In this talk, we will introduce the basic concept of the neural network
and discuss one specific type of neural network: Recurrent Neural
Network (RNN). We will discuss the backpropagation algorithm and analyze
the structure of RNN. Furthermore, we will introduce LSTM neural
network. If time permits, an example of time series forecasting by this
method will be provided.

It is a classical fact that free (discrete) groups can be embedded
in $GL_{2}(\mathbb{Z})$. In 1987, Zubkov showed that for a free pro-$p$
group $F_{\hat{p}}$, the situation changes, and when $p>2$, $F_{\hat{p}}$
cannot be embedded in $GL_{2}(\Delta)$ when $\Delta$ is a profinite
ring. In 2005, inspired by Kemer's solution to the Specht problem,
Zelmanov sketched a proof for the following generalization: For every
$d\in\mathbb{N}$ and large enough prime $p\gg d$, $F_{\hat{p}}$
cannot be embedded in $GL_{d}(\Delta)$.

The natural question then is: What can be said when $p$ is not large
enough? What can be said in the case $d=p=2$ ? In the talk I am going
to describe the proof of the following theorem: $F_{\hat{2}}$ cannot
be embedded in $GL_{2}(\Delta)$ when $char(\Delta)=2$. The main
idea of the proof is the use of trace identities in order to apply
finiteness properties of a Noetherian trace ring through the Artin-Rees
Lemma (Joint with E. Zelmanov).

Schubert calculus studies the algebraic geometry and combinatorics of matrix factorizations. I will discuss recent developments in $K$-theoretic Schubert calculus, and their connections to problems in combinatorics and representation theory.

From a probabilistic perspective, 2D quantum gravity is the study of natural probability measures on the space of all possible geometries on a topological surface. One natural approach is to take scaling limits of discrete random surfaces. Another approach, known as Liouville quantum gravity (LQG), is via a direct description of the random metric under its conformal coordinate. In this talk, we review both approaches, featuring a joint work with N. Holden proving that uniformly sampled triangulations converge to the so called pure LQG under a certain discrete conformal embedding.

Since the ground breaking work of Donaldson in the 1980s, topologists has achieved huge success in using gauge theory to study smooth 4-manifolds with nonzero second homology. The case of 4-manifolds with trivial second homology is relatively less known. In particular, when the 4-manifold have the same homology as S1 cross S3, there are several gauge theoretic invariants. The first one is the Casson-Seiberg-Witten invariant LSW(X) defined by Mrowka-Ruberman-Saveliev; the second one is the Fruta-Ohta invariant LFO(X). It is conjecture that these two invariants are equal to each other (This is an analogue of Witten's conjecture relating Donaldson and Seiberg-Witten invariants.) In this talk, I will recall the definition of these two invariants, give some applications of them (including a new obstruction for metric with positive scalar curvature), and sketch a proof of this conjecture for finite-order mapping tori. This is based on a joint work with Danny Ruberman and Nikolai Saveliev.

Since the ground breaking work of Donaldson in the 1980s, topologists has achieved huge success in using gauge theory to study smooth 4-manifolds with nonzero second homology. The case of 4-manifolds with trivial second homology is relatively less known. In particular, when the 4-manifold have the same homology as S1 cross S3, there are several gauge theoretic invariants. The first one is the Casson-Seiberg-Witten invariant LSW(X) defined by Mrowka-Ruberman-Saveliev; the second one is the Fruta-Ohta invariant LFO(X). It is conjecture that these two invariants are equal to each other (This is an analogue of Witten's conjecture relating Donaldson and Seiberg-Witten invariants.) In this talk, I will recall the definition of these two invariants, give some applications of them (including a new obstruction for metric with positive scalar curvature), and sketch a proof of this conjecture for finite-order mapping tori. This is based on a joint work with Danny Ruberman and Nikolai Saveliev.

Motivated by problems in distributed optimization and computation, we discuss a generalization of the Perron-Frobenius Theorem to products of random stochastic matrices. To do so, we introduce several objects such as infinite flow graph, infinite flow property, and show the connection of these concepts to ergodicity of chains of random stochastic matrices.

Fractal uncertainty principle states that no function can be localized to a fractal set simultaneously in position and in frequency. The strongest version so far has been obtained in one dimension by Bourgain and the speaker with recent higher dimensional advances by Han and Schlag.

I will present two applications of the fractal uncertainty principle. The first one (joint with Jin and Nonnenmacher) is a frequency-independent lower bound on mass of eigenfunctions on compact negatively curved surfaces, which in particular implies control for the Schr$\ddot{\text{o}}$dinger equation by any nonempty open set. The second application (joint with Zahl) is an essential spectral gap for convex co-compact hyperbolic surfaces, which implies exponential energy decay of high frequency waves.

We show that the average size of the automorphism group over
$\mathbb{F}_q$ of a smooth degree $d$ hypersurface in
$\mathbb{P}^{n}_{\mathbb{F}_q}$ is equal to $1$ as $d\rightarrow
\infty$. We also discuss some consequences of this result for the moduli
space of smooth degree $d$ hypersurfaces in $\mathbb{P}^n$.

