery that is computationally efficient, numerically stable, and comes with rigorous performance guarantees. phaselift is optimal in the sense that the number of amplitude measurements required for phase reconstruction scales linearly with the dimension of the signal. however, it specifically demands gaussian random measurement vectors - a limitation that restricts practical utility and obscures the specific properties of measurement ensembles that enable phase retrieval. here we present a partial derandomization of phaselift that only requires sampling from certain polynomial size vector configurations, called t-designs. such configurations have been studied in algebraic combinatorics, coding theory, and quantum information. we prove reconstruction guarantees for a number of measurements that depends on the degree t of the design. if the degree is allowed to to grow logarithmically with the dimension, the bounds become tight up to polylog-factors. beyond the specific case of phaselift, this work highlights the utility of spherical designs for the derandomization of data recovery schemes.

in this talk we discuss closed self adjoint extensions of the laplacian and fractional laplacian on $l^2$ of euclidean space minus the origin. in many cases there is a one parameter family of these operators that behave like the original operator plus a potential at the origin. using these operators, we can construct polymer measures which exhibit interesting phase transitions from an extended state to a bound state where the pinning at the origin due to the potential takes over. the talk is based on joint works with koralov, molchanov, squartini and vainberg.

i am going to present some dispersive estimates for schroedinger's equation with random time-dependent potential and their applications. this is joint work with juerg froehlich and avy soffer.

a challenge for the field of computational biophysics is to develop approaches that bridge across the disparate time and length scales sampled within different simulation regimes. methods for combining the speed of rigid-body brownian dynamics (bd) simulations with the precision of all-atom molecular dynamics (md) simulations using a markov state model (msm) attempt to improve estimates of binding kinetics. we seek to use msms to alleviate the computational burden imposed by long time processes by partitioning conformational space into manageable pieces, simulating them in parallel, and then combining the resulting statistics using the msms to yield meaningful information.

this talk will be a report of joint work in progress with v.v. shokurov. the subject of our study is birational geometry of threefold conic bundles. the talk will explain a notion of minimality for a standard conic bundle and explore the geometry of a certain class of conic bundles, called tetragonal conic bundles.

we state the yamabe conjecture and briefly review the techniques that have been used to study the yamabe problem on riemannian manifolds with smooth metrics. we review some of the basic tools from functional analysis and the theory of partial differential equations for studying the problem in the case of manifolds with non-smooth metrics. in particular, we will discuss some important results related to the yamabe problem for closed manifolds and asymptotically euclidean manifolds with rough metrics.

let c be a hyperelliptic curve defined over the rational numbers, and consider the set s of all squarefree integers d such that the quadratic twist of c by d has a rational point. in this talk we will discuss the question of whether, given a prime number p, the set s contains representatives from all congruence classes modulo p. when c has genus 0 this question can be answered using elementary number theory, but for higher genera it seems to require the use of big conjectures in arithmetic geometry.

the talk pertains to linear time invariant systems of ode. the standard engineering approach for the last 20 years has been to convert these problems to matrix inequalities. these consist of polynomials in matrices and one must find (by numerical computation) choices of matrices which make the polynomial positive definite. the trouble is polynomials which are easily produced from your system problem, by turn the crank methods, are horrible. the holy grail is to somehow convert your system of polynomial matrix inequalities to linear matrix inequalities, lmi. there are many lmi solvers and lmis are certainly convex.

the talk concerns the converse: which convex problems give lmis? the focus is on this rather than the systems engineering context.

next comes, the issue of converting a problem to a convex one. this is daunting and requires some new kind of noncommutative (free) real algebraic geometry, and will be a long time in coming, but there has been serious progress in the last 10 years.

given an algebraic variety x over a field f (e.g. number fields,
function fields), a natural question is whether the set of rational points
x(f) is non-empty. and if it is non-empty, how many rational points are
there? in particular, are they zariski dense? do they satisfy weak
approximation? for cubic hypersurfaces defined over the function field of a
complex curve, we know the existence of rational points by tsen' s theorem
or the graber-harris-starr theorem. in this talk, i will discuss the weak
approximation property of such hypersurfaces.

we continue presenting examples of how feec recasts problems into a more
geometric form, and describe extension of the method to hyperbolic problems,
by, specifically, application of feec to solving the wave equation and
maxwell's equations. we describe a choice of discretization (whitney forms) and
possible generalizations and their issues.

a famous theorem of tate implies that two smooth projective
curves over a finite field have the same zeta function if and only if
their jacobians are isogenous (in particular, the curves needn't be
isomorphic). we prove that two smooth projective curves are
isomorphic (up to automorphisms of the ground field) if and only
if certain associated dynamical systems (arising from class field theory)
are topologically conjugate. this is in turn equivalent to an equality
of all dirichlet l-series of the curves via a group isomorphism between
the groups of linear characters of their absolute galois groups.

he dynamics of a flexible filament sedimenting in a viscous fluid are explored analytically and numerically. compared with the well-studied case of sedimenting rigid rods, the introduction of filament compliance is shown to cause a significant alteration in the long-time sedimentation orientation and filament geometry. a model is developed by balancing viscous, elastic and gravitational forces in a slender-body theory for zero-reynolds-number flows, and the filament dynamics are characterized by a dimensionless elasto-gravitation number. in the weakly flexible regime, a multiple-scale asymptotic expansion is used to obtain expressions for filament translations, rotations and shapes which match excellently with full numerical simulations. furthermore, we show that trajectories of sedimenting flexible filaments, unlike their rigid counterparts, are restricted to a cloud whose envelope is determined by the elasto-gravitation number. in the highly flexible regime we show that a filament sedimenting along its long axis is susceptible to a buckling instability. a linear stability analysis provides a dispersion relation, illustrating clearly the competing effects of the compressive stress and the restoring elastic force in the buckling process. finally, we incorporate the effect of flexibility on the dynamics of a suspension of such filaments using a mean-filed model. a dilute suspension of rigid rods settling under gravity is itself unstable to density fluctuations as a result of hydrodynamic interactions; we show that introducing filament flexibility has opposing effects on this concentration instability. on the one hand, the flexibility-induced reorientation establishes a base state that is more prone to instability, while on the other hand reorientation reduces particle aggregation thereby leading to stabilization.

i will show how to find uniform finiteness results for
certain diophantine equations in terms of the laplace spectrum
of an associated graph. the method is to bound the "gonality"
of a curve (minimal degree of a map onto a line) by the
"stable gonality" of an associated stable reduction graph,
and then to bound this stable gonality of the graph
(some kind of minimal degree of a map to a tree) in terms
of spectral data. the latter bound is a graph theoretical
analogue of a famous inequality of li and yau in differential
geometry. an example of an application is to bound the degree
of the modular parametrisation of elliptic curves over
function fields. (joint work with fumiharu kato and janne kool.)

in this talk i will present some recent rigidity results for the von neumann algebras associated with actions of braid groups. we will show that any free ergodic pmp action of the central quotient of the braid group with at least five strands on a probability space is virtually w*-superrigid; this means that any such action can be (almost) completely reconstructed from its von neumann algebra. the proof uses a dichotomy theorem of popa-vaes for normalizers inside crossed products by free
groups in combination with a oe-superrigidity theorem of kida for actions of mapping class groups.
other structural results such as primeness or unique tensor factorisations for the von neumann algebras associated with braid groups will also be discussed. this is based on an initial joint work with a. ioana and y. kida and a subsequent joint work with f. goodman and s. pant.

