Geometric Deep Learning is an emerging field of research that aims to extend the success of machine learning and, in particular, convolutional neural networks, to data with non-Euclidean geometric structure such as graphs and manifolds. Despite being in its relative infancy, this field has already found great success and is utilized by, e.g., Google Maps and Amazon’s recommender systems.


In order to improve our understanding of the networks used in this new field, several works have proposed novel versions of the scattering transform, a wavelet-based model of neural networks for graphs, manifolds, and more general measure spaces. In a similar spirit to the original scattering transform, which was designed for Euclidean data such as images, these geometric scattering transforms provide a mathematically rigorous framework for understanding the stability and invariance of the networks used in geometric deep learning. Additionally, they also have many interesting applications such as discovering new drug-like molecules, solving combinatorial optimization problems, and using single-cell data to predict whether or not a cancer patient will respond to treatment.
 

In 1938, Erdos, Ko and Rado proved a foundational result on the size of intersecting families of sets. Ever since, there has been a rich body of results proving similar theorems in different contexts. Notably, in geometries over finite fields like projective and polar spaces, such results were obtained by several groups of researchers. I will explain a very successful technique that can be used to prove these results, and indicate its shortcomings. We will show how a generalization of this problem recovers all known results and how algebraic combinatorics such as Iwahori-Hecke algebras and their representations come into play. The latter is based on joint work with Jan De Beule and Klaus Metsch.

We study Markov decision processes (MDPs) in the space of state-action distributions using algebraic and differential geometric methods. We provide an explicit description of the set of feasible state-action distributions of a partially observable problems with memoryless stochastic policies through polynomial constraints. In particular, this yields a formulation of the reward optimization problem as a polynomially constrained linear objective program. This allows us to study the combinatorial and algebraic complexity of the problem and we obtain explicit upper bounds on the number of critical points over every boundary component of the feasible set for a large class of problems. We demonstrate that the polynomial programming formulation of reward optimization can be solved using tools from constrained optimization and applied algebraic geometry, which exhibit stability improvements and provide globally optimal solutions. Further, we study the convergence of several natural policy gradient (NPG) methods with regular policy parametrization. For a variety of NPGs we show that the trajectories in state-action space are solutions of gradient flows with respect to Hessian geometries, based on which we obtain global convergence guarantees and convergence rates. In particular, we show linear convergence for unregularized and regularized NPG flows proposed by Kakade and Morimura and co-authors by observing that these arise from the Hessian geometries of conditional entropy and entropy respectively.

In recent years we have experienced a remarkable growth on the number and size of available datasets. Such growth has led to the intense and challenging pursuit of estimators which are provably both computationally efficient and statistically accurate. Notably, the analysis of polynomial-time estimators has revealed intriguing phenomena in several high dimensional estimation tasks, such as their apparent failure of such estimators to reach the optimal statistical guarantees achieved among all estimators (that is the presence of a non-trivial “computational-statistical trade-off”).

In this talk, I will present new such algorithmic results for the well-studied planted clique model and for the fundamental sparse regression model. For planted clique, we reveal the surprising severe failure of the Metropolis process to work in polynomial-time, even when simple degree heuristics succeed. In particular, our result resolved a well-known 30-years old open problem on the performance of the Metropolis process for the model, posed by Jerrum in 1992. For sparse regression, we show the failure of large families of polynomial-time estimators, such as MCMC and low-degree polynomial methods, to improve upon the best-known polynomial-time regression methods. As an outcome, our work offers rigorous evidence that popular regression methods such as LASSO are optimally balancing their computational and statistical recourses.

A modular form is a certain holomorphic function on the complex upper half plane which has an infinite group of discrete symmetries. I will discuss modular forms and some of their generalizations.  In particular, I will try to answer the following questions: What is a modular form? What are some examples of modular forms? What is a simple open question about modular forms?  The examples of modular forms I will give go under the name of "harmonic theta functions".  Time permitting, I will describe some exotic variants of these harmonic theta functions that are tied up with the octonions and the compact Lie group G_2.

The talk will give an overview of categorical semantics - a magical tool that puts our wanted loose ways of reasoning on solid ground. We’ll discuss how Grothendieck generic freeness lemma is “internally” a simple one-liner, in what secret sense Spec A is free, why intuitionistic mathematics is so natural, and maybe even how to approach algebraic geometry synthetically (i.e. with no set theory references).

 I will explain the relevance of the Lichnerowicz Laplacian in several situations and how one can easily understand the zeroth order term in the expression. This leads to simple proofs of some classical results and also to new results with more general curvature conditions.

Continuous-time Markov chains are frequently used as stochastic models for chemical reaction networks, especially in the growing field of systems biology. A fundamental problem for these Stochastic Chemical Reaction Networks (SCRNs) is to understand the dependence of the stochastic behavior of these systems on the chemical reaction rate parameters. Towards solving this problem, in this talk we describe theoretical tools called comparison theorems that provide stochastic ordering results for SCRNs. These theorems give sufficient conditions for monotonic dependence on parameters in these network models, which allow us to obtain, under suitable conditions, information about transient and steady state behavior. These theorems exploit structural properties of SCRNs, beyond those of general continuous-time Markov chains. Furthermore, we derive two theorems to compare stationary distributions and mean first passage times for SCRNs with different parameter values, or with the same parameters and different initial conditions. These tools are developed for SCRNs taking values in a generic (finite or countably infinite) state space and can also be applied for non-mass-action kinetics models. We illustrate our results with applications to models of chromatin regulation and enzymatic kinetics.

