Columbia University (New York)

I will talk about several results on Hecke algebras attached to Bernstein blocks of (arbitrary) reductive p-adic groups, where we construct a local Langlands correspondence for these Bernstein blocks. Our techniques draw inspirations from the foundational works of Deligne, Kazhdan and Lusztig. 

As an application, we prove the Local Langlands Conjecture for G_2, which is the first known case in literature of LLC for exceptional groups. Our correspondence satisfies an expected property on cuspidal support, which is compatible with the generalized Springer correspondence, along with a list of characterizing properties including the stabilization of character sums, formal degree property etc. In particular, we obtain (not necessarily unipotent) "mixed" L-packets containing "F-singular" supercuspidals and non-supercuspidals. Such "mixed" L-packets had been elusive up until this point and very little was known prior to our work. I will give explicit examples of such mixed L-packets in terms of Deligne-Lusztig theory and Kazhdan-Lusztig parametrization. 

If time permits, I will explain how to pin down certain choices in the construction of the correspondence using stability of L-packets; one key input is a homogeneity result due to Waldspurger and DeBacker. Moreover, I will mention how to adapt our general strategy to construct explicit LLC for other reductive groups, such as GSp(4), Sp(4), etc. Such explicit description of the L-packets has been useful in number-theoretic applications, e.g. modularity lifting questions as in the recent work of Whitmore. 

Some parts of this talk are based on my joint work with Aubert, and some other parts are based on my joint work with Suzuki. 
 

Title: Measuring association with Wasserstein distances

Abstract: Let π ∈ Π(μ, ν) be a coupling between two probability measures μ and ν on a Polish space. In this talk we propose and study a class of nonparametric measures of association between μ and ν, which we call Wasserstein correlation coefficients. These coefficients are based on the Wasserstein distance between ν and the disintegration of π with respect to the first coordinate. We also establish basic statistical properties of this new class of measures: we develop a statistical theory for strongly consistent estimators and determine their convergence rate in the case of compactly supported measures μ and ν. Throughout our analysis we make use of the so-called adapted/bicausal Wasserstein distance, in particular we rely on results established in [Backhoff, Bartl, Beiglböck, Wiesel. Estimating processes in adapted Wasserstein distance. 2020]. Our approach applies to probability laws on general Polish spaces.

 Consider dY(t)=f(X(t))dX(t), where X(t) is a pure jump Levy process with finite p-variation norm, 1<= p < 2, and f is a Lipchitz continuous function. Following the geometric solution construction of Levy-driven stochastic differential equations in (Williams 2001), we develop a class of epsilon-strong simulation algorithms that allows us to construct a probability space, supporting both the geometric solution Y and a fully simulatable process Y_epsilon, such that Y_epsilon is within epsilon distance from Y under the uniform metric on compact time intervals with probability 1. Moreover, the users can adaptively choose epsilon’ < epsilon, so that Y_epsilon’ can be constructed conditional on Y_epsilon. This tolerance-enforcement feature allows us to easily combine our algorithm with Multilevel Monte Carlo for efficient estimation of expectations, and adding as a benefit a straightforward analysis of rates of convergence. This is joint with Jose Blanchet, Fei He and Offer Kella.

This is a report on joint work (still in progress) with Ellen Eischen, Jian-Shu Li,
and Chris Skinner.  I will describe the general structure of our construction of p-adic L-functions
attached to families of ordinary holomorphic modular forms on Shimura varieties attached to
unitary groups.  The complex L-function is studied by means of the doubling method;
its p-adic interpolation applies adelic representation theory to Ellen Eischen's Eisenstein 
measure.

A firm issues a convertible bond. At each subsequent time, the bondholder

must decide whether to continue to hold the bond, thereby collecting coupons, or

to convert it to stock. The bondholder wishes to choose a conversion strategy to

maximize the bond value. Subject to some restrictions, the bond can be called by

the issuing firm, which presumably acts to maximize the equity value of the firm

by minimizing the bond value. This creates a two-person game. We show that if

the coupon rate is below the interest rate times the call price, then conversion

should precede call. On the other hand, if the dividend rate times the call

price is below the coupon rate, call should precede conversion. In either case,

the game reduces to a problem of optimal stopping. This is joint work with Mihai

Sirbu.