Statistical bias is an inevitable factor in most measurements.  In many cases, bias transforms can be employed to counter the effect and produce (asymptotically) unbiased estimators.  The most common such transform is the size bias.  An infinitesimal version called zero bias was introduced by Goldstein and Reinert in 1997, and has become a powerful tool in Stein's method (for Gaussian and Poisson approximation).

In this talk, I will discuss recent work (arxiv.org/2403.19860) on free probability analogs of bias transforms.  I will discuss existence and regularity of free zero bias, and somewhat surprising connections to the theory of (freely) infinitely divisible laws, giving a new proof of the free Levy--Khintchine formula in the process.  I will also discuss connections between size bias and a new class: positively freely infinitely-divisible laws, and a new kind of free Levy--Khintchine formula.

Finally, time permitting, I will discuss our ongoing work developing Stein's method in free probability, using free zero bias to prove sharp quantitative free central limit theorems even for some systems with long range correlations.

This is joint work with Larry Goldstein.

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 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.