This talk is based on joint work with Simone Bruno, Felipe Campos, Domitilla Del Vecchio and Yi Fu.

Over a commutative ring of finite cardinality, how many $n
\times n$ matrices satisfy a polynomial equation? In this talk, I will explain how to solve this question when the ring is given by integers modulo a prime power and the polynomial is square-free modulo the prime.
Then I will discuss how this question is related to the distribution of the cokernel of a random matrix and the Cohen--Lenstra heuristics. This is joint work with Yunqi Liang and Michael Strand, as a result of a
summer undergraduate research I mentored.

[pre-talk at 1:20PM]

I’ll give a short intro to hyperbolic groups and small cancellation theory, and demonstrate how this theory can be used to construct groups with desirable properties. 

The Gaussian Elimination with Partial Pivoting (GEPP) is a classical algorithm for solving systems of linear equations. Empirical evidence strongly suggests that for a typical square coefficient matrix, GEPP is numerically stable. We obtain a (partial) theoretical justification of this phenomenon by showing that, given the random n×n standard Gaussian coefficient matrix A, the growth factor of the Gaussian Elimination with Partial Pivoting is at most polynomially large in n with probability close to one. This implies that with probability close to one the number of bits of precision sufficient to solve Ax=b to m bits of accuracy using GEPP is m+O(log(n)), which we conjecture to be optimal by the order of magnitude. We further provide tail estimates of the growth factor which can be used to support the empirical observation that GEPP is more stable than the Gaussian Elimination with no pivoting. Based on joint work with Han Huang.

How a Turing's reaction-diffusion process in a biological context leads to a rather surprising appearance of Ising-like colorings of the skin of Mediterranean lizards.

I will present a recent work on variational problems involving nonlocal energy functionals that appear in nonlocal mechanics. The well-posedness of variational problems is established via a careful study of the associated energy spaces, which are nonstandard. I will discuss some difficulties in proving fundamental structural properties of the function spaces such as compactness. For a sequence of parametrized nonlocal functionals in suitable form, we compute their variational limit and establish a rigorous connection with classical models.

The development of accurate and efficient numerical solutions to the wave equation is a fundamental area of scientific research with applications in several fields, including music, computer graphics, architecture, and telecommunications. A key challenge in wave simulation research concerns the proper handling of wave propagation on an unbounded domain. This challenge is known as the infinite domain problem. In this talk, I present a novel geometric framework for solving this problem based on the classical Kelvin transformation. I express the wave equation as a Laplace problem in Minkowski spacetime and show that the problem is conformally invariant under Kelvin transformations using the Minkowski metric while the boundedness of the spacetime is not. These two properties of the Kelvin transformation in Minkowski spacetime ensure that harmonic functions which span infinite spacetime can be simulated using finite computational resources with no loss of accuracy.

 In this talk we will discuss several stochastic epidemic models recently developed to account for general infectious durations, infection-age dependent infectivity and/or progress loss of immunity/varying susceptibility, extending the standard epidemic models. We construct individual based stochastic models, and prove scaling limits for the associated epidemic dynamics in large populations. Each individual is associated with a random function/process that represents the infection-age dependent infectivity force to exert on other individuals. We extend this formulation to associate each individual with a random function that represents the loss of immunity/varying susceptibility. A typical infectivity function first increases and then decreases from the epoch of becoming infected to the time of recovery, while a typical immunity/susceptibility function gradually increases from the time of recovery to the time of losing immunity and becoming susceptible. The scaling limits are deterministic and stochastic Volterra integral equations. We also discuss some new PDEs models arising from the scaling limits. (This talk is based on joint work with Etienne Pardoux, Raphael Forien, and Arsene Brice Zosta Ngoufack.)

It is a folklore conjecture that the sup norm of a Laplace eigenfunction on a compact hyperbolic surface grows more slowly than any positive power of the eigenvalue.  In dimensions three and higher, this was shown to be false by Iwaniec-Sarnak and Donnelly.  I will present joint work with Farrell Brumley that strengthens these results, and extends them to locally symmetric spaces associated to $\mathrm{SO}(p,q)$.

[pre-talk at 1:20PM]

Expander graphs are perhaps one of the most widely useful classes of graphs ever considered. In this talk, we will focus on a fairly weak notion of expanders called sublinear expanders, first introduced by Komlós and Szemerédi around 25 years ago. They have found many remarkable applications ever since. In particular, we will focus on certain robustness conditions one may impose on sublinear expanders and some applications of this very recent idea, which include: 

- recent progress on one of the most classical decomposition conjectures in combinatorics, the Erdős-Gallai Conjecture,

- Rainbow Turan problem for cycles, raised by Keevash, Mubayi, Sudakov and Verstraete, including an application of this result to additive number theory and 

- essentially tight answers to the classical Erdős unit distance and distinct distances problems in "almost all" real normed spaces of any fixed dimension.