The properties of a commutative scheme are strongly reflected in its category of quasi-coherent sheaves.  One approach to noncommutative geometry is to consider arbitrary categories with similar properties (e.g. Grothendieck categories) as geometric objects in their own right.  We discuss how one might to define an analog of closed subscheme in this context and give lots of examples of how the definition behaves in both reasonable and non-intuitive ways.

I will discuss the role of regularity properties in the structure and classification of C*-algebras, singling out the property of strict comparison of positive elements by traces. A well-understood source of strict comparison is through tensorial absorption of the Jiang-Su C*-algebra. This property is, however, absent from naturally occurring examples such as the reduced group C*-algebras of free groups. Thus, for some time this notion was hindered by a lack of concrete examples (particularly non-nuclear ones). This situation changed after the recent breakthrough work of Amrutam, Gao, Kunnawalkam Elayavalli, and Patchel. This work exploited the connection between Voiculescu's free independence and strict comparison, encapsulated in the concept of "selfless C*-algebra", to confirm that large classes of reduced group C*-algebras indeed obey strict comparison. I will discuss ongoing joint work with Hayes and Kunnawalkam Elayavalli, where we continue the program of verifying selflessness, and thus strict comparison, for new classes of C*-algebras, this time those arising as reduced free products.

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.

We will discuss the following result. For every geometrically finite Kleinian group $\Gamma < SL_2(\mathbb C)$ there is $\epsilon_\Gamma$ such that for every $g \in SL_2(\mathbb C)$ the intersection $g \Gamma g^{-1} \cap SL_2(\mathbb R)$ is either a lattice or has critical exponent $\delta(g \Gamma g^{-1} \cap SL_2(\mathbb R)) \leq 1 - \epsilon_\Gamma$. This result extends Margulis-Mohammadi and Bader-Fisher-Milier-Strover. We will discuss some ideas of the proof. We will focus on the applications of a new ergodic component: the preservation of entropy in a direction.

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.}

I will discuss join work with Flores, Kunnawalkam Elayavalli and Patchell where we introduce the property of subexponential decay, generalizing Haagerup-Jolissaint's property RD. I will provide examples of interest and also various applications. 

In this defense, I will motivate von Neumann algebras and give several examples of constructions inspired by group theory, highlighting the similarities and differences between the study of tracial von Neumann algebras and countable discrete groups. I will state recent results about how various combinations of these group-inspired constructions yield structural results, including: absence of tensor decomposition, sequential commutation, single generation, and the existence of exotic non-separable algebras. 

How many isomorphism classes of genus g curves are there over a finite field $\mathbb{F}_q$? In joint work with Samir Canning, Sam Payne, and Thomas Willwacher, we prove that the answer is a polynomial in q if and only if g is at most 8. One of the key ingredients is our recent progress on understanding low-degree odd cohomology of moduli spaces of stable curves with marked points.