Abstract: We investigate the large time behavior of $N$ particles restricted to a smooth closed curve in $\mathbb{R}^d$ and subject to a gradient flow with respect to Euclidean hyper-singular repulsive Riesz $s$-energy with $s>1$. We show that regardless of their initial positions, for all $N$ and time $t$ large, their normalized Riesz $s$-energy will be close to the $N$-point minimal possible energy. Furthermore, the distribution of such particles will be close to uniform with respect to arclength measure along the curve.
Abstract: I will describe the use of ensemble based particle methods to solve Bayesian inverse problems, including ensemble Kalman methods, methods based on multiscale SDEs, and conjunctions of the two approaches.
Abstract: In this talk we will discuss two central problems in algebraic number theory and their interconnections: explicit class field theory (also known as Hilbert's 12th Problem), and the special values of L-functions. The goal of explicit class field theory is to describe the abelian extensions of a ground number field via analytic means intrinsic to the ground field. Meanwhile, there is an abundance of conjectures on the special values of L-functions at certain integer points. Of these, Stark's Conjecture has special relevance toward explicit class field theory. I will describe two recent joint results with Mahesh Kakde on these topics. The first is a proof of the Brumer-Stark conjecture away from p=2. This conjecture states the existence of certain canonical elements in CM abelian extensions of totally real fields. The second is a proof of an exact formula for Brumer-Stark units that has been developed over the last 15 years. We show that these units together with other easily written explicit elements generate the maximal abelian extension of a totally real field, thereby giving a p-adic solution to Hilbert's 12th problem.
Abstract: Our aim is to obtain precise information on the asymptotic behaviour of various dynamical systems by an improved understanding the discrete spectrum of the associated transfer operators. I'll discuss the general principle that has come to light in recent years and which often allows us to obtain substantial spectral information. I'll then describe several settings where this approach applies, including affine expanding Markov maps, monotone maps, hyperbolic diffeomorphisms. (Joint work with: Niloofar Kiamari & Carlangelo Liverani.)
Abstract: In this talk first, we discuss the notion of asymptotic classes which was introduced by Macpherson and Steinhorn in 2008. Then we discuss some examples. Finally, as long as time permits, I will try to explain an idea of how to characterize N-dimensional asymptotic classes of finite trees (and their expansions). This is joint work (in progress) with Cameron Hill.
Abstract: In this talk we shall discuss dynamics of systems of particles that allow interactions beyond binary, and their behavior as the number of particles goes to infinity. In particular, an example of such a system of bosons leads to a quintic nonlinear Schrodinger equation, which we rigorously derived in a joint work with Thomas Chen. An example of a system of classical particles that allows instantaneous ternary interactions leads to a new kinetic equation that can be understood as a step towards modeling a dense gas in non-equilibrium. We call this equation a ternary Boltzmann equation and we rigorously derived it with Ioakeim Ampatzoglou. Time permitting, we will also discuss the recent work with Ampatzoglou on a derivation of a binary-ternary Boltzmann equation describing the kinetic properties of a dense hard spheres gas, where particles undergo either binary or ternary instantaneous interactions, while preserving momentum and energy. An important challenge we overcome in deriving this equation is related to providing a mathematical framework that allows us to detect both binary and ternary interactions. Furthermore, this work introduces new algebraic and geometric techniques in order to eventually decouple binary and ternary interactions and understand the way they could succeed one another in time.
Abstract: It is known that the additive hazards model is collapsible, in the sense that when omitting one covariate from a model with two independent covariates, the marginal model is still an additive hazards model with the same regression coefficient. In contrast, for the proportional hazards model under the same covariate assumption, it is well-known that the marginal model is no longer a proportional hazards model and hence not collapsible. These results, however, relate to the model specification and not necessarily to the regression estimates.
I point out that if covariates in risk sets at all event times are independent then both Cox and Aalen regression estimates are collapsible, in the sense that there is no systematic change in the parameter estimates. Vice-versa, if this assumption fails, then the estimates will change systematically both for Cox and Aalen regression. In particular, if the data are generated by an Aalen model with censoring independent of covariates both Cox and Aalen regression are collapsible, but if generated by a proportional hazards model neither estimators are. We will also discuss settings where survival times are generated by proportional hazards models with censoring patterns providing uncorrelated covariates and hence collapsible Cox and Aalen regression estimates. Furthermore, possible consequences for instrumental variable analyses with survival data are discussed.