it is known that the distribution of a random unitary matrix, under the heat kernel measure on the unitary group u(n), converges as n tends to infinity. i will discuss the convergence of the distribution of a random unitary matrix arising from a levy process on the unitary group u(n). the approach is based on the schur-weyl duality, and we will see that the asymptotic distribution is closely related to the counting of certain paths in the symmetric group.

in 1975, szemeredi proved that any set of integers of positive density must contain arbitrarily long arithmetic progressions. this result solved a 40 year old conjecture of erdos and turan. furthermore, it was one of the main ingredients used by green and tao in their proof that the primes contain arbitrarily long arithmetic progressions. in this talk we will discuss the easiest case of szemeredi's theorem: arithmetic progressions of length 3.

the theory of \emph{supercharacters}, which generalizes classical character theory, was recently developed in an axiomatic fashion by p. diaconis and i.m. isaacs, based upon earlier work of c. andre. when this machinery is applied to abelian groups, a wide variety of applications emerge. in particular, we develop
a generalization of the discrete fourier transform along with several combinatorial tools. this perspective illuminates several classes of exponential sums (e.g., gauss, kloosterman, and ramanujan sums) that are of interest in number theory. we also consider certain exponential sums that produce visually striking patterns of great
complexity and subtlety. (partially supported by nsf grants dms-1265973, dms-1001614, and the fletcher jones foundation.)

the riemann mapping theorem says that every simply connected proper plane domain can be conformally mapped to the unit disk. this result is over a 100 years old, but the study and computation of such maps is still an active area. in this talk, i will discuss the computational complexity of constructing a conformal map from the disk to an n-gon and show that it is linear in n, with a constant that depends only on the desired accuracy. as one might expect, the proof uses ideas from complex analysis, quasiconformal mappings and numerical analysis, but i will focus mostly on the surprising roles played by computational planar geometry and 3-dimensional hyperbolic geometry.

if time permits, i will discuss how this conformal mapping algorithm implies new results in discrete geometry, e.g., every simple polygon can be meshed in linear time using quadrilaterals with all new angles between 60 and 120 degrees. a closely related result states that any planar triangulation of n points can be refined by adding vertices and edges into a non-obtuse triangulation (no angles bigger than 90 degrees) in time $o(n^(5/2))$. no polynomial bound was previously known.

questions like the nagata conjecture seek to determine when certain zero dimensional schemes impose independent conditions on sections of a line bundle on a surface. understanding analogous questions for vector bundles instead amounts to studying the birational geometry of moduli spaces of sheaves on a surface. we explain how to use higher-rank interpolation problems to compute the cone of effective divisors on any moduli space of sheaves on the plane. this is joint work with izzet coskun and matthew woolf.

the (higher order) johnson homomorphism embeds the associated graded lie algebra for the mapping class group of a once punctured surface into a certain lie algebra. calculating the image of the johnson homomorphism is a challenging problem. shigeyuki morita was the first to define obstructions to lying in the image back in the 90s. more recently enomoto and satoh have defined a new series of obstructions, and work of conant-kassabov-vogtmann has provided a rich family of obstructions, involving classical modular forms, stemming from the abelianization of the target lie algebra. in this talk, i will present joint work with martin kassabov which simultaneously generalizes all of these obstructions, making use of an apparently new action of $aut(f_n) on h^{\otimes n}$ for any cocommutative hopf algebra h.

this talk compares local and global optimality conditions for multivariate polynomial optimization problems. first, we prove that the constraint qualification, strict complementarity and second order sufficiency conditions are all satisfied at each local minimizer, for generic cases. second, we prove that if such optimality conditions hold at each global minimizer, then a global optimality certificate must be satisfied. third, we show that lasserre's hierarchy almost always has finite convergence in solving polynomial optimization under the archimedeanness.

we will talk about how a combinatorial approach to multiplying polynomials is useful in the study of gromov-witten invariants (constants arising in string theory) and macdonald coefficients (polynomials in 2 parameters tied to representation theory and geometry). no background beyond undergraduate math is needed.

we will discuss a type of partial rigidity result for proper holomorphic mappings of certain $\ell$-concave domains in projective space into model quadratic $\ell$-concave domains. the main technical result is a degree estimate for proper holomorphic mappings into the model domains, provided that the mappings extend to projective space as rational mappings, and the source domain contains algebraic varieties and has a boundary with low cr complexity.

in their attempt to mimic the proof of fermat's last theorem for gl(n), clozel, harris, and taylor, were led to a conjectural analogue of ihara's lemma -- which is still open for n>2. in this talk we will revisit their conjecture from a more modern point of view, and reformulate it in terms of local langlands in families, as currently being developed by emerton and helm. at the end, we hope to sketch how this can be used to obtain a factorization of completed cohomology for u(2). [the last part is joint work with p. chojecki.]

the (higher order) johnson homomorphism embeds the associated graded lie algebra for the mapping class group of a once punctured surface into a certain lie algebra. calculating the image of the johnson homomorphism is a challenging problem. shigeyuki morita was the first to define obstructions to lying in the image back in the 90s. more recently enomoto and satoh have defined a new series of obstructions, and work of conant-kassabov-vogtmann has provided a rich family of obstructions, involving classical modular forms, stemming from the abelianization of the target lie algebra. in this talk, i will present joint work with martin kassabov which simultaneously generalizes all of these obstructions, making use of an apparently new action of $aut(f_n) on h^{\otimes n}$ for any cocommutative hopf algebra h.

(joint work with a tapay) we discuss a model for studying spontaneous phenomena in the 2d euler equations for incompressible fluid flow. we tie the behavior of the model to the behavior of the actual euler equations.

we describe toric varieties, some of the most delightful objects in algebraic geometry, as quotients of open subsets of affine space by reductive groups. many explicit examples will be shown. time permitting, we will explore the toric varieties package in sage.

the interface between protein solute and aqueous solvent exhibits complex geometries, and can undergo conformational changes by combined influences from electrostatic force, surface tension, and hydrodynamic force. such a combined force on the interface can be calculated via an energy variation approach together with an addition of hydrodynamic interaction. in this talk we present the linear stability analysis for a cylindrical solute-solvent interface, where the linearization system can be solved analytically. the asymptotic dispersion relation satisfies a power law. examples have been given that has long wave (in)stability and short wave stability. bifurcation diagram with multiple steady states are captured in these examples. the role of each part (electrostatics, surface tension, hydrodynamics) in the dispersion relation has also been clarified.

let f be a totally real number field, p a prime number, and m the (splitting model of) hilbert modular variety for f (of some fixed level) defined over a finite field of characteristic p. i will explain how exploiting the geometry of m, and in particular the existence of the partial hasse invariants, one can attach galois representations to hecke eigensystems occurring in the coherent cohomology of m. this is a joint work with matthew emerton and davide reduzzi.

we construct anticyclotomic p-adic l-functions of hida families in a more controlled way using a multiplicity one result arising from arithmetic of shimura curves. using this construction, we can calculate the difference of iwasawa invariants of p-adic l-functions of congruent modular forms in different weights. as an application, we can see the equivalence of the main conjectures of two congruent forms under certain conditions. this is joint work in progress with francesc castella and matteo longo.

the speaker will discuss the recent work(-in-progress) on unifying various mirror constructions of various authors, such as batyrev-borisov and berglund-hübsch-krawitz. this talk hopes to focus on questions, conjectures, and examples involved in this more generalized framework.

i'll try to explain what tcft is and what, if anything, it has to do with jim conant's work from the previous talks.

