Hecke and Galois Properties of Special Cycles on Unitary Shimura Varieties

Speaker: 

Dimitar Jetchev

Institution: 

EPFL (Lausanne)

Time: 

Tuesday, February 25, 2014 - 4:00pm

Host: 

Alice Silverberg

Location: 

RH 340P

We define a collection of special 1-cycles on certain Shimura 3-folds associated to U(2,1) x U(1,1) and appearing in the context of the Gan--Gross--Prasad conjectures. We study and compare the action of the Hecke algebra and the Galois group on these cycles via distribution relations and congruence relations that would ultimately lead to the construction of a novel Euler system for these Shimura varieties. The comparison is achieved adelically using Bruhat--Tits theory for the corresponding buildings.

Harrington's Principle and remarkable cardinals

Speaker: 

Ralf-Dieter Schindler

Institution: 

Muenster University, Germany and UC Berkeley

Time: 

Monday, February 24, 2014 - 4:00pm to 5:30pm

Host: 

Martin Zeman

Location: 

RH 440R

It is open whether \Pi^1_1 determinacy implies the existence of 0^\# in 3rd order arithmetic, call it Z_3. We compute the large cardinal strength of Z_3 plus "there is a real x such that every x-admissible is an L-cardinal." This is joint work with Yong Cheng.
 

An Eulerian-Lagrangian Method for Optimization Problems Governed by Multidimensional Nonlinear Hyperbolic PDEs

Speaker: 

Alina Chertock

Institution: 

North Carolina State Unviersity

Time: 

Monday, April 21, 2014 - 4:00pm to 5:00pm

Host: 

Hongkai Zhao

Location: 

RH306

In this talk, I will present a numerical method for solving tracking-type optimal control problems subject to scalar nonlinear hyperbolic balance laws in one and two space dimensions. The approach is based on the formal optimality system and requires numerical solutions of the hyperbolic balance law forward in time and its nonconservative adjoint equation backward in time. To this end, we develop a hybrid method, which utilizes advantages of both the Eulerian finite-volume scheme (for solving the balance law) and the Lagrangian discrete characteristics method (for solving the adjoint transport equation). Experimental convergence rates as well as numerical results for optimization problems with both linear and nonlinear constraints and a duct design problem will also be presented and discussed.

Units in function rings

Speaker: 

Daniel Bertrand

Institution: 

Université Pierre et Marie Curie and MSRI

Time: 

Thursday, April 24, 2014 - 2:00pm

Host: 

Alice Silverberg

Location: 

RH 440R

Contrary to their classical namesakes over the ring of integers, Pell equations over function rings in characteristic zero need not have infinitely many solutions. How often this occurs has been the theme of recent work of D. Masser and U. Zannier. The case of smooth curves is governed by the relative Manin-Mumford conjecture on abelian schemes. We pursue this study by considering singular curves and the associated generalized jacobians.

 

Stochastic Differential Equations, Intermittent Diffusion, and Shortest Path

Speaker: 

Haomin Zhou

Institution: 

Georgia Institute of Technology

Time: 

Monday, April 28, 2014 - 4:00pm to 5:00pm

Host: 

Hongkai Zhao

Location: 

RH306

In this talk, I will use the shortest path problem as an example to
illustrate how one can connect optimization, stochastic differential
equations and partial differential equations together to solve some
challenging real world problems. On the other end, I will show what
new and challenging mathematical problems can be raised from those applications. The talk is based on a joint work with Shui-Nee Chow
and Jun Lu.

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