Scientific

Since Alan Turing's pioneering publication on morphogenetic pattern formation obtained with reaction-diffusion (RD) systems, it has been the prevailing belief that two-component reaction diffusion systems have to include a fast diffusing inhibiting component (inhibitor) and a much slower diffusing activating component (activator) in order to break symmetry from a uniform steady-state. This time-scale separation is often unbiological for cell signal transduction pathways.
We modify the traditional RD paradigm by considering nonlinear reaction kinetics only inside compartments with reactive boundary conditions to the extra-compartmental space that provides a two-species diffusive coupling. The construction of a nonlinear algebraic system for all existing steady-states enables us to derive a globally coupled matrix eigenvalue problem for the growth rates of eigenperturbations from the symmetric steady-state, on finite domains in 1-D and 2-D and a periodically extended version in 1-D.

We show that the membrane reaction rate ratio of inhibitor rate to activator rate is a key bifurcation parameter leading to robust symmetry-breaking of the compartments. Illustrated with Gierer-Meinhardt, FitzHugh-Nagumo and Rauch-Millonas intra-compartmental reaction kinetics, our compartmental-reaction diffusion system does not require diffusion of inhibitor and activator on vastly different time scales.
Our results elucidate a possible mechanism of the ubiquitous biological cell specialization observed in nature.

Optimal transportation, at its core, is a powerful framework for obtaining structured yet general couplings between general probability measures based on matching underlying characteristics. This framework lends itself naturally to applications in causal inference: from matching estimation, to difference-in-differences and synthetic controls approaches, to instrumental variable estimation, just to name a few. In this talk, I will provide a non-exhaustive overview of potential applications of optimal transport approaches to causal inference. I will focus on providing an overview of general concepts and ideas.

The talk is based on joint work with Rex Hsieh, Myung Jin Lee, Philippe Rigollet, William Torous, and Yuliang Xu.

In this talk we try to understand the Bowen-Margulis measure for geodesic flow on manifolds of (variable) negative curvature. In the first half-hour (pre-talk), I will discuss the necessary backgrounds from hyperbolic geometry and dynamics, and during the next hour (main talk), I will explain a construction due to Hamenstädt, which relates Bowen-Margulis measure to the Hausdorff measure with respect to a certain metric.

The Hurwitz zeta function $\zeta(s, \alpha)$ is a shifted integer analogue of the Riemann zeta function which shares many of its properties, but is not an ”L-function” under any reasonable definition of the word. We will first review the basics of the value distribution of the Riemann zeta function in the critical strip (moments, Bohr–Jessen theory...) and then contrast it with the value distribution of the Hurwitz zeta function.

Our focus will be on shift parameters $\alpha / \in \mathbb{Q}$, i.e., algebraic irrational or transcendental. We will present a new result (joint with Winston Heap) on moments of these objects on the critical line.

Mathieu Dutour (University of Alberta, Canada)

In the finite-dimensional situation, Lie's third theorem provides a correspondence between Lie groups and Lie algebras. Going from the latter to the former is the more complicated construction, requiring a suitable representation, and taking exponentials of the endomorphisms induced by elements of the group.

As shown by Garland, this construction can be adapted for some Kac-Moody algebras, obtained as (central extensions of) loop algebras. The resulting group is called a loop group. One also obtains a relevant infinite-rank Chevalley lattice, endowed with a metric. Recent work by Bost and Charles provide a natural setting, that of pro-Hermitian vector bundles and theta invariants, in which to study these objects related to loop groups. More precisely, we will see in this talk how to define theta-finite pro-Hermitian vector bundles from elements in a loop group. Similar constructions are expected, in the future, to be useful to study loop Eisenstein series for number fields.

This is joint work with Manish M. Patnaik.