two lines of attack in the spectral theory of n x n matrix polynomials of degree d will be outlined. the first is an algebraic approach based on the notion of isospectral linear systems in $c^dn$ (the linearizations) and the second on analysis of associated matrix-valued functions acting on $c^n$.

the first approach leads to canonical forms for real symmetric systems consisting of real matrix triples, and thence to canonical triples. furthermore, for real selfadjoint systems we describe selfadjoint canonical triples of real matrices and illustrate their properties.

it turns out that, in this context, there is a fundamental orthogonality property associated with the spectrum. it will be shown how this can play a role in inverse (spectral) problems, i.e. constructing systems with prescribed spectral properties.

we establish global well-posedness and scattering for the cubic dirac equation for small data in the critical space $h^1(\mathbb{r}^3)$. the main ingredient is obtaining a sharp end-point strichartz estimate for the klein-gordon equation. in a classical sense this fails and it is related to the failure of the endpoint strichartz estimate for the wave equation in space dimension three. we construct systems of coordinate frames in which endpoint strichartz estimates are recovered and energy estimates are established.

the functor of p-typical witt vectors is most well known for lifting perfect fields of characteristic p into complete discrete valuation rings. however, it is a well-defined functor on arbitrary rings; we will indicate how applying this functor to local rings of mixed characteristic gives some new perspectives on p-adic hodge theory. we will also touch briefly upon some mysterious links to algebraic k-theory coming from the work of hesselholt. based on joint papers with chris davis (copenhagen).

let $c$ be a smooth irreducible projective curve. we can consider the brill-noether locus $b(2,\omega_c,r)$ of stable rank two vector bundles with canonical determinant and at least $r$ linearly independent sections on $c$. there is a complete description of $b(2,\omega_c,r)$ when $c$ has general moduli and genus $g\leq 12$. for higher genus several basic questions like non-emptyness, irreducibility, etc, are still open. the talk will focus in these questions and some conjectures.

satisfaction and optimization problems subject to random constraints are a well-studied area in the theory of computation. these problems also arise naturally in combinatorics, in the study of sparse random graphs. while the values of limiting thresholds have been conjectured for many such models, few have been rigorously established. in this context we study the size of maximum independent sets in random d-regular graphs. we show that for d exceeding an absolute constant, there exist explicit constants (a,c) depending on d such that the maximum size has constant fluctuations around (an - c(log n)), establishing the one-step replica symmetry breaking heuristics developed by statistical physicists. as an application of our method we also prove an explicit satisfiability threshold in random regular k-nae-sat. this is joint work with jian ding and allan sly.

motivated by turbulence theories, we address the behavior in the infinite reynolds number limit of invariant measures for the 2d stochastic navier-stokes equations. we prove that the limiting inviscid invariant measures are supported on bounded vorticity solutions of the 2d euler equations. we also prove the that ergodic invariant measures for the fractionally damped stochastic 2d euler equations are unique.

i'll present an assortment of non-associative algebras that are considered special, exceptional, or even magical. necessary background will be provided.

first, i will describe how our results (joint with greither) on the brumer-stark conjecture lead to a new construction of hecke characters for cm number fields, generalizing a. weil's jacobi sum hecke characters. second, i will show how the values of these characters can be used to construct special elements in the even
k-groups of cm and totally real number fields. several applications ensue: a general construction of euler systems in the odd k-theory of cm and totally real number fields; a k-theoretic reformulation (and potential proof strategy) of a classical and wide open conjecture of iwasawa on class groups of cyclotomic fields; potential new insights into hilbert's 12th problem for cm number fields etc. time permitting, i will touch upon some of these applications as well. this is based on joint work with g. banaszak (poland).

when each subject in a study provides a vector of numbers/features for analysis, and one wants to standardize, then for each coordinate of the resulting rectangular array one may subtract the mean by subject and divide by the standard deviation by subject. each feature then has mean 0 and standard deviation 1. data from expression arrays and protein arrays often come as such rectangular arrays, where typically column denotes "subject" and the other some measure of "gene". when analyzing these data one may ask that subjects and genes "be on the same footing". thus, there may be a need to standardize across rows and columns of the matrix. we investigate the convergence of a successive approach to standardization, which we learned from colleague bradley efron. limit matrices exist on a borel set of full measure; these limits have row and column means 0, row and column standard deviations 1. we study implementation on simulated data and data that arose in cardiology. the procedure can be shown not to work with simultaneous standardization. results make contact with previous work on large deviations of lipschitz functions of gaussian vectors and with von neumann's algorithm for the distance between two closed, convex subsets of a hilbert space. new insights regarding inference are enabled.

efforts are joint with colleague bala rajaratnam and have been helped by conversations with many others.

we study how to solve semi-infinite polynomial programming (sipp) problems by semidefinite relaxation method. we first introduce two sdp relaxation methods for solving polynomial optimization problems with finitely many constraints. then we propose an exchange algorithm with sdp relaxations to solve sipp problems with compact index set. at last, we extend the proposed method to sipp problems with noncompact index set via homogenization. numerical results show that the algorithm is efficient in practice.

agilent technologies and the university of california san diego are pleased to offer the first in a series of lectures by leading researchers in diverse, cutting-edge fields of biological science.

traffic inside a cell is as complicated as rush hour near any metropolitan area. but drivers know how to follow the signs and roadways to reach their destinations. how do different cellular proteins "read" molecular signposts to find their way inside or outside of a cell? for the past three decades, randy schekman has been characterizing the traffic drivers that shuttle cellular proteins as they move in membrane-bound sacs, or vesicles, within a cell. his detailed elucidation of cellular travel patterns has provided fundamental knowledge about cells and has enhanced understanding of diseases that arise when bottlenecks impede some of the protein flow.

sign up for this informative lecture today at: http://event.signup4.com/ucsd

a smooth complex projective surface x always has picard number at most equal to the hodge number $h^{1,1}$. if equality holds, we say that x has maximal picard number. the known examples of such surfaces (recently surveyed by beauville) are rare and sporadic. we try to explain this rarity by studying the hodge structure of such a surface.

in order to perform their functions, many cells must crawl through a complex environment, including neighboring cells and a confining extracellular matrix. this environment can modify their behavior. i will show that simple physical models can describe a wide variety of these complex cell motions, including collective rotations of confined cells, anomalously large persistence of cells in microchannels, and
turning instabilities in crawling cells. these models, which link cell shape, polarity (an internal cell compass), and physical forces, show how different environments (and a few other factors) can lead to distinct types of cell motility.

studying lie superalgebras and supergroups was initially motivated by applications in physics. in the recent years interesting connections with other branches of mathematics were discovered. the goal of the talk is to review some of these results.

i start with describing four series of algebraic supergroups, which are natural generalizations of general linear, orthogonal and symplectic groups. we shall see different superanalogues of schur-- weyl duality, which reveal connections with universal tensor categories constructed by deligne. then we discuss geometric methods in representation theory of algebraic supergroups: associated variety and borel-weil-bott theory. finally, i will talk about categorification and weight diagram technique and try to explain how they can be used for calculating the characters of irreducible representations of classical supergroups.

i will briefly review the material covered in last week's lecture (may 1) and will continue with a more detailed description of the k-theoretic constructions and their arithmetic applications mentioned in last week's abstract. joint work with g.
banaszak (poland.)

it is a mystery which groups can occur as fundamental groups of smooth complex projective varieties. it is conceivable that whenever the fundamental group is infinite, the variety has some "negative curvature" properties. we discuss a result in this direction, in terms of "symmetric differentials". there are interesting open questions even about the special case of compact quotients of the unit ball in $c^n$. (joint work with yohan brunebarbe and bruno klingler.)

