## Eliza O’Reilly : Stochastic and Convex Geometry for Complex Data Analysis

- Colloquium Seminar,Colloquium,Uploaded Videos ( 823 Views )Many modern problems in data science aim to efficiently and accurately extract important features and make predictions from high dimensional and large data sets. Naturally occurring structure in the data underpins the success of many contemporary approaches, but large gaps between theory and practice remain. In this talk, I will present recent progress on two different methods for nonparametric regression that can be viewed as the projection of a lifted formulation of the problem with a simple stochastic or convex geometric description, allowing the projection to encapsulate the data structure. In particular, I will first describe how the theory of stationary random tessellations in stochastic geometry can address the computational and theoretical challenges of random decision forests with non-axis-aligned splits. Second, I will present a new approach to convex regression that returns non-polyhedral convex estimators compatible with semidefinite programming. These works open many directions of future work at the intersection of stochastic and convex geometry, machine learning, and optimization.

## Evita Nestoridi : Mixing times and the cutoff phenomenon.

- Colloquium ( 236 Views )Markov chains are random processes that retain no memory of the past. The mixing time of a Markov chain is the time it takes for it to reach equilibrium. During the last three decades, there has been a lot of progress in developing various techniques to estimate mixing times for various chains and to understand the cutoff phenomenon which means that the Markov chain has an abrupt convergence to equilibrium. We will present recent work establishing cutoff for the random to random card shuffle which confirms a 2001 conjecture of Diaconis. We will also present a proof of uniform lower bounds for Glauber dynamics for the Ising model, extending a result of Ding and Peres. The proofs employ both probabilistic and algebraic techniques.

## Jim Arthur : The principle of functorialty: an elementary introduction

- Colloquium ( 38 Views )The principle of functoriality is a central question in present day mathematics. It is a far reaching, but quite precise, conjecture of Langlands that relates fundamental arithmetic information with equally fundamental analytic information. The arithmetic information arises from the solutions of algebraic equations. It includes data that classify algebraic number fields, and more general algebraic varieties. The analytic information arises from spectra of differential equations and group representations. It includes data that classify irreducible representations of reductive groups. The lecture will be a general introduction to these things. If time permits, we shall also describe recent progress that is being made on the problem.

## Tom Kepler : Microevolution in the Immune System: A Computational Systems Approach

- Colloquium ( 32 Views )Vaccines protect their recipients by inducing long-term structural changes in populations of immune cells. Part of that restructuring is exactly analogous to Darwinian Selection. New antibody molecules are created by somatic mutation of existing antibody genes. Subsequently, the immune cell populations that possess these mutated receptors overtake the "wild-type" immune cells due to the selective advantage they have acquired. Thus the immune system is vastly better prepared to recognize and eliminate the eliciting pathogen the next time around.

New sequencing and biosynthesis technologies, together with mathematical and computational tools, now allow us to investigate this fascinating and important phenomenon more deeply than ever before. I will illustrate this development with examples from the immune response to HIV infection.