Absolute irreducibility is a fundamental property of algebraic varieties: a variety is absolutely irreducible if it remains irreducible after extending the base field to its algebraic closure. Determining absolute irreducibility is an important problem in algebraic geometry, particularly over finite fields, where it has applications to the study of rational points, coding theory, cryptography, and finite geometry. In particular, absolute irreducibility plays an important role in establishing bounds for the number of rational points on varieties and in the study of exceptional functions, such as almost perfect nonlinear (APN) and permutation functions. In this talk, I will discuss criteria for establishing the absolute irreducibility of hypersurfaces defined by multivariate polynomials over finite fields. I will present a criterion that, under a square-free hypothesis on the leading homogeneous component, reduces the problem to computations involving multivariate greatest common divisors and avoids the need to test irreducibility over the ground field or its extensions. Since square-free polynomials constitute a large and natural class, this provides an effective method for proving absolute irreducibility in many situations. I will also discuss applications of these techniques to polynomial families arising from problems over finite fields.

We consider a system of elliptic equations, depending on a small parameter, that models long-range segregation of populations. The diffusion is governed by the Laplacian. This system was previously investigated by Caffarelli, Patrizi, and Quitalo as a model in population dynamics, and they established the regularity of the free boundary in dimension two. In the present work we study the free boundary in the higher dimensional case. We extend the concept of angles and asymptotic cones to higher dimensions, and give a characterization of regular and singular points in terms of their densities and angles. We obtain a structure result of the free boundary and show that, if the angles at the singular points are away from $\frac{n\omega_n}{2}$, the regular set is open in the free boundary and locally a $C^1$ manifold of dimension $n-1$. We also show that, if the supports of the populations are convex, they are convex polytopes. A weak form of the equality of angles for the convex configuration is also derived. The talk is based on a joint work with Professor Monica Torres.

Stochastic mean-field control coordinates large populations of interacting agents under noise, from swarm planning and systemic risk to Schrödinger bridges. Classical solvers discretize the Fokker-Planck equation on a mesh, so their cost grows exponentially with the state dimension and a tensor-product grid is already out of reach past two dimensions. Deep neural network solvers lift that ceiling but still handle one instance per run, re-optimizing whenever the initial law, the target, or a cost weight changes. This talk presents a mesh-free solution operator for a whole family of such problems, in which a task enters as a prompt and the optimal transport map comes out in one forward pass. We pass to the probability-flow ODE and parameterize its flow map by a prompt-conditioned coupling flow whose exact inverse and analytic log-determinant give the score at linear cost in the dimension, against the cubic cost of a generic map.

Cookie random walks are a type of self-interacting random walk. In the most basic model, an (M,p)-cookie random walk, M cookies of strength p are placed at each site on Z. When a random walker reaches a site with a cookie, he steps right with probability p>1/2 and eats the cookie there. At sites with no cookies, the walker steps left and right with equal probability. Pinsky proposed a variation in which only one cookie is placed at each site, but the random walker only eats it after stepping left from that site. He was able to compute the speed of this walk for small values of p and sufficiently large values of p using martingale arguments. To close the gap he left open, we study a certain branching process. We calculate the mean of this process's stationary distribution, obtaining formulae for the speed of both Pinsky's random walk and other related walks as a consequence. For a further application of our methods, we obtain an explicit (but not particularly attractive) formula for the speed of (3,p)-cookie random walk. We discuss how beautifying this formula is equivalent to establishing an asymptotic expansion of several orders for certain generalized Hypergeometric-like functions.

We examine the exponential sum $S_k(\alpha;K)$ over the k-free numbers in the short interval $(N-K,N]$ as $N$ goes to infinity. It has been known since work of Bruedern et al. in 1998 that $S_k(\alpha;N)$ displays better-than-square-root cancellation in the $L^1$-mean, revealing underlying additive structure. Keil (2013) gave an essentially complete characterization of this subconvex behavior, again for $K = N$. We prove an analogous result whenever $K \gg N^{1/2-\theta_k}$, where $\theta_k>0$ is some small explicit constant, improving a result of Sun (2023) and notably going beyond the natural barrier of $N^{1/2}$. Furthermore, we prove similar results for all $L^s$-means with $s>0$. Time permitting, we will discuss the extension of this result into the function field setting.

Can waves reveal an object that we cannot see? In inverse scattering, waves interact with an unknown inhomogeneous medium—an object whose material properties differ from its surroundings—and the scattered waves are used to learn about it. We will explore several spectral phenomena arising from this process, including transmission eigenvalues, non-scattering frequencies, and scattering poles.  Transmission eigenvalues are special frequencies at which the scattering can become unusually weak and can provide information about the material properties of the object, while scattering poles describe resonant behavior at complex frequencies. We will discuss what these quantities reveal about the hidden medium and how they can be extracted from scattering data. 

The mathematics community is facing an unsustainable deluge of new articles circulated online and submitted to journals. Meanwhile, AI companies are strongly incentivized to bulldoze their way through major open problems across mathematics without waiting for community acceptance. At the intersection of these pressures stands Lean, an interactive theorem prover in which mathematical statements and proofs can be expressed in a precise language and checked by a computer. Through live demonstrations, we will explore the conceptual underpinnings of Lean and its role in our evolving conversations about an artificially intelligent future. No prior experience with Lean or programming is assumed.

he classification of C*-algebras has deployed K-theory and tracial data in order to form a total invariant for the class of regular C*-algebras. We focus on the pathological inductive limit algebras, introduce a finer invariant, and provide the first partial computation of notable subset. The results are motivated by the classical rank problem in C*-algebras, and its interplay with dimension functions.

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