semidefinite programming (sdp) is the problem of optimizing a linear objective function of a symmetric matrix variable, with the requirement that the variable also be positive semidefinite. duality theory is a central concept in sdp, just like it is in linear programming. however, in sdp pathological phenomena occur, such as nonattainment of the optimal values, and positive gaps.

this research was motivated by the curious similarity of pathological sdps that appear in the literature.

we give exact characterizations of when a semidefinite system is badly behaved from the standpoint of duality, and show that "all bad semidefinite programs look the same", as they are characterized by the presence of certain excluded matrices.

we find certain combinatorial characterizations: we prove that all badly behaved semidefinite systems can be brought to a certain standard form, on which it is trivial to recognize their bad behavior, using only elementary linear algebra, without referring to any theorem. we prove analogous results for well-behaved systems, which are not badly behaved.

as a byproduct, we present an algorithm to generate all well behaved systems; in particular, we present a method to generate all linear maps under which the image of the cone of psd matrices is closed.

a theorem of whitehead asserts that the topological 2-type of a (connected) space is uniquely characterised by the triple ($\pi_1, \pi_2, k_3$), where the
$\pi_i, i\leq 2$ are the homotopy groups $\pi_i, i\leq 2, k_3$ is the postnikov class
$\in h^3$($pi_1, \pi_2$), and, indeed all such triples may be realised. such triples
are synonymous with a 2-group, $\pi_2$, i.e. a group `object' in the category of categories, which plays the same role for 2-types as the fundamental group does
for 1-types. in particular, there is a 2-galois correspondence between the
2-category of champs which are etale fibrations over a space and $\pi_2$
equivariant groupoids generalising the usual 1-galois correspondence between spaces which are etale fibrations over a given space and $\pi_1$ equivariant sets. the talk will explain the pro-finite analogue of this correspondence, so, albeit only for the 2-type, a much simpler and more generally valid description of the etale homotopy than that of artin-mazur.

i will present joint work with mark berman, uri onn, and pirita paajanen showing that given a chevalley group, for large p, the number of conjugacy classes of all the congruence quotients of the group of rational points over the valuation ring of a non-archimedean local field of residue characteristic p depends only on the cardinality of the residue field and not on the ring. this reduces to proving that the conjugacy class zeta function is motivic in the sense that it is given uniformly (across all local fields) by a formula of the model-theoretic language of denef-pas-loeser for valued fields, and then to use a so-called motivic transfer principle.

i will then discuss an analogue of this question for the case of a global field and related issues in algebra and number theory.

finally, i will discuss a related general perspective involving a model theory for adeles of a number field and a model theory for finite fields (joint works with angus macintyre).

in this talk i will describe an approach to questions about the relationship between causal sets and smooth lorentzian geometries based on considering both types of structures as examples of partially ordered measure spaces. there is a natural definition of closeness between any two objects of this type. i will introduce this definition, describe some of its properties, and comment on its application to manifoldlike causal sets.

evolution by natural selection can act at multiple biological levels, often in opposing directions. this is particularly the case for pathogen evolution, which occurs both within the host it infects and via transmission between hosts, and for the evolution of cooperative behavior, where individually advantageous strategies are disadvantageous at the group level. in mathematical terms, these are multiscale systems characterized by stochasticity at each scale. we show how a simple and natural formulation of this can be viewed as a measure-valued process. this equivalent process has very nice mathematical properties, namely it converges weakly to either the solution of an analytically tractable integro-partial differential equation or a fleming-viot process. we can then study properties of these limiting objects to infer general properties of multilevel selection.

gradient steady ricci solitons are potential singularity models for the ricci flow. also, they are natural generalizations of ricci-flat manifolds, hence share many interesting properties. in this talk i'll present some $\epsilon$-rigidity results for noncompact ricci-flat manifolds and the generalization to gradient steady ricci solitons.

mori fiber spaces (mfs) are one of the building blocks in the minimal model program.
these are maps $x \to y$ between normal varieties with nice singularities, such that $\dim y <\dim x$,
$\rho(x/y)=1$ and $-k_x$ is ample on every fiber. in particular, most fibers will be $q$-fano varieties.
starting from classical results on the topology of fibrations, i will try to explain how the above conditions
place strong restrictions on what varieties can appear as fibers of mfs. i will give characterizations
for low-dimensional varities and explain what happens in the toric category. moreover, we will show that
this question can be connected to the question of existence of kaehler-einstein metrics.
joint work with g. codogni, a. fanelli, l. tasin.

a symmetric matrix a is completely positive (cp) if there exists an entrywise nonnegative matrix b such that $a = bb^t$. we characterize the interior of the cp cone. we formulate the problem as linear optimizations with cones of moments. a semidefinite algorithm is proposed for checking interiors of the cp cone, and its properties are studied. a cp-decomposition of a matrix in dickinson's form can be obtained if it is an interior of the cp cone. some computational experiments are also presented.

ozawa and popa proved a unique factorization property for tensor products of free group factors. roughly speaking, this means these tensor products "remember" each tensor component. in this seminar, we study similar factorization results for free quantum group factors. in the proof, we use a condition (ao) type phenomena for continuous cores of the tensor products, and prove first a weak factorization property on the cores. then we deduce the desired property for the original tensor
products.

i will survey some recent combinatorial developments on the anti-concentration of random multilinear forms and provide some applications.

(based on joint works with s. o'rourke and v. vu)

by using a simple method, we show that a random $\pm 1$ polynomial of degree n does not have double roots with probability tending to one (as $n$ tends to infinity). as a consequence, we deduce that the expected number of real roots is $(2/\pi)(\log n) + c + o(1)$ for some absolute constant $c$. the method extends to more general coefficient distributions. (based on joint work with o. nguyen and v. vu)

a discrete group is said to be inner amenable if it admits an atomless mean which is invariant under conjugation. in this talk i will provide a satisfying characterization of inner amenability for linear groups over an arbitrary field. i will also discuss a complete characterization of linear groups which are stable in the sense of jones and schmidt. the analysis of stability leads to many new examples of (non-linear) stable groups; notably, all nontrivial countable subgroups of monod's group h(r) are
stable. this includes nonamenable groups constructed by monod and by lodha
and moore, as well as thompson's group f.

it is desirable for the spectrum of a uniform banach algebra to have a structure sheaf. this happens when the algebra is acyclic. we will discuss what it means for a uniform banach algebra to be acyclic. i will explain how to show preperfectoid algebras are acyclic.

we will discuss two graph coloring problems and some applications to extremal graph theory and combinatorial number theory.

in this work, we further extend the recently developed adaptive data analysis method, the sparse time-frequency representation (stfr) method. this method is based on the assumption that many physical signals inherently contain am-fm representations. we propose a sparse optimization method to extract the am-fm representations of such signals. we prove the convergence of the method for periodic signals under certain assumptions and provide practical algorithms specifically for the non-periodic stfr, which extends the method to tackle problems that former stfr methods could not handle, including stability to noise and non-periodic data analysis. this is a significant improvement since many adaptive and non-adaptive signal processing methods are not fully capable of handling non-periodic signals. in particular, we present a simplified and modified version of the stfr algorithm that is potentially useful for the diagnosis and monitoring of some cardiovascular diseases.

