The symmetric group $\mathfrak{S}_n$ acts naturally on the polynomial ring of rank $n$ by variable permutation. The classical coinvariant ring $R_n$ is the quotient of this action by the ideal generated by invariant polynomials with vanishing constant term. The ring $R_n$ has deep ties to the combinatorics of permutations and the geometry of the flag variety. The superspace coinvariant ring $SR_n$ is obtained by an analogous construction where one considers the action of $\mathfrak{S}_n$ on the algebra $\Omega_n$ of polynomial-valued differential forms on $n$-space. We describe the Macaulay-inverse system associated to $SR_n$, give a formula for its bigraded Hilbert series, and give an explicit basis of $SR_n$. The basis of $SR_n$ will be derived using Solomon-Terao algebras associated to free hyperplane arrangements. Joint with Robert Angarone, Patty Commins, Trevor Karn, Satoshi Murai, and Andy Wilson.

The Hilbert transform is a cornerstone of  the classical analysis. A key approach to establishing its Lp-boundedness is through Cotlar's identity, a powerful equation that not only yields optimal constants for the Lp bounds of the Hilbert transform but also generalizes to broader settings where the notion of "analytic functions" is meaningful. In this talk, I will revisit Cotlar’s identity and explore how modified versions  extend to branches of free groups and hyperbolic groups.

Given $r$-uniform hypergraphs $G$ and $F$ and an integer $n$, let $f_{F,G}(n)$ be the maximum $m$ such that every $n$-vertex $G$-free $r$-graph has an $F$-free induced subgraph on $m$ vertices. We show that $f_{F,G}(n)$ is polynomial in $n$ when $G$ is a subgraph of an iterated blowup of $F$. As a partial converse, we show that if $G$ is not a subgraph of an $F$-iterated blowup and is $2$-tightly connected, then $f_{F,G}(n)$ is at most polylogarithmic in $n$. Our bounds generalize previous results of Dudek and Mubayi for the case when $F$ and $G$ are complete. Joint work with Xiaoyu He.

Given $r$-uniform hypergraphs $G$ and $F$ and an integer $n$, let $f_{F,G}(n)$ be the maximum $m$ such that every $n$-vertex $G$-free $r$-graph has an $F$-free induced subgraph on $m$ vertices. We show that $f_{F,G}(n)$ is polynomial in $n$ when $G$ is a subgraph of an iterated blowup of $F$. As a partial converse, we show that if $G$ is not a subgraph of an $F$-iterated blowup and is $2$-tightly connected, then $f_{F,G}(n)$ is at most polylogarithmic in $n$. Our bounds generalize previous results of Dudek and Mubayi for the case when $F$ and $G$ are complete. Joint work with Xiaoyu He.

The explosion of real-time data in the physical world requires new generations of tools to model complex dynamical systems. Deep learning, the foundation of modern AI, offers highly scalable models for spatiotemporal data. On the other hand, deep learning is opaque and complex. Dynamical system theory plays a key role in describing the emerging behavior of deep neural networks. It provides new paths towards understanding the hidden structures in these complex systems. In this talk, I will give an overview of our research to explore the interplay between the two. I will showcase the applications of these approaches to different science and engineering tasks. 

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The moduli space of genus g Riemann surfaces equipped with the Teichmuller metric exhibits rich geometric, analytic, and dynamical properties. A major challenge is to understand the totally geodesic submanifolds -- these share many properties with the moduli space itself. For many decades, research focused on the one (complex) dimensional case, i.e. the fascinating Teichmuller cuves. The discovery of interesting higher-dimensional examples in recent years has led to new questions. In this talk, I will discuss joint work with Benirschke and Rached in which we study the boundary of a totally geodesic subvariety in the Deligne-Mumford compactification, showing that the boundary is itself totally geodesic.

Consider the random bipartite Erdos-Renyi graph $G(n, m, p)$, where each edge with one vertex in $V_{1}=[n]$ and the other vertex in $V_{2}=[m]$ is connected with probability $p$ with $n \geq m$. For the centered and normalized adjacency matrix $H$, it is well known that the empirical spectral measure will converge to the Marchenko-Pastur (MP) distribution. However, this does not necessarily imply that the largest (resp. smallest) singular values will converge to the right (resp. left) edge when $p = o(1)$, due to the sparsity assumption. In Dumitriu and Zhu 2024, it was proved that almost surely there are no outliers outside the compact support of the MP law when $np = \omega(\log(n))$. In this paper, we consider the critical sparsity regime with $np =O(\log(n))$, where we denote $p = b\log(n)/\sqrt{mn}$, $\gamma = n/m$ for some positive constants $b$ and $\gamma$. For the first time in the literature, we quantitatively characterize the emergence of outlier singular values. When $b > b_{\star}$, there is no outlier outside the bulk; when $b^{\star}< b < b_{\star}$, outlier singular values only appear outside the right edge of the MP law; when $b < b^{\star}$, outliers appear on both sides. Meanwhile, the locations of those outliers are precisely characterized by some function depending on the largest and smallest degrees of the sampled random graph. The thresholds $b^{\star}$ and $b_{\star}$ purely depend on $\gamma$. Our results can be extended to sparse random rectangular matrices with bounded entries.

Stationary varifolds generalize minimal surfaces and can exhibit singularities. The most general regularity theorem in this context is the celebrated Allard's Regularity Theorem, which asserts that the set of singular points has empty interior. However, it is believed that the set of singular points should have codimension (at least) one. Despite more than 50 years having passed since Allard's breakthrough, stronger results have remained elusive. In this talk, after a brief discussion about the regularity theory for stationary varifolds, I will discuss the principle of unique continuation and the topic of rectifiability, both of which are linked to understanding the structure of singularities. This discussion is based on joint works with Stefano Decio, Camillo De Lellis, and Federico Franceschini.

The concept of ultraproducts in the context of tracial von Neumann algebras was effectively introduced by Wright in 1954. Since then, it has been used as a central technique in several important works on the classification and structure theory of von Neumann algebras, including works of McDuff and Connes. Developments beginning in the 2010s also connected the concept to ultraproducts in model theory. In this talk, I will be presenting a general overview of the technique and relevant results, both from a von Neumann algebra and from a continuous model theory perspective. I will also present several of my works, with various collaborators, that apply the technique and related techniques in C*-algebras and group theory.