The Brill-Noether theory of curves plays a fundamental role in the theory of curves and their moduli and has been intensively studied since the 19th century. In contrast, Brill-Noether theory for vector bundles and higher dimensional varieties is less understood. It is hard to determine when Brill-Noether loci are nonempty and these loci can be reducible and of larger than the expected dimension. In this talk, we will study Brill-Noether loci for vector bundles on the projective plane in the case where the number of sections is close to the largest possible number. When the number of sections is very large, Brill-Noether problems are all "trivial"--the Brill-Noether loci are either empty or the entire moduli space. As the number of sections decreases, we find that there is a "first" nontrivial Brill-Noether locus, and we discuss its geometry.

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. We will introduce the class of NSOP_1 theories, which contains the class of simple theories while allowing for additional complexity, and to which many of the independence phenomena originating from stable and simple theories extend in new ways. Specifically, Chernikov, Kaplan and Ramsey show that Kim-independence, a version of forking-independence "at a generic scale," behaves similarly in NSOP_1 theories to how forking-independence behaves in simple theories. We will define Kim-independence and give the main ideas of the proofs of some of Kaplan and Ramsey's results on NSOP_1 theories, including Kim's lemma for Kim-dividing, the equivalence of Kim-forking and Kim-dividing, and potentially the symmetry of Kim-independence.

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.

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.

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.

TBA

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.

TBA

TBA