what's the general methodology behind ai for math?  ai is disrupting all industries in an unprecedented way. what might happen for mathematics? are mathematicians ever going to be "replaced"?

ehrhart theory—the study of lattice point enumeration in polytopes with rational vertices—can be used to study various combinatorial objects, including posets and graphs. in this talk, we explore two weighted versions of ehrhart theory. we first ask which polynomial weights we can apply to our lattice so that the associated weighted h*-polynomials retain some of their classical properties, such as nonnegativity and monotonicity. we also study a second weighted ehrhart theory, chapoton’s q-analog ehrhart theory, and discuss its relationship to the principal specialization of stanley’s chromatic symmetric function. the first project is joint work with robert davis, jesús a. de loera, alexey garber, sofía garzón mora, katharina jochemko, and josephine yu, and the second project is joint work with matthias beck and andrés r. vindas meléndez.

i will report on recent results stemming from the analysis of schrödinger operators on manifolds. i will first describe results dealing with isoperimetric inequalities and optimal (aka extremal) metrics on closed manifolds. these issues have been instrumental in the study of the spectrum of several classical operators, and are motivated by the understanding of the behaviour of the spectrum under changes of metrics. then, motivated by conjectures of yau on measures of nodal sets (but which are actually related to the first part of the talk), i will describe how eigenfunctions are concentrating in terms of lp norms (with an explicit dependence on the eigenvalues). my goal is to emphasize on the case of schrödinger operators with rough potentials. i will also state several open problems. 

topological order is a notion in theoretical condensed matter physics describing new phases of matter beyond landau's symmetry breaking paradigm. bravyi, hastings, and michalakis introduced certain topological quantum order (tqo) axioms to ensure gap stability of a commuting projector local hamiltonian and stabilize the ground state space with respect to local operators in a quantum spin system. in joint work with corey jones, pieter naaijkens, and daniel wallick (arxiv:2307.12552), we study nets of finite dimensional c*-algebras on a 2d $\mathbb{z}^2$ lattice equipped with a net of projections as an abstract version of a quantum spin system equipped with a local hamiltonian. we introduce a set of local topological order (lto) axioms which imply the tqo conditions of bravyi-hastings-michalakis in the frustration free commuting projector setting, and we show our lto axioms are satisfied by known 2d examples, including kitaev's toric code and levin-wen string net models associated to unitary fusion categories (ufcs). from the lto axioms, we can produce a canonical net of algebras on a codimension 1 $\mathbb{z}$ sublattice which we call the net of boundary algebras. we get a canonical state on the boundary net, and we calculate this canonical state for both the toric code and levin-wen string net models. surprisingly, for the levin-wen model, this state is a trace on the boundary net exactly when the ufc is pointed, i.e., all quantum dimensions are equal to 1. moreover, the boundary net for levin-wen is isomorphic to a fusion categorical net arising directly from the ufc. for these latter nets, corey jones' category of dhr bimodules recovers the drinfeld center, leading to a bulk-boundary correspondence where the bulk topological order is described by representations of the boundary net.

polynomial optimization is an important class of non-convex optimization problems, and has a powerful modelling ability for both continuous and discrete optimization. over the past two decades, the moment-sos hierarchy has been well developed for globally solving polynomial optimization problems. however, the rapidly growing size of sdp relaxations arising from the moment-sos hierarchy makes it computationally intractable for large-scale problems. in this talk, i will show that there are plenty of algebraic structures to be exploited to remarkably improve the scalability of the moment-sos hierarchy, which leads to the new active research area of structured polynomial optimization.

we study variational inequality problems (vips) with involved mappings and feasible sets characterized by polynomial functions (namely, polynomial vips). we propose a numerical algorithm for computing solutions to polynomial vips based on lagrange multiplier expressions and the moment-sos hierarchy of semidefinite relaxations. we also extend our approach to finding more or even all solutions to polynomial vips. we show that the method proposed in this paper can find solutions or detect the nonexistence of solutions within finitely many steps, under some general assumptions. in addition, we show that if the vip is given by generic polynomials, then it has finitely many karush-kuhn-tucker points, and our method can solve it within finitely many steps. numerical experiments are conducted to illustrate the efficiency of the proposed methods.

a borel equivalence relation on a polish space is called hyperfinite if it can be approximated by equivalence relations with finite classes. this notion has long been studied in descriptive set theory to measure complexity of borel equivalence relations. although group actions on hyperbolic spaces don't always induce hyperfinite orbit equivalence relations on the gromov boundary, some natural boundary actions were recently found to be hyperfinite. examples of such actions include actions of hyperbolic groups and relatively hyperbolic groups on their gromov boundary and acylindrical group actions on trees. in this talk, i will show that any acylindrically hyperbolic group admits a non-elementary acylindrical action on a hyperbolic space with hyperfinite boundary action. 

the memory capacity of a statistical model is the largest size of generic data that the model can memorize and has important implications for both training and generalization. in this talk, we will prove a tight memory capacity result for two-layer neural networks with polynomial or real analytic activations. in order to do so, we will use tools from linear algebra, combinatorics, differential topology, and the theory of real analytic functions of several variables. in particular, we will show how to get memorization if the model is a local submersion and we will show that the jacobian has generically full rank. the perspective that is developed also opens up a path towards deeper architectures, alternative models, and training.

the work of gromov, lawson, schoen, yau on scalar curvature inspired progress in metric geometry. in turn, this progress inspired new results about geometry and topology of manifolds with positive scalar curvature. in my talk i will give some examples of how ideas travel between these two worlds. the talk will be based on joint works with lishak-nabutovsky-rotman, maximo, chodosh-li, wang.

 in this talk, we study a tannakian category of admissible pairs, which arise naturally when one is comparing etale and de rham cohomology of p-adic formal schemes. admissible pairs are parameterized by local shimura varieties and their non-minuscule generalizations, which admit period mappings to de rham affine grassmannians. after reviewing this theory, we will state a result characterizing the basic admissible pairs that admit cm in terms of transcendence of their periods. this result can be seen as a p-adic analogue of a theorem of cohen and shiga-wolfhart characterizing cm abelian varieties in terms of transcendence of their periods. all work is joint with sean howe.

[pre-talk at 1:20pm]

i will give an introduction to the $\overline{\partial}$-neumann problem and the bergman kernel, topics that are studied in several complex variables. i will conclude by discussing two open questions about the bergman kernel which motivate research in several complex variables today.