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.

This defense explores the computation of the minimal exponent of a hypersurface singularity using birational geometry. Although the minimal exponent is originally defined through the Bernstein–Sato polynomial, we show that in several important cases it can be detected directly on a log resolution.

For isolated quasi-homogeneous singularities, we demonstrate that a single weighted blow-up produces an exceptional divisor that computes the minimal exponent. Building on this, we utilize the Mustață–Chen birational formula and Chen’s inversion of adjunction to formulate a squeeze criterion. Finally, we apply this criterion to ADE singularities and extend the results to certain Newton-degenerate cases, such as Cayley cubic singularities.

Geometric mechanics describes Lagrangian and Hamiltonian mechanics geometrically, and information geometry formulates statistical estimation, inference, and machine learning in terms of geometry. A divergence function is an asymmetric distance between two probability densities that induces differential geometric structures and yields efficient machine learning algorithms that minimize the duality gap. The connection between information geometry and geometric mechanics will yield a unified treatment of machine learning and structure-preserving discretizations. In particular, the divergence function of information geometry can be viewed as a discrete Lagrangian, which is a generating function of a symplectic map, that arise in discrete variational mechanics. This identification allows the methods of backward error analysis to be applied, and the symplectic map generated by a divergence function can be associated with the exact time-h flow map of a Hamiltonian system on the space of probability distributions. We will also discuss how time-adaptive Hamiltonian variational integrators can be used to discretize the Bregman Hamiltonian, whose flow generalizes the differential equation that describes the dynamics of the Nesterov accelerated gradient descent method.

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.