Alison Etheridge (Kai Lai Chung Lecturer)

Oxford University, UK

 Modelling evolution in a fluctuating environment

Abstract:  Since the pioneering work of Fisher, Haldane and Wright at the beginning of the 20th Century, mathematics has played a central role in theoretical population genetics. In turn, population genetics has provided the motivation both for important classes of probabilistic models, such as coalescent processes, and for deterministic models, such as the celebrated Fisher-KPP equation. Whereas coalescent models capture `relatedness’ between genes, the Fisher KPP equation captures something of the interaction between natural selection and spatial structure.  What has proven remarkably difficult is to combine the two, at least in the biologically relevant setting of a two-dimensional spatial continuum.  Moreover, whereas the Fisher-KPP equation assumes that the selective advantage of a particular genetic type is constant in time and space, it has long been recognized that fitnesses of different genetic types will fluctuate in space and time, driven by temporal and spatial fluctuations in the environment. We briefly discuss some problems and (fewer) solutions related to modelling populations undergoing fluctuating selection.

Rami Atar

Technion, Israel 

Robustness quantification at the large deviations scale

Abstract: An approach to quantifying robustness of large deviations estimates to the probability distribution underlying the model will be described. Its basis is an identity which extends the well-known duality between relative entropy and exponential integrals, to one that involves Renyi divergence. When viewed at large deviation scale, this identity corresponds to working with a reference measure under which the probability of an event of interest may decay exponentially, rather than remain order one as in the conventional use of change of measure. Moreover, it gives rise to bounds that are uniform within a family of models specified in terms of this divergence. The main application areas to be discussed are queueing networks and risk sensitive control.

The talk is based on joint work with Amarjit Budhiraja, Paul Dupuis and Ruoyu Wu.

Antonio Auffinger

Northwestern University, USA 

Energy levels of random functions in high dimensions

Abstract:  What does a random smooth function look like in high-dimensions? How many peaks, saddles or critical values of given index at a given energy level? What can be said about the topology of its level sets? In this talk, we will address these questions by looking at the low temperature limit of the corresponding Gibbs measure of spin glass models. In particular, we will provide the first examples of two-step replica symmetry breaking (2-RSB) models for the spherical mixed p-spin glass. These examples largely contrast with the early prediction that random functions could be classified into two different categories: one-step replica symmetry breaking (1RSB) or full-step replica symmetry breaking (FRSB). Based on a joint works with Wei-Kuo Chen (Univ. of Minnesota) and Qiang Zeng (Northwestern).

Fabrice Baudoin

University of Connecticut, USA 

Hypocoercive diffusion processes and gradient bounds

Abstract: We will study several methods, both probabilistic and analytic, yielding gradient bounds for the semigroups associated to Kolmogorov type operators. Some of those bounds will be shown to imply convergence to equilibrium with explicit rates. Several models will be discussed in details. The talk will be based on joint works with Camille Tardif, Maria Gordina and Phanuel Mariano.

Mykhaylo Shkolnikov

Princeton University, USA

Particles interacting through their hitting times: neuron firing, supercooling and systemic risk 

Abstract: I will discuss a class of particle systems that serve as models for supercooling in physics, neuron firing in neuroscience and systemic risk in finance. The interaction between the particles falls into the mean-field framework pioneered by McKean and Vlasov in the late 1960s, but many new phenomena arise due to the singularity of the interaction. The most striking of them is the loss of regularity of the particle density caused by the the self-excitation of the system. In particular, while initially the evolution of the system can be captured by a suitable Stefan problem, the following irregular behavior necessitates a more robust probabilistic approach. Based on joint work with Sergey Nadtochiy 

Gregory Miermont 

Ecole Normale Superieure de Lyon, France

Exploring random maps: slicing, peeling and layering

Abstract: The combinatorial theory of maps, or graphs on surfaces, is rich of many different approaches (recursive decompositions, algebraic approaches, matrix integrals, bijective approaches) which often have probabilistic counterparts that are of interest when one wants to study geometric aspects of random maps. In these lectures, I will review parts of this theory by focusing on three different decompositions of maps, namely, the slice decomposition, the peeling process, and the decomposition in layers, and by showing how these decompositions can be used to give access to quite different geometric properties of random maps.