a prevailing question in the study of von neumann algebras asks to what extent certain algebras constructed from groups and their actions "remember" the original group and action. pursuing this question led naturally to the study of von neumann algebras coming from certain equivalence relations as well. though a large class of groups and actions which produce "forgetful" algebras have been known since the 1970s (due to connes and zimmer), very little progress was made outside of this class until a breakthrough by sorin popa some 30 years later. we will give an overview of popa's powerful deformation/rigidity theory, state a recent result for von neumann algebras of equivalence relations, and discuss future directions of research.

in this talk i'll discuss how lagrangian particle methods are being used to study the dynamics of fluid vortices. these methods use the biot-savart integral to recover the velocity from the vorticity and they track the flow map using adaptive particle discretizations. i'll present computations of vortex sheet motion in 2d flow, with reference to kelvin-helmholtz instability, the moore singularity, spiral roll-up, and chaotic dynamics. other examples include vortex rings in 3d flow, and vortex dynamics on a rotating sphere.

inside the moduli space of curves of genus 2 with 2 marked points, the loci of curves admitting a map to p1 of degree d totally ramified at the two marked points have codimension two. in this talk i will show how to compute the classes of the compactifications of such loci in the moduli space of stable curves. i will also discuss the relation with the related work of hain, grushevsky-zakharov, chen-coskun, cavalieri-marcus-wise.

recent results on syzygies of projective varieties concentrate on their asymptotics. in this talk, i will discuss results on the asymptotics of syzygies under group actions. in particular, we study two cases. when the underlying space is the projective space, we give the asymptotic growth of syzygy modules with respect to the general linear group. when the underlying space is a toric variety, we give a sharp asymptotic description of the distribution of torus weights.

we discuss analytic-numeric methods for solving stiff scenarios in computer animation. classical explicit numerical integration schemes have the shortcoming that step sizes are limited by the highest frequency that occurs within the solution spectrum of the governing equations, while implicit methods suffer from an inevitable and mostly uncontrollable artificial viscosity that often leads to non-physical behavior. to overcome these specific detriments, an appropriate class of so-called exponential integrators that solves the stiff part of the governing equations by employing a closed-form solution is presented. with these techniques, up to three orders of magnitude greater time steps can be handled compared to conventional methods, and, at the same time, a tremendous increase in overall long-term stability is achieved. this advantageous behavior is demonstrated across a broad spectrum of stiff scenarios that include deformable solids, trusses, and textiles, including damping, collision responses, and friction.

to realize an efficient and physically accurate simulation of stiff fiber-based systems such as human hair, wool infills, and brushes, an appropriate approach for the physically accurate simulation of densely packed fiber assemblies is presented.

this will be a survey/introduction to recently emerging applications of continuous logic to the theory of $ii_1$ factors. no background in continuous logic will be assumed.

in 2006, kostant and wallach constructed an integrable system on the n x n complex matrices $m_{n}( c)$ using gelfand-zeitlin theory. this system can be viewed as a complexified version of the one studied by guillemin and sternberg on the n x n hermitian matrices, which is related to the classical gelfand-zeitlin basis for irreducible representations of the unitary group via geometric quantization.

in this talk, we discuss joint work with sam evens in which we develop a geometric description of the fibres of the moment map for the complexified gelfand-zeitlin system. our approach uses the theory of orbits of a symmetric subgroup k of the group g of all invertible n x n complex matrices on the flag variety of $m_{n}( c)$ . these orbits play a central role in the geometric construction of harish-chandra modules for the pair $(m_{n} (c ), k)$ using the beilinson-bernstein correspondence. we indicate how our work provides the foundation for the geometric construction of a category of generalized harish-chandra modules studied by drozd, futorny, and ovsienko.

the talk will describe joint work with l. berger and b. xie in which we build, for a finite extension l of $\bbb{q}_p$, a new theory of $(\phi,\gamma)$-modules whose coefficient ring is the ring of holomorphic functions on the rigid character variety of the additive group $o_l$, resp. a "robba" version of it.

in the local langlands program the (smooth) representation theory of p-adic reductive groups g in characteristic zero plays a key role. for any compact open subgroup k of g there is a so called hecke algebra $h(g,k)$. the representation theory of g is equivalent to the module theories over all these algebras $h(g,k)$. very important examples of such subgroups k are the iwahori subgroup i and the pro-p iwahori subgroup $i_p$. by a theorem of bernstein the hecke algebras of these subgroups (and many others) have finite global dimension.

in recent years the same representation theory of g but over an algebraically closed field of characteristic p has become more and more important. but little is known yet. again one can define analogous hecke algebras. their relation to the representation theory of g is still very mysterious. moreover they are no longer of finite global dimension. in joint work with r. ollivier we prove that the characteristic p version of $h(g,i_p)$ is gorenstein.

atherosclerosis is an important component of cardiovascular disease and usually occurs in medium-sized and large arteries. there are many different types of atherosclerotic plaque, each with its own distinctive composition and characteristics. for example, late-stage plaques tend to be highly inflamed and contain interior regions of necrosis. on the other hand, intimal thickenings are more homogeneous and consist mainly of smooth muscle cells and their products. in this talk i will present some simple pde and free boundary models to describe the growth, and change in composition of, different types of atherosclerotic plaque. these models aim to give some physical insight into a highly complex disease.

splines on simplicial complexes in 1, 2, and 3 dimensions are well studied objects. as the dimension is raised, there is increased complexity of both the connectivity and geometric information. abstract simplicial complexes provide a means to separate connectivity from geometric information. they are well studied objects, and are a standard tool used to construct and study topologies. in this talk we present the theory of abstract simplicial complexes and splines necessary to understand a proof that the conditions for continuity on the lower order
sub-simplices are contained in the conditions for continuity on the connected, higher order sub-simplices.

we study the generating function for the simplest frame pattern called the
$\mu$-pattern in $n$-cycles. given a cycle $c =(c_1, \ldots, c_n)$, we say that
$(c_i,c_j)$ matches the $\mu$-pattern if $c_i < c_j$ and there is no $c_k$ which lies
cyclicly between $c_i$ and $c_j$ such that $c_i < c_k < c_j$. we will show that the study of $\mu$-patterns in $n$-cycles give rise to a new $q$-analogue of the derangement numbers and has a rather surprising connection with the charge
statistic of lascoux and schutzenberger.

i will discuss three questions in this talk.

(existence) given a smooth hypersurface y of a projective manifold x with numerically trivial normal bundle, does there exist a codimension one foliation on x having y as a compact leaf?

(abelian holonomy) what can we say about foliations having a compact leaf with abelian holonomy?

(factorization) it is rather easy to construct foliations on projective surfaces having compact leaves with non-solvable holonomy. in higher dimensions, the only known examples are pull-backs of foliations on surfaces through rational morphism. is this a general phenomenon? in particular, does the holonomy of compact leaves factor through curves when non solvable? (joint work with b.claudon, f. loray, f. touzet)

we will discuss our on-going investigation of adaptive strategies for finite element equations. we will examine a posteriori error estimates based on superconvergent
derivative recovery. we then survey and compare h, p, r, and h-p adaptive approaches. some numerical examples will be provided.

a crystal is a shape which tessellates the plane (or space) in a periodic way, so that the pattern repeats at regular intervals in all directions. the group of symmetries (translational, rotational, reflectional) of such a tessellation is called a crystallographic group. in two dimensions there are exactly 17 different kinds of symmetry, the so-called "wallpaper groups", which i'll describe. i'll also mention what happens in three dimensions, in hyperbolic space, and how you can make "quasiperiodic" tessellations (penrose tilings) with five-fold symmetry.