It is a classical theorem of Roth that every dense subset of $\left\{1,\ldots,N\right\}$ contains a nontrivial three-term arithmetic progression. Quantitatively, results of Sanders, Bloom, and Bloom-Sisask tell us that subsets of relative density at least $1/(\log N)^{1-\epsilon}$ already have this property. In this talk, we will discuss about some sets of $N$ integers which unlike $\left\{1,\ldots,N\right\}$ do not contain nontrivial four-term arithmetic progressions, but which still have the property that all of their subsets of density at least $1/(\log N)^{1-\epsilon}$ must contain a three-term arithmetic progression. Perhaps a bit surprisingly, these sets turn out not to have as many three-term progressions as one might be inclined to guess, so we will also address the question of how many three-term progressions can a four-term progression free set may have. Finally, we will also discuss about some related results over $\mathbb{F}_{q}^n$. Based on joint works with Jacob Fox and Oliver Roche-Newton.

Modular forms are holomorphic functions invariant under a
certain group action. They have a surprising amount of number theoretic
information. We introduce their basic theory and explain how their
connection to Galois theory can be used to study congruences between
modular forms.

In this talk, we present recent results on the geometry of centrally-symmetric random polytopes, generated by $N$ independent copies of a random vector $X$ taking values in ${\mathbb{R}}^n$. We show that under minimal assumptions on $X$, for $N \gtrsim n$ and with high probability, the polytope contains a deterministic set that is naturally associated with the random vector -- namely, the polar of a certain floating body. This solves the long-standing question on whether such a random polytope contains a canonical body.

Moreover, by identifying the floating bodies associated with various random vectors we recover the estimates that have been obtained previously, and thanks to the minimal assumptions on $X$ we derive estimates in cases that had been out of reach, involving random polytopes generated by heavy-tailed random vectors (e.g., when $X$ is $q$-stable or when $X$ has an unconditional structure). Finally, the structural results are used for the study of a fundamental question in compressive sensing -- noise blind sparse recovery.

This is joint work with the speaker's PhD student Christian K$\ddot{\text{u}}$mmerle (now at Johns Hopkins University) as well as Olivier Gu{\'e}don (University of Paris-Est Marne La Vall{\'e}e), Shahar Mendelson (Sorbonne University Paris), and Holger Rauhut (RWTH Aachen).

The 2-party political system defines a natural partition of a network of individuals into 2 teams. One can view these individuals as players in a network-sized game, and the utility (or equivalently cost) functions for each player can be realized as wanting to be connected to individuals on the same political party and distanced from those in the opposing political party. When considered from a game theoretic point of view, the ``greedy'' (or myopic/optimal) strategies can be examined. Thus, the game turns into a dynamical system, which can be investigated to understand when each political party will become totally connected. When the original network is sampled from an Erdos-Renyi graph G(n, q), we find a one-sided threshold of when a political party will become completely connected.

Until recently, it was unclear if there was a natural way to construct (compactified) moduli spaces of Fano varieties. One approach to solving this problem is the K-moduli Conjecture, which predicts that K-polystable Fano varieties of fixed dimension and volume are parametrized by a projective good moduli space. In this talk, I will survey recent progress on this conjecture and discuss a result with Yuchen Liu and Chenyang Xu proving the openness of K-stability (a step in constructing K-moduli spaces).

Invariant random subgroups (IRS) are conjugacy invariant probability measures on the space of subgroups of a given locally compact group G. They arise naturally as point stabilizers of probability measure preserving actions. Invariant random subgroups can be regarded as a generalization both of normal subgroups and of lattices in topological groups. As such, it is interesting to extend results from the theories of normal subgroups and of lattices to the IRS setting.
Stuck-Zimmer proved that for higher rank simple Lie groups, any nontrivial IRS comes from a lattice. In rank 1 however the situation is far more complex. Indeed, the space of invariant random subgroups of $SL_{2}R$ contains all moduli spaces of Riemann surfaces, and can be used to obtain an interesting compactification thereof related to the Deligne-Mumford compactification.

Nevertheless, jointly with Arie Levit, we prove a different type of rigidity result valid in the rank 1 setting. We show that the critical exponent (exponential growth rate) of an infinite IRS in an isometry group of a Gromov hyperbolic space (such as a rank 1 Lie group, or a hyperbolic group) is almost surely greater than half the Hausdorff dimension of the boundary. This can be reinterpreted by saying that for any probability measure preserving action of such a group, stabilizers are almost surely either trivial or ``very big''.
This generalizes an analogous result of Matsuzaki-Yabuki-Jaerisch for normal subgroups.
As a corollary, we obtain that if $\Gamma$ is a typical subgroup and $X$ a rank 1 symmetric space then $\lambda_{0}(X/\Gamma)\< \lambda_{0}(X)$ where $\lambda_0$ is the smallest eigenvalue of the Laplacian. The proof uses ergodic theorems for actions of hyperbolic groups.