Given a topological group G, one considers the space of its closed subgroups, called the Chabauty space. I will talk about the structure and features of this space, and show how various algebraic and topological properties of a group are expressed there. 

The Garsia-Procesi module $R_\lambda$ has a well known basis of Artin monomials indexed by λ-subYamanouchi words, which correspond to the inv-statistic of the Haglund-Haiman-Loehr combinatorial formula for the modified Macdonald polynomials $H_\lambda(X;q,t)$ at $t=0$. We introduce a new basis for $R_\lambda$ of Garsia-Stanton descent monomials, giving a major-index type formula of the modified Hall-Littlewood polynomial $H_\lambda(x;q,t)$, and discuss the subtle connection to $H_\lambda(x;q,t)$ at $q=0$ via Robinson-Schensted-Knuth insertion. Our formula was discovered while searching for a basis of the Garsia-Haiman module by extending a similar result of Carlsson and Oblomkov for the diagonal coinvariants $DH_n$. This is joint work with E. Carlsson.

 A hybrid system is a dynamical system that exhibits both continuous and discrete dynamic behavior. The state of a hybrid system changes either continuously by the flow described by a differential equation or discretely following some jump conditions. A canonical example of a hybrid system is the bouncing ball, imagined as a point-mass, under the influence of gravity. In this talk, we explore the solutions and algorithms to the extensions of this example in 3-dimension, where the body of interest is rigid and convex in general and the plane may be tilted. In particular, the solutions utilize the theory of nonsmooth Lagrangian mechanics to derive the differential equations and jump conditions, which heavily depend on the collision detection function. The proposed algorithm called Lie group variational collision integrator (LGVCI) is developed using the combination of techniques and knowledge from variational collision integrators and Lie group variational integrators. Furthermore, we also developed a sensible and practical regularization (by analysis and applying $\epsilon$-rounding on signed distance functions) for collision response for convex rigid bodies with corners, and this completely avoids the need for nonsmooth convex analysis, and computations of tangent and normal cones. We have extensive numerical experiments and animations from our algorithm demonstrating that LGCVI are symplectic-momentum preserving and have long-time, near energy conservation.

This is a joined work with Professor Melvin Leok, and we are looking to apply and extend this work in the fields of control & optimal control theory and robotics, especially in the realm of bipedal robots. There will be further discussions on these topics in the section of future directions of the talk.

A separable $C^*$-algebra $A$ is said to be $\ell$-open ( or $\ell$-closed) when the image of Hom(A, B) is open (or closed) in Hom(A, B/I), for all separable $C^*$-algebras B and ideals I. The concept of semiprojectivity has been used many times in the classification of C*-algebras. Bruce Blackadar introduced $\ell$-open and $\ell$-closed $C^*$-algebras as a superclass of semiprojective $C^*$-algebras.

In recent work with A. Tikuisis, we characterize $\ell$-open and $\ell$-closed $C^*$-algebras and deduce that $\ell$-open $C^*$-algebras are $\ell$-closed as conjectured by Blackadar. Moreover, we show that the notion of $\ell$-open $C^*$-algebras and semiprojective $C^*$-algebras coincide for commutative unital $C^*$-algebras.

Topological K-theory is one of the classical motivating examples of a commutative ring spectrum, and it has a natural equivariant generalization. The equivariant structure here has the strongest possible type of compatibility with the multiplication, making K-theory an example of a ``genuine-commutative" ring spectrum.  There's quite a lot of structure involved here, so in order to understand it, we employ a classic strategy and rationalize.   After rationalizing, we can use algebraic models due to Barnes--Greenlees--Kedziorek and to Wimmer to show that all of the additional ``norm" structure is determined by the equivariant homotopy groups and the underlying multiplication.  This is joint work with Christy Hazel, Jocelyne Ishak, Magdalena Kedziorek, and Clover May.

