Departmental Colloquium

How complicated can successive manifolds get in a tower of covering spaces? Specifically, how large can the dimension of the first cohomology get? We will begin with a tour of possible behaviors for low-dimensional spaces, and then focus on arithmetic manifolds. Specifically, for towers of complex-hyperbolic manifolds, I will describe how to bound the rates of growth using known instances of Langlands functoriality.

In the 1970s, seminal work of Margulis showed that higher rank lattices have superrigid representations, which in particular implies that all such lattices are arithmetic. Since then Gromov--Piatetski-Shapiro and Deligne--Mostow have shown that a similar superrigidity theorem cannot hold in the hyperbolic setting. In this talk, we will survey the work of Margulis on superrigidity and then go on to discuss how one can prove certain superrigidity/arithmeticity theorems for hyperbolic manifolds provided that the associated manifolds satisfy additional geometric hypothesis.

One of the most fundamental unsolved problems in representation theory is to classify the set of irreducible unitary representations of a semisimple Lie group. In this talk, I will define a class of such representations coming from filtered quantizations of certain graded Poisson varieties. The representations I construct are expected to form the "building blocks" of all unitary representations.

The study of closed geodesics on hyperbolic surfaces is a subject that shares deep connections with many areas of mathematics. We survey classic and recent results in the subject, emphasizing the relations between three different perspectives: topology, geometry, and arithmetic. In particular, we discuss recently discovered connections with mapping class groups and Teichmüller dynamics. The talk assumes no previous knowledge on the subject and is aimed at a wide mathematical audience.

It is possible to learn a lot about a group by studying how it acts on various metric spaces. One particularly interesting (and ubiquitous) class of groups are those that act nicely on negatively curved spaces, called hyperbolic groups. Since their introduction by Gromov in the 1980s, hyperbolic groups and their generalizations have played a central role in geometric group theory. One fruitful tool for studying such groups is their boundary at infinity. In this talk, I'll discuss two generalizations of hyperbolic groups, relatively hyperbolic groups and hierarchically hyperbolic groups, and describe boundaries of each. I'll describe various relationships between these boundaries, and explain how the hierarchically hyperbolic boundary characterizes relative hyperbolicity among hierarchically hyperbolic groups. This is joint work with Jason Behrstock and Jacob Russell.

The immune system is able to fight cancer by mustering and training an army of effector "killer" cells. Mathematical models of tumor-immune interactions must describe the proliferation, recruiting and killing rates of immune cells. Earlier work surprisingly showed that the functions describing the kill rates distinguish between two types of immune cells. The mechanisms behind these differences have been a mystery, however. In an attempt to unravel this mystery, we have created a cell-based fixed-lattice model that simulates immune cell and tumor cell interaction involving tumor recognition and two killing mechanisms. These mechanisms play a big role in the effectiveness of many cancer immunotherapies. Results from model simulations, along with theories developed by ecologists, can help to illuminate which mechanisms are at work in different conditions.

Data assimilation and inverse methods for ill-posed problems find optimal estimates of states or parameters. Methods for both combine observations with a model, which here we assume is a partial differential equation (PDE). Finding a compromise between observations and model is challenging because the actual observations often have values significantly different than the corresponding PDE estimates. Neither the observations nor the PDE exactly characterize the state because each has error, and in the case of the PDE, this can be due to unknown forcings, initial or boundary conditions. State estimates from data assimilation can vary significantly depending on specified errors in the PDE. In this work we estimate PDE errors by developing a common framework between variational data assimilation and regularization for ill-posed problems. This framework arises when weakly constrained variational data assimilation is viewed as regularizing the severely underdetermined data fitting problem in data assimilation. Within this framework we derive error estimates for data assimilation using regularization parameter selection methods including the L-curve, Generalized Cross Validation (GCV) and the Chi-squared method. Data assimilation results will be shown from a one dimensional transport model with simulated data, where the resulting state estimates can be viewed as air quality estimates.