The moduli of semisimple generic L-parameters, introduced by Hansen, is a dense open substack of the moduli stack of L-parameters, and it is expected to be the largest open substack for which Fargues-Scholze’s categorical local Langlands correspondence can be understood reasonably explicitly.  In this talk we show that much of the correspondence over this locus can indeed be made explicit, assuming certain properties of the correspondence such as geometric Eisenstein compatibility (currently known for GL(2)).

[pre-talk at 3:00PM]

 In this talk, I will discuss gradient descent for rank-1 matrix factorization at large step sizes. The main idea is to construct a parameterized quadratic certificate $I(\delta;\cdot)$ whose level sets shrink along the discrete-time dynamics, thereby producing a monotone state variable $\delta_t$. This state-dependent Lyapunov perspective gives a geometric mechanism for convergence in the certified regime and explains why, in the post-critical regime, trajectories are driven toward a balanced terminal manifold. I will also describe how these certificates can be derived from structural monotonicity axioms: in the scalar case, the certificate is uniquely determined, and the same local Lagrange-multiplier analysis constrains rank-1 extensions through their signal and noise blocks. Finally, I will present numerical evidence suggesting that the same certificate mechanism may extend beyond the proved settings, including two-dimensional rank-1 approximation and quartic perturbations of scalar factorization.

We determine the $\tau^n$-torsion in the first 5 lines of the $E_2$ page of the $\mathbb{C}$-motivic Adams spectral sequence using the techniques of Burklund-Xu. In particular, every element in this range is either $\tau^1$-torsion or $\tau$-free. We also show that $\tau^n$-torsion elements can appear only in Adams filtration at least $2n+2$ and give further evidence of a possible $3n$ bound.

In Part I, we provide an explicit version of the Eichler--Selberg trace formula for Shimura curves with level structure over the rationals. As an application, we provide an algorithm to compute the isogeny decomposition of the Jacobian of Shimura curves into modular abelian varieties using the method that Rouse--Sutherland--Zureick-Brown developed for classical modular curves. We also give a trace formula for definite quaternionic modular forms over the rationals.

In Part II, we compile a complete list of isomorphism class representatives of curves of genus 6 over $\mathbb{F}_2$. We use explicit descriptions of canonical curves in each stratum of the Brill--Noether stratification of the moduli space $\mathcal{M}_6$, due to Mukai in the generic case. Our computed value of $\#\mathcal{M}_6(\mathbb{F}_2)$ agrees with the Lefschetz trace formula as recently computed by Bergstrom--Canning--Petersen--Schmitt.

We also report progress on compiling a corresponding list in genus 7 over $\mathbb{F}_2$ (for which explicit descriptions of canonical curves in each stratum of the Brill--Noether stratification of the moduli space $\mathcal{M}_7$ are also available) and genus 5 over $\mathbb{F}_3$, where the censuses are complete in all except for the generic strata in both cases.

In this talk, we discuss a method using Zariski localization to study how singularities of certain algebras such as Rees algebras or rational localizations behave under perturbation of defining ideals. If time permits, I will talk about a potential application to the almost purity theorem. 

Approximating data sets by graphs and simplicial complexes has been shown to be a very useful way to obtain qualitative information about data, and more recently has been shown to similarly contribute to artificial intelligence.  I will discuss the mathematics around this, with examples from various domains.  

BIO: Gunnar Carlsson is the Ann and Bill Swindells Professor Professor of Mathematics, Emeritus, at Stanford University, and a pioneer in the field of computational topology. His research focuses on the application of topological methods  to the analysis of high-dimensional, complex data, a discipline known as Topological Data Analysis (TDA). Professor Carlsson is perhaps best known for leading the "Topological Methods in Data Analysis" project (supported by DARPA), which catalyzed the development of persistent homology and mapper algorithms. Beyond his academic contributions, he co-founded Ayasdi, a company dedicated to utilizing TDA for industrial-scale machine learning and data science. He holds a Ph.D. from Stanford and has previously held faculty positions at the University of Chicago, the University of California, San Diego, and Princeton University.