AI is taking off and we could say we are living in “the AI Era”.  Progress in AI today is based on mathematics and statistics under the covers of machine learning models.  This talk will explain recent work on AI4Crypto, where we train AI models to attack Post Quantum Cryptography (PQC) schemes based on lattices. I will use this work as a case study in training ML models to solve hard math problems in practice.  Our AI4Crypto project has developed AI models capable of recovering secrets in post-quantum cryptosystems (PQC).  The standardized PQC systems were designed to be secure against a quantum computer, but are not necessarily safe against advanced AI!  

Understanding the concrete security of these standardized PQC schemes is important for the future of e-commerce and internet security.  So instead of saying that we are living in a “Post-Quantum” era, we should say that we are living in a “Post-AI” era!

The Lévy-Khintchine theorem is a classical result in Diophantine approximation that describes the growth rate of denominators of convergents in the continued fraction expansion of a typical real number. We make this theorem effective by establishing a quantitative rate of convergence. More recently, Cheung and Chevallier (Annales scientifiques de l'ENS, 2024) established a higher-dimensional analogue of the Lévy-Khintchine theorem in the setting of simultaneous Diophantine approximation, providing a limiting distribution for the denominators of best approximations. We also make their result effective by proving a convergence rate, and in addition, we establish a central limit theorem in this context. Our approach is entirely different and relies on techniques from homogeneous dynamics.

Starting  with a transient irreducible diffusion process $X^0$ on a locally compact separable metric space $(D, d)$ (for example, absorbing Brownian motion in a snowflake domain), one can construct a canonical symmetric reflected diffusion process $\bar X$ on a completion $D^*$ of $(D, d)$ through the theory of  reflected Dirichlet spaces. The boundary trace process $\check X$ of $X$ on the boundary $\partial D:=D^*\setminus D$ is the reflected diffusion process $\bar X$ time-changed by a smooth measure $\nu$ having full quasi-support on $\partial D$. The Dirichlet form of the trace process $\check X$ is called the trace Dirichlet form. In this talk, I will address the following two fundamental questions:

1) How to characterize the boundary trace Dirichlet space in a concrete way?

2) How does the boundary trace process behave? 

Based on a joint work with Shiping Cao.

The study of singularities of minimal submanifolds has a long history, with isolated singularities being the best understood case. The next simplest case is that of minimal submanifolds with families with singularities locally modeled on the product of an isolated conical singularity and a Euclidean space — such submanifolds are said to have cylindrical tangent cones at these singularities. Despite work in many contexts on minimal submanifolds with such singularities, the only known explicit examples at present are global products or involve extra structure (e.g. Kahler subvarieties). In this talk, I will describe a method for constructing infinite-dimensional families of non-product minimal submanifolds in arbitrary codimension whose singular set is itself an analytic submanifold. The construction uses techniques from the analysis of singular elliptic operators and Nash-Moser theory. This talk is based on joint work with Rafe Mazzeo.

The Heilbronn triangle problem is a classical problem in discrete geometry with several old and new close connections to various topics in extremal and additive combinatorics, graph theory, incidence geometry, harmonic analysis, and number theory. In this talk, we will survey a few of these stories, and discuss some recent developments. Based on joint works with Alex Cohen and Dmitrii Zakharov. 

Deep neural networks have been widely adopted for climate prediction tasks and have achieved high prediction accuracy across many problems. However, their decision-making processes remain opaque, and the complexity of these models poses significant challenges for interpretation. A recent theoretical breakthrough, "Recursive Feature Machine" (RFM), provides an alternative methodology for climate prediction that is interpretable and data efficient. Applying RFM to El Niño–Southern Oscillation (ENSO) prediction yields promising interpretability results and offers insights into the most influential geographical features that the model learns from training data. The method is clean, easy to implement, and can be generalized to a broad range of scientific fields.

How in the world did they get those crazy pictures of electron orbitals? Those chemists had to have talked to somebody about it! It turns out they talked to math people (probably physicists, but physicists talk to math people, and so on). These orbitals can actually be derived in not-too-bad a way using representation theory. We'll go over what electron orbitals are, how they show up in the periodic table, how representation theory gets involved, and how to derive the electron orbitals ourselves. We will even find orbitals that are bigger than the highest electron on Oganesson! We'll hopefully also understand what physicists and engineers mean when they say they have a "tensor." I've also been studying the periodic table using mnemonic devices lately, so you'll be sure to hear about that.

Elliptic surfaces are complex surfaces with two discrete invariants, $g$ and $d>0$. We will discuss the moduli and Hodge theory of these surfaces for small values of $(g,d)$. The case $(g,d)=(1,1)$ is particularly interesting, in view of a new conjectural Fourier-Mukai type correspondence. It also provides a test case of the Hodge Conjecture in dimension 4.

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 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.

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.

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]