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Thu Oct 04

Special Events and Seminars

1:25pm - Vincent Hall 313
The Langlands-Kottwitz-Scholze method for Shimura varieties of abelian type
Alex Youcis, University of California, Berkeley

The local (and global) Langlands conjectures attempt to bridge the major areas of harmonic analysis and number theory by forming a correspondence between representations which naturally appear in both areas. A key insight due to Langlands and Kottwitz is that one could attempt to understand such a conjectural correspondence by comparing the traces of natural operators on both sides of the bridge. Moreover, it was realized that Shimura varieties present a natural means of doing this. For global applications, questions of reduction type (at a particular prime p) for these Shimura varieties can often be avoided, and for this reason the methods of Langlands and Kottwitz focused largely on the setting of good reduction. But, for local applications dealing with the case of bad reduction is key. The setting of bad reduction was first dealt with, for some simple Shimura varieties, by Harris and Taylor which they used, together with the work of many other mathematicians, to prove the local Langlands conjecture for GL_n. A decade later Scholze gave an alternative, more geometric, way to understand the case of bad reduction for certain Shimura varieties and was able to reprove the local Langlands conjecture for GL_n. In this talk we will discuss an extension of the ideas of Scholze to a wider class of Shimura varieties, as well as the intended application of these ideas to the local Langlands conjectures for more general groups.

Thu Nov 08

Special Events and Seminars

1:25pm - Vincent Hall 313
Irreducible components of affine Deligne--Lusztig varieties and orbital integrals
Rong Zhou, Institute for Advanced Study

Affine Deligne--Lusztig varieties (ADLV) naturally arise in the study of Shimura varieties and Rapoport--Zink spaces. Their irreducible components provide an interesting class of cycles on the special fiber of Shimura varieties. We prove a conjecture of Miaofen Chen and Xinwen Zhu, which relates the number of irreducible components of ADLV's to a certain weight multiplicity for a representation of the Langlands dual group. Our approach is to count the number of F_q points as q goes to infinity; this boils down to computing a certain twisted orbital integral. After applying techniques from local harmonic analysis, we reduce to computing a particular coefficient of the matrix for the inverse Satake transform. Using an interpretation of this coefficient in terms of a q-analogue of Kostant's partition function, we are able to reduce the problem to the previously known special cases of the conjecture proved by Hamacher--Viehmann and Nie. This is joint work with Yihang Zhu.

Mon Nov 26

Special Events and Seminars

3:35pm - Vincent Hall 207
Inverse transport theory and related applications
Ru-Yu Lai, University of Minnesota

The inverse transport problem consists of reconstructing the optical properties of a medium from boundary measurements. It finds applications in a variety of fields. In particular, radiative transfer equation (a linear transport equation) models the photon propagation in a medium in optical tomography. In this talk I will address results on the determination of these optical parameters. Moreover, the connection between the inverse transport problem and the Calderon problem will also be presented.

Thu Dec 06

Special Events and Seminars

11:00am - Tate Hall B20
p-adic local systems in p-adic geometry
Koji Shimizu, University of California, Berkeley

An etale p-adic local system on a rigid analytic variety can be regarded as a family of p-adic Galois representations parametrized by the variety, and p-adic Hodge theory has brought many results and applications on such objects, including a p-adic Riemann-Hilbert correspondence by Diao, Lan, Liu and Zhu. I will discuss constancy of a key invariant (generalized Hodge-Tate weights) of general p-adic local systems

Mon Dec 10

Special Events and Seminars

3:35pm - Vincent Hall 207
Optimal transport on graphs with Applications
Wuchen Li

In recent years, optimal transport has many applications in evolutionary dynamics, statistics, and machine learning. In this talk, we introduce dynamical optimal transport on finite graphs. We proposed to study the probability simplex as a Riemannian manifold with a Wasserstein metric. We call it a probability manifold. Various developments, especially the Fokker-Planck equation, will be introduced. The entropy production on graphs related to Shannon entropy will be established. Its connection with Fisher information and Yano’s formula will be studied. Many examples, including Mean field games, geometry of graphs, statistical learning problems, will be presented.

Tue Mar 12

Special Events and Seminars

11:00am - Vincent 213
Special Algebraic Geometry Seminar
Weizhe Zheng, Chinese Academy of Sciences and Princeton University
Thu Mar 14

Special Events and Seminars

1:25pm - Vincent Hall 113
Special Algebraic Geometry Seminar
Weizhe Zheng, Chinese Academy of Sciences and Princeton University
Tue Apr 09

Special Events and Seminars

11:00am - Vincent Hall 213
Special Number Theory Seminar
Yifeng Liu, Yale University
Thu Apr 11

Special Events and Seminars

1:25pm - Vincent Hall 113
Special Number Theory Seminar
Yifeng Liu, Yale University