Linear algebra is the workhorse of modern data science and machine learning, but none of the fun applications are ever mentioned in Math 18. We remedy this by discussing Principal Component Analysis,,the best* of these applications, and show how it follows quickly from the Singular Value Decomposition, the best* theorem in linear algebra. We present a few mathematical perspectives, explain the equivalent formulations of PCA, and ultimately use PCA to build an elementary image classifier without any fancy tools from machine learning.

^{*The speaker does not necessarily believe any of these claims but will nonetheless defend them vehemently if heckled.}

The study of p-adic Lie groups and their representations is a central piece of the p-adic Langlands program.  One tool which is used to study these is the notion of a saturated pro-p group, and the famous result of Lazard which states that every p-adic Lie group contains an open saturable subgroup.  In this talk, we will demonstrate a family of open saturated subgroups of G(F) for G a reductive group over a p-adic field F, which is indexed by the semisimple Bruhat-Tits building of G, given a mild assumption on G.  We will then review some group-theoretic consequences of this result.

The exchange property, introduced by Crawley and Jonsson in 1964 in the study of direct decompositions of algebraic systems and later extended to modules and rings by Warfield, plays a central role in modern decomposition theory. One of the main open problems in the area is whether the finite exchange property implies the full exchange property. This talk surveys the development of exchange theory from its module-theoretic origins to its ring-theoretic formulation via exchange rings. The last part of the talk is based on joint work with A. Cigdem Ozcan and focuses on lifting theory, including idempotent, regular, and unit lifting ideals and morphisms, and their interaction with local morphisms.

We discuss construction of a derived Lagrangian intersection theory of moduli spaces of perfect complexes, with support on divisors on compact Calabi-Yau threefolds. Our goal is to compute deformation invariants associated to a fixed linear system of divisors in CY3. We apply a Tyurin degeneration of the CY3 into a normal-crossing singular variety composed of Fano threefolds meeting along their anti-canonical divisor. We show that the moduli space over the Fano 4 fold given by total space of the degeneration family satisfies a relative Lagrangian foliation structure which leads to realizing the moduli space as derived critical locus of a global (-1)-shifted potential function. We construct a flat Gauss-Manin connection to relate the periodic cyclic homology induced by matrix factorization category of such function to the derived Lagrangian intersection of the corresponding “Fano moduli spaces”. The latter provides one with categorification of DT invariants over the special fiber (of degenerating family). The alternating sum of dimensions of the categorical DT invariants of the special fiber induces numerical DT invariants. If there is time, we show how in terms of “non-derived” virtual intersection theory, these numerical DT invariants relate to counts of D4-D2-D0 branes which are expected to have modularity property by the S-duality conjecture. This talk is based on joint work with Jacob Krykzca.

In 2024, Hikita showed that the chromatic symmetric functions of incomparability graphs of (3+1)-free posets expand with positive coefficients in the basis of elementary symmetric functions. This result resolved the long-standing Stanley–Stembridge conjecture. Finding a combinatorial interpretation of the $e$-coefficients remains a major open problem. In this talk I will define powerful and strong $P$-tableaux and conjecture that they give upper and lower bounds for the $e$-coefficients of chromatic symmetric functions. As evidence for these conjectures, we obtain combinatorial interpretations for various e-coefficients which live in between strong and powerful $P$-tableaux. Additionally, we show how Hikita’s theorem relates to strong $P$-tableaux and the Shareshian–Wachs inversion statistic.

The mathematical core of deep learning is function approximation by neural networks trained on data using stochastic gradient descent. I will explain an emerging geometric framework for the analysis of this process. This includes a collection of rigorous results on training dynamics for the deep linear network (DLN) as well as general principles for arbitrary neural networks. The mathematics ranges over a surprisingly broad range, including geometric invariant theory, random matrix theory, and minimal surfaces. However, little background in these areas will be assumed and the talk will be accessible to a broad audience. The talk is based on joint work with several co-authors: Yotam Alexander, Nadav Cohen (Tel Aviv), Kathryn Lindsey (Boston College), Alan Chen, Zsolt Veraszto and Tianmin Yu (Brown).

In this talk, I will discuss symmetrized geometric last passage percolation. This is a prototypical model in the half-space KPZ universality class that has a natural interpretation as a line ensemble (a collection of random continuous functions). The model depends on parameters q,c, which control the background noise strength away from and at the boundary of the model, respectively. Depending on whether c < 1(subcritical), c = 1(critical), or c> 1(supercritical), the model exhibits different asymptotic behavior, and I will discuss some recent progress in understanding this from a line ensemble perspective.

^{* Optimization for Machine Learning Seminar}