the truncated estimation method of ratio type functionals based on a dependent sample of finite size will be presented. this method makes it possible to obtain estimators with guaranteed accuracy in the sense of the $l_{2m}$-norm for $m\geq 1.$
as an illustration, the parametric and non-parametric estimation problems on a time interval of a fixed length are considered. in addition to non-asymptotic properties, the limit behavior of presented estimators is investigated. it is shown that all the truncated estimators have asymptotic properties of basic estimators.
in particular, the asymptotic efficiency in the mean square sense of the truncated estimator of the dynamic parameter of a stable autoregressive process is established. as an application, the problem of asymptotic efficiency of adaptive one-step predictors for a stable multivariate first order autoregressive process with unknown parameters is considered. the predictors are based on the truncated estimators of the dynamic matrix parameter. the criterion of optimality is based on the loss function, defined as a sum of sample size and squared prediction error's sample mean.

the gaussian unitary and orthogonal ensembles (gue, goe) are some of the most studied hermitian random matrix models. when appropriately rescaled the eigenvalues in the interior of the spectrum converge to a translation invariant limiting point process called the sine process. on large intervals one expects the sine process to have a number of points that is roughly the length of the interval times a fixed constant (the density of the process). we solve the large deviation problem which asks about the asymptotic probability of seeing a different density in a large interval as the size of the interval tends to infinity. our proof works for a one-parameter family of models called beta-ensembles which contain the gaussian orthogonal, unitary and symplectic ensembles as special cases.

given a global field k and a rational function f(x) defined over k, one may take pre-images of 0 under successive iterates of f, and thus obtain an infinite tree by assigning edges according to the action of f. the absolute galois group of k acts on the tree, giving a subgroup of the group of all tree automorphisms.

beginning in the 1980s with work of odoni, and developing especially over the past decade, a significant body of work has emerged on the size and structure of this galois representation. the analogy here is to serre's finite index results for galois representations arising from elliptic curves.

i will discuss the contributions of several researchers, including boston and jones, along with my own work (joint with jones) on these questions.

modular operads were introduced by getzler and kapranov to formalize the structure of gluing maps between moduli of stable marked curves. we present a construction of analogous gluing maps between moduli of pluri-log-canonically embedded marked curves, which fit together to give a modular operad of embedded curves. this is joint work with satoshi kondo and jesse wolfson.

i will present a new method to analyze and design iterative optimization algorithms, built on the framework of integral quadratic constraints (iqc) from robust control theory. iqcs provide sufficient conditions for the stability of complicated interconnected systems, and these conditions can be checked by semidefinite programming. i will discuss how to adapt iqc theory to study optimization algorithms, proving new inequalities about convex functions. using these inequalities, i will derive upper bounds on convergence rates for the gradient method, the heavy-ball method, nesterov's accelerated method, and related variants by solving small, simple semidefinite programs. i will close with a discussion of how these techniques can be used to search for algorithms with desired performance characteristics, establishing a new methodology for algorithm design.

given u an n by m matrix, the aim is to extract a large number of linearly independent columns of u and estimate the smallest and the largest singular value of the restricted matrix. for that, we give two deterministic algorithms: one for a normalized version of the restricted invertibility principle of bourgain-tzafriri, and one for the norm of coordinate restriction problem due to kashin-tzafriri. merging the two algorithms, we are able to extract a well-conditioned block inside u, improving a previous result due to vershynin. we use this to attempt a proof of the paving conjecture which is known to be equivalent to the kadison-singer problem and was recently solved by marcus-spielman-srivastava. their proof is only existential. in our attempt, we fail to solve the conjecture; we give however a deterministic algorithm for the best previously known result on it due to bourgain-tzafriri.

let g denote an algebraic group over q and k and l two number fields. assume that there is a group isomorphism of points on g over the adeles of k and l, respectively. we establish conditions on the group g, related to the structure and the splitting field of its borel groups, under which k and l have isomorphic adele rings. under these conditions, if k or l is a galois extension of q and $g(a_k)$ and $g(a_l)$ are isomorphic, then k and l are isomorphic as fields. as a corollary, we show that an isomorphism of hecke algebras for $gl(n)$ (for fixed $n > 1$), which is an isometry in the $l^1$ norm over two number fields k and l that are galois over q, implies that the fields k and l are isomorphic. this can be viewed as an analogue in the theory of automorphic representations of the theorem of neukirch that the absolute galois group of a number field determines the field if it is galois over q.

the nearest point map of a real algebraic variety with respect to euclidean distance is an algebraic function. the euclidean distance degree is the number of critical points for this optimization problem. we focus on projective varieties seen in engineering applications, and we discuss tools for exact computation. our running example is the eckart-young theorem which relates the nearest point map for low rank matrices with the singular value decomposition. this is joint work with jan draisma, emil horobet, giorgio ottaviani, rekha thomas.

a central problem in curve theory is to describe the extrinsic geometry of algebraic curves in a given projective space with fixed genus and degree. koszul cohomology groups in some sense carry everything one ever wants to know about the defining equations of a curve $x$ in $\mathbb{p}^r$: the number of independent equations
of each degree vanishing on $x$ , the relations between the generators of the ideal $i_x$ of $x$, etc.

in this talk, i will describe an inductive approach to study koszul cohomology groups of general curves. in particular, we show that to prove the maximal rank conjecture (for quadrics), it suffices to check all cases with the brill-noether number $\rho=0$. as a consequence, the maximal rank conjecture holds if the embedding line bundles $l$ on $x$ satisfies the condition $h^1(l)<3$.

this talk is intended for a general audience. no knowledge of infinite dimensional lie theory is needed, and the affine algebras are an "excuse" to discuss, mostly by concrete examples, a bridge between infinite dimensional lie theory and sga3. the title of this talk is (intentionally) misleading: kac-moody lie algebras did not exist in 1963. that said, over the last decade substantial results on infinite dimensional lie theory have been proven using the theory of reductive group schemes [sga3] developed by demazure and grothendieck. one can therefore ask, a posteriori, what are the affine algebras in the language of [sga3]. it is an intriguing question with an elegant answer that naturally leads to a (new) family of infinite dimensional lie algebras related to grothendieck's dessins d'enfants.

the finite element exterior calculus (feec) has been useful for the numerical solution of elliptic pdes that more properly accounts for the geometric and topological structures, leading to better numerical stability. our principal goal is to examine how these methods can be extended to evolutionary problems on manifolds, primarily in the parabolic case. we do, however, describe the interesting issues that a hyperbolic generalization poses.

suppose we have a continuous process in some filtration. even it is not a brownian motion (bm) in the given filtration in some cases it is a bm in its own filtration. assume now, that we have $s$ a brownian motion with constant drift in some filtration and we can take an integral $ \beta=h\cdot s $ with respect to this process $s$. is it possible to choose the integrand $h$ in such a way that the result is a bm in its own filtration. the idea of the solution is to take an additional uniform random variable $u$ independent of $s$ and define the integrand such that it takes plus or minus one according to $u$ is smaller or bigger than its conditional median given $f^\beta_t$. it is possible to derive a solution in this way, however the integrand obtained from this median rule will not be adapted to the filtration of $s$. it turns out that the existence of a strong solution, that is, $h$ adapted to the filtration of $s$, is related to the strong solvability of an sde. this later problem is also related to the lack of the semimartingale property of reflected bm in the orthant and also to some classical local time method of proving strong uniqueness of sde. finally, it raises the question that in what generality is it true that a strong enough additive noise restores the strength of the solution of an sde.

