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. 

Please register at https://forms.gle/yDcUa9LAmpY1F2178.

Please register here:

https://forms.gle/yDcUa9LAmpY1F2178.

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.

We propose a novel framework for solving nonlinear PDEs using sparse radial basis function (RBF) networks. Sparsity-promoting regularization is employed to prevent over-parameterization and reduce redundant features. This work is motivated by longstanding challenges in traditional RBF collocation methods, along with the limitations of physics-informed neural networks (PINNs) and Gaussian process (GP) approaches, aiming to blend their respective strengths in a unified framework. The theoretical foundation of our approach lies in the function space of Reproducing Kernel Banach Spaces (RKBS) induced by one-hidden-layer neural networks of possibly infinite width. We prove a representer theorem showing that the solution to the sparse optimization problem in the RKBS admits a finite solution and establishes error bounds that offer a foundation for generalizing classical numerical analysis. The algorithmic framework is based on a three-phase algorithm to maintain computational efficiency through adaptive feature selection, second-order optimization, and pruning of inactive neurons. Numerical experiments demonstrate the effectiveness of our method and highlight cases where it offers notable advantages over GP approaches. This work opens new directions for adaptive PDE solvers grounded in rigorous analysis with efficient, learning-inspired implementation.

We study the algorithmic problem of finding a large independent set in an Erdős–Rényi random graph $G(n,p)$. For constant $p$ and $b=1/(1-p)$, the largest independent set has size $2\log_b n$, while a simple greedy algorithm -- where vertices are revealed sequentially and the decision at any step depends only on previously seen vertices -- finds an independent set of size $\log_b n$. In his seminal 1976 paper, Karp challenged to either improve this guarantee or establish its hardness. Decades later, this problem remains one of the most prominent algorithmic problems in the theory of random graphs.

In this talk we provide some evidence for the algorithmic hardness of Karp's problem. More concretely, we establish that a broad class of online algorithms, which we shall define, fails to find an independent set of size $(1+\epsilon)\log_b n$ for any constant $\epsilon>0$, with high probability. This class includes Karp’s algorithm as a special case, and extends it by allowing the algorithm to also query additional `exceptional' edges not yet `seen' by the algorithm. For constant~$p$ we also prove that our result is asymptotically tight with respect to the number of exceptional edges queried, by  designing an online algorithm which beats the half-optimality threshold when the number of exceptional edges is slightly larger than our bound.

Our proof relies on a refined analysis of the geometric structure of tuples of large independent sets, establishing a variant of the Overlap Gap Property (OGP) commonly used as a barrier for classes of algorithms. While OGP has predominantly served as a barrier to stable algorithms, online algorithms are not stable, i.e., our application of OGP-based techniques to the online setting is novel.

Based on joint work with D. Gamarnik and E. Kızıldağ; see arXiv:2504.11450.

I will begin by reviewing results about the evolution of the roots of real-rooted polynomials under repeated differentiation. In this case, the limiting evolution of the (real) roots can be described in terms of the concept of fractional free convolution, which in turn is equivalent to the operation of taking corners of Hermitian random matrices. 

Then I will present new results about the evolution of the complex roots of random polynomials under repeated differentiation—and more generally under repeated applications of differential operators. In this case, the limiting evolution of the roots has an explicit form that is closely connected to free probability and random matrix theory. 

The talk will be self-contained and will have lots of pictures and animations.

In this talk, we will address areas of recent work centered around the themes of fairness and foundations in machine learning as well as highlight the challenges in this area. We will discuss recent results involving linear algebraic tools for learning, such as methods in non-negative matrix factorization that include tailored approaches for fairness. We will showcase our approach as well as practical applications of those methods.  Then, we will discuss new foundational results that theoretically justify phenomena like benign overfitting in neural networks.  Throughout the talk, we will include example applications from collaborations with community partners, using machine learning to help organizations with fairness and justice goals. This talk includes work joint with Erin George, Kedar Karhadkar, Lara Kassab, and Guido Montufar.

^{Prof. Deanna Needell earned her PhD from UC Davis before working as a postdoctoral fellow at Stanford University. She is currently a full professor of mathematics at UCLA, the Dunn Family Endowed Chair in Data Theory, and the Executive Director for UCLA's Institute for Digital Research and Education. She has earned many awards including the Alfred P. Sloan fellowship, an NSF CAREER and other awards, the IMA prize in Applied Mathematics, is a 2022 American Mathematical Society (AMS) Fellow and a 2024 Society for industrial and applied mathematics (SIAM) Fellow. She has been a research professor fellow at several top research institutes including the SLMath (formerly MSRI) and Simons Institute in Berkeley. She also serves as associate editor for several journals including Linear Algebra and its Applications and the SIAM Journal on Imaging Sciences, as well as on the organizing committee for SIAM sessions and the Association for Women in Mathematics.}

Diagonals of multivariate rational functions are an important class of functions arising in number theory, algebraic geometry, combinatorics, and physics. For instance, many hypergeometric functions are diagonals as well as the generating function for Apery's sequence. A natural question is to determine the diagonal grade of a function, i.e., the minimum number of variables one needs to express a given function as a diagonal. The diagonal grade gives the ring of diagonals a filtration. In this talk we study the notion of diagonal grade and the related notion of Hadamard grade (writing functions as the Hadamard product of algebraic functions), resolving questions of Allouche-Mendes France, Melczer, and proving half of a conjecture recently posed by a group of physicists. This work is joint with Andrew Harder.

[pre-talk at 3:00PM]