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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.

Thu Feb 14

Special Events and Seminars

1:25pm - Vincent Hall 113
Fontaine-Mazur conjecture in the residually reducible case (I)
Lue Pan, University of Chicago

We prove the modularity of some two-dimensional residually reducible p-adic Galois representations over Q when p is at least 5. In the first talk, I will present a generalization of Emerton's local-global compatibility result. In the second talk, I will use this compatibility result to make a patching argument for completed homology in this setting.

Fri Feb 15

Special Events and Seminars

3:35pm - Vincent Hall 313
Fontaine-Mazur conjecture in the residually reducible case (II)
Lue Pan, University of Chicago

We prove the modularity of some two-dimensional residually reducible p-adic Galois representations over Q when p is at least 5. In the first talk, I will present a generalization of Emerton's local-global compatibility result. In the second talk, I will use this compatibility result to make a patching argument for completed homology in this setting.

Tue Mar 12

Special Events and Seminars

11:00am - Vincent 213
Compatible systems along the boundary
Weizhe Zheng, Chinese Academy of Sciences and Princeton University

A theorem of Deligne says that compatible systems of l-adic sheaves on a smooth curve over a finite field are compatible along the boundary. I will present an extension of Deligne's theorem to schemes of finite type over the ring of integers of a local field. This has applications to the equicharacteristic case of some conjectures on l-independence. I will also discuss the relationship with compatible wild ramification. This is joint work with Qing Lu.

Thu Mar 14

Special Events and Seminars

1:25pm - Vincent Hall 113
Nearby cycles over general bases and duality
Weizhe Zheng, Chinese Academy of Sciences and Princeton University

Over one-dimensional bases, Gabber and Beilinson proved theorems on the commutation of the nearby cycle functor and the vanishing cycle functor with duality. In this talk, I will explain a way to unify the two theorems, confirming a prediction of Deligne. I will also discuss the case of higher-dimensional bases and applications to local acyclicity, following suggestions of Illusie and Gabber. This is joint work with Qing Lu.

Fri Mar 15

Special Events and Seminars

2:30pm - Vincent Hall 1
Mathematical Physics, Algebraic Geometry, and Commutative Algebra
Nadia Ott
Tue Apr 09

Special Events and Seminars

11:00am - Vincent Hall 213
Arithmetic level raising for unitary groups and Beilinson-Bloch-Kato conjecture (I)
Yifeng Liu, Yale University

In this series of two talks, we will introduce the recent progress on Beilinson-Bloch-Kato conjecture for Rankin-Selberg motives of arbitrary rank. We will discuss an important technique used in the proof, namely, the arithmetic level raising for unitary groups of even rank. We will also mention other interesting results we obtained during the course of proof. This is based on a joint work with Y. Tian, L. Xiao, W. Zhang, and X. Zhu.

Wed Apr 10

Special Events and Seminars

2:30pm - Vincent Hall 213
Arithmetic level raising for unitary groups and Beilinson-Bloch-Kato conjecture (II)
Yifeng Liu, Yale University

In this series of two talks, we will introduce the recent progress on Beilinson-Bloch-Kato conjecture for Rankin-Selberg motives of arbitrary rank. We will discuss an important technique used in the proof, namely, the arithmetic level raising for unitary groups of even rank. We will also mention other interesting results we obtained during the course of proof. This is based on a joint work with Y. Tian, L. Xiao, W. Zhang, and X. Zhu

Thu Apr 11

Special Events and Seminars

2:30pm - Vincent Hall 570
Informal Fluids Seminar with the speaker Samuel Punshon-Smith
Samuel Punshon-Smith
Tue Apr 23

Special Events and Seminars

3:30pm - Vincent Hall 364
Vortex filaments in the 3D Navier-Stokes equations
Jacob Bedrossian, Maryland

We consider solutions of the Navier-Stokes equations in 3d with vortex filament initial data of arbitrary circulation, that is, initial vorticity given by a divergence-free vector-valued measure of arbitrary strength supported on a smooth curve. First, we prove global well-posedness for perturbations of the Oseen vortex column in scaling-critical spaces. Second, we prove local well-posedness (in a sense to be made precise) when the filament is a smooth, closed, non-self-intersecting curve. Besides their physical interest as a model for the coherent vortex filament structures observed in 3d fluids, these results are the first to give well-posedness (in a certain sense) in a neighborhood of large self-similar solutions of 3d Navier-Stokes, as well as solutions which are locally approximately self-similar. Joint work with Pierre Germain and Benjamin Harrop-Griffiths.

Tue May 14

Special Events and Seminars

3:30pm - Vincent Hall 570
Singularity formation for some solutions of the incompressible Euler equation
Tarek Elgindi

We describe a recent construction of self-similar blow-up solutions of the incompressible Euler equation. A consequence of the construction is that there exist finite-energy $C^{1,a}$ solutions to the Euler equation which develop a singularity in finite time for some range of $a>0$. The approach we follow is to isolate a simple non-linear equation which encodes the leading order dynamics of solutions to the Euler equation in some regimes and then prove that the simple equation has stable self-similar blow-up solutions.

Wed May 29

Special Events and Seminars

1:30pm - Vincent Hall 206
Informal Fluids Seminar
Raj Beekie
Wed Jun 05

Special Events and Seminars

2:00pm - Vincent Hall 570
AWM Talk
Mimi Boutin
Mon Jun 10

Special Events and Seminars

1:30pm - Vincent Hall 364
Basic Operations on Representations
Andy Hardt

We continue our crash course in finite group representation theory by looking at some important operations on representations.

We start by defining the group algebra of a finite group; group representations naturally biject with modules over the group algebra.

After that, we'll talk through a variety of ways to construct new representations from old, such as restriction, induction, inflation, tensor product, and we may even squeeze in symmetric and exterior powers.