for $n > 1$, let $\pi, \pi'$ be two irreducible cuspidal automorphic representations of gl(n, a) where a denotes the adeles over q. let $l(s, \pi \times \pi')$ be the rankin-selberg l-function. if one of $\pi$ or $\pi'$ is self dual then it was shown by moreno and sarnak that the rankin-selberg l-function does not vanish at s = c+it when 1-c is less than a positive fixed constant times a negative power of log(|t| +2). this is also called a standard zero free region. a standard zero free region for the riemann zeta function was first obtained by de la vallee poussin (prime number theorem).

currently, the best known zero free region for rankin selberg l-functions on gl(n) (in the non self dual case) is due to brumley who has proved 1-c is less than a fixed constant times a negative power of |t| +2. in joint work with xiaoqing li we obtain a standard zero free region in the non self dual case.

in this talk, we review various subspace techniques that have been used in constructing numerical methods for solving nonlinear optimization problems. as large scale optimization problems are attracting more and more attention in recent years, subspace methods are getting more and more important due to the fact that subspace methods do not need to solve large scale subproblems in each iteration. the essential parts of a subspace method are how to construct subproblems defined in lower dimensional subspaces and how to choose the subspaces in which the subproblems are defined. various subspace methods for unconstrained optimization, constrained optimization, nonlinear equations and nonlinear least squares, and matrix optimization problems are given respectively, and different proposals are made on how to choose the subspaces.

in this talk we will use the classification results of pelayo-vu ngoc to define a family of metrics on the space of semitoric integrable systems. the family is parameterized by two choices, but the induced topology does not depend on these choices. the resulting metric space is disconnected and incomplete. we will construct the completion and if time allows discuss the connectedness.

the security of genus $3$ curves in public key cryptography has long been somewhat unclear. for non-hyperelliptic genus $3$ curves claus diem found a way to exploit the geometry of the curve to speed up index calculus on the jacobian, achieving an impressive running time of $\widetilde{o}(q)$. unfortunately the algorithm suffers from massive memory requirements.

we have our own variation of non-hyperelliptic genus $3$ index calculus, which improves diem’s approach in several ways. we study both the computational complexity and the memory cost of our method in great detail and make the results completely explicit. combining this with some techniques to alleviate the memory cost, we get a very clear understanding of the security and show that for certain field sizes of practical interest the non-hyperelliptic genus $3$ index calculus is a threat worth taking into account. the so-called isogeny attacks make genus $3$ hyperelliptic curves equally vulnerable.

let x be our favorite space of continuous functions on $r^n$, and let f be a real-valued function defined on some awful subset e of $r^n$. how can we decide whether f extends to a function f in x? if f exists, then how small can we take its norm? what can we say about the derivatives of f (if they exist)? can we take f to depend linearly on f?

suppose e is finite. can we compute an f with close to least-possible norm? how many computer operations does it take? what if f is required merely to agree approximately with f on e? which points of e should we delete as "outliers"?

the subject goes back to whitney. the recent results are joint work with arie israel, bo'az klartag and garving luli.

let $f \in w^{s1,p1}$ and $g \in w^{s2,p2}$ ($s1,s2\ge 0)$. what can be said about $fg$? to which sobolev spaces does the product $fg$ belong? this is the question that we want to talk about. why do we care about this question? as we will discuss, one of the main applications of such results is in the theory of partial differential equations (pdes) and in particular nonlinear elliptic pdes. we will review some of the well-known results and present alternate proofs of those results. in particular we will point out a common mistake in some of the existing literature as we discuss the question of existence of holder-type inequalities for the product of two functions in sobolev spaces.

the celebrated langlands program is anchored in number theory, representation theory, and algebraic geometry, but involves just about all areas of mathematics. one of its cornerstones is a conjectural link between harmonic analysis and algebraic geometry, mediated by an equality of so-called l-functions, which generalize the riemann zeta function. a famous case is the relation between modular forms and elliptic curves (which implies fermat's last theorem). the goal of the talk is to give the general audience some idea of what this is all about, and introduce some of the fundamental players. (familiarity with basic complex analysis, group actions, and quadratic residues will be an advantage.)

following various ideas in the literature, we try to provide a measure of the predictive quality of a model that corresponds to the aic criterion when used for model selection. i will discuss the proposed measure and our motivation, and depending on time, describe some related ideas and issues as background.

around 1960, i. csisz\'ar introduced the notion of divergence, which generalizes the previously known distances between probability measures, including the kullback-leibler relative entropy. an extended kullback-leibler inequality leads to characterizations of various versions of sufficiency which altogether can be rephrased in analytic terms within le cam's decision-theoretic approach to the comparison of statistical experiments.

connections (or `bridges') between pi theory (polynomial identities) and group gradings on associative algebras are quite well known for more than 30 years, where applications appear in both directions. for instance, kemer applied the theory of `super algebras' in order to solve the famous specht problem (to be
explained in the lecture) for nonaffine pi algebras. in the other direction, pi theory is used in order to solve a conjecture of bahturin and regev on `regular gradings' on associative algebras over a field of characteristic zero.

in the lecture i'll recall both subjects (pi theory and g-gradings) and explain how they are related. as an application, i'll present a (jordan's like) theorem on g-gradings on associative algebras. the last part is joint work with ofir david.

we cover a number of combinatorial objects which will be beneficial to the study of the cyclic sieving phenomenon--a recent combinatorial field of research first introduced by reiner, stanton and white. by introducing the study of dyck paths, set partitions, and q-analogues, we will study simple examples of how groups exhibit the cyclic sieving phenomenon. we will conclude this week's presentation with a buildup to an open conjecture and our current research program progress.

unipotent quantum groups are non-commutative versions of unipotent group schemes in algebraic geometry. finite unipotent quantum groups only appear in positive characteristic. in this talk, we will provide a complete classification of such objects up to prime-cube dimension, which can be thought as a generalization of the well-known fact about the structure of p-groups of small orders. in contrast to the finiteness of isomorphism classes for each fixed order in group theory, we obtain nine infinite families among these prime-cube ones, which are all naturally parameterized by finite group quotients of the affine line. further topics regarding representations, cohomology and invariant theory of small unipotent quantum groups will also be discussed during the talk.

during the past decade, much progress has made in refining the principle that high-dimensional statistical problems are tractable when they exhibit some form of low-dimensional structure. however, in practice, it is often unclear whether or not structural assumptions are justified by data, and the problem of validating such assumptions is unresolved in many contexts. in this talk, i will focus on the context of compressed sensing (cs) --- a signal processing framework that is built on the structural assumption of sparsity. although the theory of cs offers strong guarantees for recovering sparse signals, many aspects of the recovery process depend on prior knowledge of the signal's sparsity level --- a parameter which is rarely known in practice. towards a resolution of this issue, i will introduce a generalized family of sparsity parameters that can be estimated in a way that is free of structural assumptions. in connection with signal recovery, i will show that the error rate of the basis pursuit denoising algorithm can be bounded tightly in terms of these parameters. lastly, i will present consistency results for the proposed sparsity estimation procedure, including a clt, which allows for the hypothesis of sparsity to be tested in a precise sense.

in this talk, it is shown that a rank decomposition of symmetric tensors must be its symmetric rank decomposition when the tensor's rank is less than its order. furthermore, when the rank of symmetric tensors equals the order, the symmetric rank must be the rank. as a corollary, for symmetric tensors, rank and symmetric rank coincide when rank is at most order. this partially gives a positive answer to the comon's conjecture. finally, a sufficient condition under which a symmetric decomposition of symmetric tensors is a symmetric rank decomposition is presented. some examples are presented to show the efficiency of the condition.

