In this talk, I will discuss semi-parametric estimation when nuisance parameters cannot be estimated consistently, focusing in particular on the estimation of average treatment effects, conditional correlations, and linear effects under high-dimensional GLM specifications. In this challenging regime, even standard doubly-robust estimators can be inconsistent. I describe novel approaches which enjoy consistency guarantees for low-dimensional target parameters even though standard approaches fail. For some target parameters, these guarantees can also be used for inference. Finally, I will provide my perspective on the broader implications of this work for designing methods which are less sensitive to biases from high-dimensional prediction models.

Quantum graphs are a non-commutative analogue of classical graphs related to operator algebras, quantum information, quantum groups, etc. In this talk, I will give a brief introduction to quantum graphs and talk about spectral characterizations of properties of quantum graphs. We introduce connectedness and bipartiteness of quantum graphs in terms of graph homomorphisms, and these properties have algebraic characterizations in the same way as classical cases. We also see the equivalence between bipartiteness and two-colorability of quantum graphs defined by two notions of graph homomorphisms: one respects adjacency matrices, and the other respects edge spaces.

This talk is based on arXiv:2310.09500.

Let $\mathrm{ex}(G_{n,p}^r,F)$ denote the maximum number of edges in an $F$-free subgraph of the random $r$-uniform hypergraph $G_{n,p}^r$.  Following recent work of Conlon, Lee, and Sidorenko, we prove non-trivial lower bounds on $\mathrm{ex}(G_{n,p}^r,F)$ whenever $F$ is not Sidorenko. This connection between Sidorenko's conjecture and random Tur\'an problems gives new lower bounds on $\mathrm{ex}(G_{n,p}^r,F)$ whenever $F$ is not Sidorenko, and further allows us to bound how ``far'' from Sidorenko an $r$-graph $F$ is whenever upper bounds for $\mathrm{ex}(G_{n,p}^r,F)$ are known.  This is joint work with Jiaxi Nie.

 How to understand the Galois group of Q-bar over Q? We want to analyze its action on the nonsingular complex varieties defined over finite extensions of Q. This action preserves the underlying etale homotopy type but permutes the manifold structures over it. In 1970, Sullivan proposed that there is an abelianized Galois symmetry on higher dimensional simply-connected TOP manifolds by the Adams conjecture and it is compatible with the Galois symmetries on varieties. It is still an ongoing project to describe this mysterious Galois symmetry in a more geometric way by branched coverings. Indeed, this agrees with Grothendieck's discussion of dessin d'enfants on Riemann surfaces in the 1980's. I will report our ongoing works on a generalization to higher dimensions.

An {\em increasing subsequence} of a permutation $w \in S_n$ is a sequence of numbers $1 \leq i_1 < \cdots < i_k \leq n$ such that $w(i_1) < \cdots < w(i_k)$. Increasing subsequences have appeared in various guises in combinatorics, probability, and representation theory. We present an algebraic interpretation in terms of a quotient ring inspired by a problem in cryptography. A link between standard monomial bases and Viennot's `shadow line' construction for the Schensted correspondence will play a key role.

I will present a rather elementary inequality and discuss its application to dbar equations with weights on the whole complex Euclidean space and to subelliptic estimates for the dbar-Neumann problem.

The latter is joint work with Samuele Mongodi (Univ. Milano-Bicocca, Italy).

The interplay between deformation of complex structures and stability of spectrum for the complex Laplacian on compact complex manifolds was studied extensively by Kodaira and Spencer in the 1950s.

In this talk, we will discuss analogous problems for complex manifolds with boundaries and for compact CR manifolds. This talk is based on joint work with Howard Jacobowitz and Siqi Fu.

We will introduce a local notion of Hilbert-Schmidt stability (HS-stability), partially motivated by the recent introduction of local permutation stability by Bradford. We will discuss some basic properties, and then establish a local character criterion for local HS-stability of amenable groups, by analogy with the character criterion for HS-stability of Hadwin and Shulman. We will then discuss further examples of (flexible versions of) local HS-stability. Finally, we show that infinite sofic (resp. hyperlinear) property (T) groups are never locally permutation (resp. HS-) stable, answering a question by Lubotzky. This is based on joint work with Francesco Fournier-Facio and Maria Gerasimova.

Estimation and learning algorithms are dramatically improving our capacity to extract information from massive datasets, with impressive consequences for technology and society at large. Although these algorithms have had widespread empirical success, we have yet to find a coherent mathematical foundation that can explain why these algorithms succeed on such a wide array of problems. The challenge is that the two assumptions that underpin classical optimization theory---smoothness and convexity---rarely hold in contemporary settings. Nonetheless, simple optimization algorithms often do succeed, and over the last few years, I have studied when and why this happens. In this talk, I will survey some recent work in this area covering optimization theory, algorithms, and applications in signal processing and machine learning. In the process, we will encounter a surprisingly rich array of mathematical tools spanning nonsmooth analysis, semi-algebraic geometry, and high dimensional probability and statistics.

Representations of p-adic groups and Hecke algebras Abstract: An explicit understanding of the category of all (smooth, complex) representations of p-adic groups provides an important tool in the construction of an explicit and a categorical local Langlands correspondence and also has applications to the study of automorphic forms. The category of representations of p-adic groups decomposes into subcategories, called Bernstein blocks, which are indexed by equivalence classes of so called supercuspidal representations of Levi subgroups. In this talk, I will give an overview of what we know about an explicit construction of supercuspidal representations and about the structure of the Bernstein blocks. In particular, I will discuss a joint project in progress with Jeffrey Adler, Manish Mishra and Kazuma Ohara in which we show that general Bernstein blocks are equivalent to much better understood depth-zero Bernstein blocks. This is achieved via an isomorphism of Hecke algebras and allows to reduce a lot of problems about the (category of) representations of p-adic groups to problems about representations of finite groups of Lie type, where answers are often already known or easier to achieve.