We will discuss a proof that the integral Hodge conjecture is false for a very general abelian variety of dimension ≥ 4. Associated to any regular matroid is a degeneration of principally polarized abelian varieties. We introduce a new combinatorial invariant of regular matroids, which obstructs the algebraicity of the minimal curve class, on the very general fiber of the associated degeneration. In concert with a result of Voisin, one deduces (via the intermediate Jacobian) the stable irrationality of a very general cubic threefold. This is joint work with Olivier de Gaay Fortman, and Stefan Schreieder.

The Erdős–Rogers function f_{K_s​, K_t}(n) is defined as the minimum possible value of the s-independence number over all n-vertex K_t​ -free graphs. Introduced by Erdős and Rogers in 1962, it has since become an important topic in Ramsey theory, with numerous papers achieving significant improvements on the bounds for various pairs (s, t). In this talk, we will discuss two recent generalizations of the Erdős–Rogers function: the multicolor case and the case of arbitrary pairs of graphs. We will also present some open problems.

We will have a research seminar this semester on forking, broadly construed, particularly in the setting of unstable first-order theories. Graduate students are particularly encouraged to attend. In this session, we will introduce the stable forking conjecture, with the goal of recounting Brower's striking argument about forking with types of rank two.

The Bernoulli property is the strongest statistical property that a measure preserving system can exhibit. It not only implies other important statistical properties such as ergodicity, mixing and the K-property, but, as shown by Ornstein, Bernoulli systems with the same entropy are measurably conjugate. We prove that, for $C^{1+\alpha}$ diffeomorphisms of compact manifolds, exponentially mixing SRB measures are Bernoulli. This extends a recent result by Dolgopyat, Kanigowski and F. Rodriguez Hertz. Using similar techniques, we also show that if volume is almost exponentially mixing, then the limit SRB measure constructed by Ben Ovadia and F. Rodriguez Hertz is Bernoulli.

Testing composite null hypotheses is fundamental to many scientific applications, including mediation and replicability analyses, and becomes particularly challenging in high-throughput settings involving tens of thousands of features. Existing high-dimensional composite null hypotheses testing often ignores the dependence structure among features, leading to overly conservative or liberal results. To address this limitation, we develop a four-state hidden Markov model (HMM) for bivariate $p$-value sequences arising from two-study replicability analysis. This model captures local dependence among features and accommodates study-specific heterogeneity. Based on the HMM, we propose a multiple testing procedure that asymptotically controls the false discovery rate (FDR). Extending this framework to more than two studies is computationally intensive, with complexity growing exponentially in the number of studies $n.$ To address this scalability issue, we introduce a novel e-value framework that reduces computational complexity to quadratic in $n,$ while preserving asymptotic FDR control. Extensive simulations demonstrate that our method achieves higher power than existing approaches at comparable FDR levels. When applied to genome-wide association studies (GWAS), the proposed approach identifies novel biological findings that are missed by current methods.

In 2000, Langlands proposed a method to weight the Arthur-Selberg trace formula with automorphic L-functions, whose analytic behaviour is expected to detect when an automorphic form is a functorial transfer from a smaller group. An immediate obstruction that arises is the presence of nontempered representations, i.e., representations that do not satisfy the Ramanujan conjecture. It was later proposed in 2010 that an appropriate Poisson summation might be used to remove the contribution of such representations. So far, this has been carried out successfully in limited cases involving GL(2). In this talk I will introduce a general method for GL(2) and its connection to prehomogeneous vector spaces. Time permitting, I will also discuss potential consequences for the analytic behaviour of GL(2) L-functions.

Due to a vast amount of work over the last decades, the fundamental groups of 3-manifolds are by now very well understood. I will focus on the following (wide open) question: When is a discrete group the fundamental group of a compact 3-manifold? I'll discuss the background to this question, what is known in various dimensions, and then focus on the case of greatest interest in 3 dimensional topology - the hyperbolic case. Finally, I'll report on some recent work around this question in joint work with Haissinsky, Manning, Osajda, Sisto, and Walsh.

Given a class in the cohomology of a projective manifold, one can ask whether the class can be represented by an irreducible subvariety. If the class is represented by an irreducible subvariety, we say that the class is realizable. One can further ask whether the subvariety can be taken to satisfy additional properties such as smooth, nondegenerate, rational, etc. These questions are closely related to central problems in algebraic geometry such as the Hodge Conjecture or the Hartshorne Conjecture. Recently, June Huh and collaborators have made significant progress in understanding realizable classes in products of projective spaces. In this talk, I will give a survey of this circle of ideas and discuss recent joint work with Julius Ross on realizable classes in Grassmannians.

In this talk, I will present the construction of solutions to the one-dimensional Dirac equation on the half-line with Dirichlet boundary conditions. While the Dirac equation is a four-dimensional system arising in quantum field theory, I will focus on the one- dimensional initial-boundary value problem. The primary analytical tool is the unified transform method (or known as Fokas method), which provides an explicit representation of the solution. To introduce the method, I will first demonstrate it in the context of the heat equation on the half-line. I will then apply it to the Dirac equation to derive explicit solution formulas and analyze the associated boundary behavior at the origin. Furthermore, I will discuss the long-time dynamics of these solutions. If time permits, I will conclude with a discussion of Sobolev-space energy estimates for the solutions, including control of both spatial norms and time-regularity.

A/B testing is an effective method to assess the potential impact of two treatments. For A/B tests conducted by IT companies like Meta and LinkedIn, the test users can be connected and form a social network. Users’ responses may be influenced by their network connections, and the quality of the treatment estimator of an A/B test depends on how the two treatments are allocated across different users in the network. In this talk, I will discuss optimal design criteria based on some commonly used outcome models, under assumptions of network-correlated outcomes or network interference. I will show that the optimal design criteria under these network assumptions depend on several key statistics of the random design vector. I will discuss a framework to develop algorithms that generate rerandomization designs meeting the required conditions of those statistics. I further talk about asymptotic distributions to guide the specification of algorithmic parameters and validate the proposed approach using both synthetic and real-world networks.

Active Learning techniques (IBL, POGIL, ...) rely on teamwork to promote student learning. Far too often, assessments don't follow suit and instead ask students to perform computational tasks without assessing whether or not students can solve problems, apply their knowledge to new settings, or think creatively. This talk provides a detailed examination of how the University of Minnesota has revised its precalculus sequence to incorporate communication skills as a significant component of the grading scheme.

Five Internship Alums will discuss their internship experiences with other MSCS Grads. Please bring your questions!

TBA

TBA

We introduce the dynamical commensurator group for a generalized odometer action, that is for minimal equicontinuous group actions on Cantor sets. We show there is a map from the pointed mapping class group of a solenoidal manifold (ie a weak solenoid) to a dynamical commensurator group, and give conditions for when this map is either surjective or an isomorphism. Odden proved that this map is an isomorphism for the mapping class of the universal hyperbolic solenoid; Bering and Studenmund proved that the mapping class group of a universal solenoid over a compact K(G,1) manifold maps onto the commensurator group of G. We extend the results of both of these papers to arbitrary solenoidal manifolds. This work is joint with Olga Lukina.