we will continue our talk about the cyclic sieving phenomenon involving the relationship between q-rational catalan number and the set of non crossing partitions. by utilizing the injective mapping between rational deck paths and non-crossing partitions, we have constructed small cases of how to observe that the set $(nc(a,b),z_{b-1},cat_q(a,b))$ exhibits the cyclic sieving phenomenon. we will also demonstrate a few of our matlab programs that helped aid us in the progress of trying to prove the cyclic sieving phenomenon.

one of the many ways to represent juggling patterns is through so-called \emph{juggling cards}. these are templates which describe the spatial ordering of a set of balls at each point in a juggling pattern. in this talk we describe a number of new combinatorial and probabilistic results in the study of these objects, and state some related, unsolved problems. attendees are welcome to bring their own chainsaws or lit torches for the interactive portion of the talk.

last february, mike cranston spoke in this seminar about a polymer model based on three-dimensional brownian motion conditioned to hit (and keep returning to) the origin. i will discuss the construction and certain properties of this conditioned brownian motion from two points of view (i) dirichlet forms, and (ii) excursion theory. the latter gives a nice interpretation of the johnson-helms example from martingale theory. it turns out that this diffusion process is not a semimartingale, even though its radial part is just a one-dimensional brownian motion reflected at the origin.

based on joint work with liping li of fudan university.

the representation theory of subfactors generalizes the representation theory of quantum groups, and thus we think of subfactors as objects which encode quantum symmetries. in one sense, subfactors of small index are the simplest examples of subfactors, and we have a complete classification of their standard invariants to index 5. i will discuss recent joint work with morrison which classifies certain examples at index $3+\sqrt{5}$. one important ingredient is a new variation of bigelow's jellyfish algorithm which is universal for finite depth subfactors.

the inverse problem in spectral theory was proposed by bochner and weyl in the early 20th century. one formulation of the question is: how much of the dynamics of a classical dynamical system can be detected from the spectrum of its quantization? i will describe this question and review recent results for the case integrable dynamical system, where in certain fundamental cases going back to the work of atiyah and guillemin-sternberg a full solution can be given. the talk will emphasize the interplay between symplectic geometry and semiclassical spectral theory and is intended for a general audience.

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.

we present five novel algorithms for partitioning domains that utilize information from the underlying pde. when a pde has strong convection or anisotropic diffusion, 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
dependance. in our hundreds of experiments, the number of dd iterations required for convergence reduces by a factor between 0.25 and 0.75. in some cases, these methods improve the final finite element solution's accuracy also.

proof of thurston's geometrization conjecture has been a major achievement in the study of 3-manifolds and proves the existence of natural geometric structures on compact 3-manifolds. however this proof and the mere knowledge of the existence of such structures does not give a description of the geometry and therefore fails to answer many of the existing questions. we discuss a project that attempts to find an effective solution to the geometrization and therefore produces models of the promised hyperbolic structures. we explain how this can be used to answer a number of unanswered questions and relate topological and geometrical properties of the manifold.

the main result presented in this talk is a plethystic formula for the
specialization at $t = 1/q$ of the $q_{u,v}$ operators studied in [math
arkiv:1405.0316]. this discovery yields elementary and direct
derivations of several identities relating these operators at $t =1/q$
to the rational compositional shuffle conjecture of [math arkiv:
1404.4616]. in particular we are able to give a direct derivation of a
simple formula for the symmetric polynomial
$$q_{km,kn}1|_{t=1/q} \ \mbox{(for all $m,n$ co-prime and $k \geq 1
$.)}$$

we also give an elementary proof that this polynomial is schur positive.
moreover, by combining our main result with the rational compositional
shuffle conjecture, we obtain a completely elementary derivation of the
identity expressing this polynomial in terms of parking functions in the
$km \times km$ rectangle.

the study of the outer automorphism group out(f) of a free group has been a very active area of geometric group theory in the past few years, driven on many fronts by natural parallels that exist between out(f) and the mapping class group mod(s) of a surface. i will provide an overview of some of the recent developments in the theory of out(f), while emphasizing distinguishing features of out(f) that make it an often more challenging group to understand than its mod(s) counterpart.

the study of risk measures began in a static environment with the papers of artzner et al. (1999) and follmer and schied (2002). to incorporate information structure over time, static risk measures were extended to a dynamic setting in barrieu and el karoui (2009), jobert and rogers (2008), yong (2007) and many others.

we are interested in a specific class of dynamic risk measures, namely dynamic risk measures that arise as solutions of certain types of backward stochastic differential equations (bsdes) or backward stochastic volterra integral equations (bsvies). we will discuss this connection between risk measures, capital allocations and bsdes/bsvies and provide representation results for dynamic risk measures and dynamic capital allocations. these results are based on classical differentiability results for bsdes/bsvies and girsanov-type change of measure arguments.

joint work with ludger overbeck.

this talk will be about random lozenge tilings of a class of planar domains which i like to call "sawtooth domains." the basic question is: what does a uniformly random tiling of a large sawtooth domain look like? at the first order of randomness, a remarkable form of the law of large numbers emerges: the height function of the tiling converges to a deterministic "limit shape." my talk is about the next order of randomness, where one wants to analyze the fluctuations of tiles around their eventual positions in the limit shape. quite remarkably, this ostensibly analytic problem can be solved in an essentially combinatorial way, using a desymmetrized version of the double hurwitz numbers from enumerative algebraic geometry.

i will provide a broad overview of the kinds of problems that i've worked on with particular emphasis on three subjects: understanding the distribution of statistics of random set partitions; the distribution of ranks of selmer groups of elliptic curves; and the study of polynomials in large numbers of variables with random inputs.

understanding the coupling of physical models at different scales is important and challenging. in this talk, we focus on the issue of kinetic-fluid coupling, in particular, the half-space problems for kinetic equations coming from the boundary layer. we will present some recent progress in algorithm development and analysis for the linear half-space kinetic equations, and its application in coupling of neutron transport equations with diffusion equations. (joint work with jianfeng lu and
weiran sun).

i will discuss some of the major problems and results pertaining to dehn surgery, with a highlight on the application of combinatorial methods and heegaard floer homology. in particular, i will report on progress on two guiding conjectures, the cabling conjecture and the berge conjecture.

the main result presented in this talk is a plethystic formula for the
specialization at $t = 1/q$ of the $q_{u,v}$ operators studied in [math
arkiv:1405.0316]. this discovery yields elementary and direct
derivations of several identities relating these operators at $t =1/q$
to the rational compositional shuffle conjecture of [math arkiv:
1404.4616]. in particular we are able to give a direct derivation of a
simple formula for the symmetric polynomial
$$q_{km,kn}1|_{t=1/q} \ \mbox{(for all $m,n$ co-prime and $k \geq 1
$.)}$$

we also give an elementary proof that this polynomial is schur positive.
moreover, by combining our main result with the rational compositional
shuffle conjecture, we obtain a completely elementary derivation of the
identity expressing this polynomial in terms of parking functions in the
$km \times km$ rectangle.

we investigate the role of eigenvector norms in spectral graph theory to various combinatorial problems including the densest subgraph problem, the cheeger constant, among others. we introduce randomized spectral algorithms that produce guarantees which, in some cases, are better than the classical spectral techniques. in particular, we will give an alternative cheeger “sweep” (graph partitioning) algorithm which provides a linear spectral bound for the cheeger constant at the expense of an additional factor determined by eigenvector norms. finally, we apply these ideas and techniques to problems and concepts unique to directed graphs.