The Kardar-Parisi-Zhang (KPZ) equation is a fundamental stochastic PDE related to many important models like random growth processes, Burgers turbulence, interacting particles system, random polymers etc. In this talk, we focus on how the tall peaks and deep valleys of the KPZ height function grow as time increases. In particular, we will ask what is the appropriate scaling of the peaks and valleys of the (1+1)-d KPZ equation and whether they converge to any limit under those scaling. These questions will be answered via the law of iterated logarithms and fractal dimensions of the level sets. The talk will be based on joint works with Sayan Das and Jaeyun Yi. If time permits, I will also mention an interesting story about the (2+1)-d and (3+1)-d case (work in progress with Jaeyun Yi).

There are two types of algorithms in Reinforcement Learning (RL): value-based and policy-based. As nonlinear function approximations, such as Deep Neural Networks, become popular in RL, algorithmic instability is often observed in practice for both types of algorithms. One reason is that most algorithms are based on the contraction property of the Bellman operator, which may no longer hold in nonlinear approximation. In this talk, we will introduce two algorithms based on the Bellman residual whose performance is independent of the contraction property of the Bellman operator. In both algorithms, we formulate the RL into an unconstrained optimization problem. The first algorithm is value-based, where we assume the underlying dynamics is smooth. We proposed an algorithm called Borrowing From the Future (BFF), and we proved that it has an exponentially fast convergence rate in model-free control. The second algorithm is policy-based. We proposed an algorithm called variational actor-critic with flipping gradients. We prove that it is guaranteed to converge to the optimal policy when the state space is finite. 

This talk will cover some foundational frameworks, strategies, and empirical findings related to undergraduate mathematics education and mathematics education more broadly. Specifically, I will present some evidence-based research on how to promote student engagement (what is active learning?), instructional design theory, teacher talk moves, and equity frameworks. Lastly, I’ll give some resources related to teaching undergraduate mathematics. 

We present families of origamis of genus 3 that have arithmetic Kontsevich-Zorich monodromy in the sense of Sarnak. It is known this is true for origamis of genus 2, however the techniques for higher genera should be different. We present an outline of the proof for the existence of these families.

A lot of graph inference problems consist in finding a low-rank structure planted in the adjacency matrix of the graph. When sparse enough, the simple study of the adjacency matrix is not enough; the individual variance of each vertex influences too much the overall spectrum of $A$. In contrast, we show how the non-backtracking matrix $B$ recovers these low-rank structures more consistently. This generalizes the results of Bordenave et al. (2015) to a much wider range of settings, beyond the classical stochastic block model.

Consider a countable group G acting on the unit sphere S in the space of L^2 functions on G by the regular representation. Given a homology class h in the quotient space S/G, one defines the spherical Plateau solutions for h as the intrinsic flat limits of volume minimizing sequences of cycles representing h. In some special cases, for example when G is the fundamental group of a closed hyperbolic manifold of dimension at least 3, the spherical Plateau solutions are essentially unique and can be identified. However not much is known about the properties of general spherical Plateau solutions. I will discuss the questions of existence and structure of non-trivial spherical Plateau solutions.

The Emerton-Gee stack for $\mathrm{GL}_2$ is a stack of $(\varphi, \Gamma)$-modules whose reduced part $\mathcal{X}_{2, \mathrm{red}}$ can be viewed as a moduli stack of mod $p$ representations of a $p$-adic Galois group. We compute criteria for codimension one intersections of the irreducible components of $\mathcal{X}_{2, \mathrm{red}}$, and interpret them in sheaf-theoretic terms. We also give a cohomological criterion for the number of top-dimensional components in a codimension one intersection.

[pre-talk at 1:20PM]

Mirror symmetry was a phenomenon discovered by physicists
around 1989: they observed that certain kinds of six-dimensional
geometric objects known as Calabi-Yau manifolds seemed to come in
pairs, with a strange relationship between different kinds of
geometric objects on the pairs. Since then, the subject has blossomed
into a vast field, with many different approaches and philosophies. I
will give a brief introduction to the subject, and explain how one of
these approaches, developed with Bernd Siebert, has led to a general
construction of mirror pairs.