Tom Alberts, University of Utah
" Conformal Field Theory for Multiple SLEs "
Schramm-Loewner evolutions (SLEs) are the scaling limits of interface curves in planar statistical models, characterized by a conformal Markov property and Loewner's growth process. Imposing alternating
boundary conditions on the model leads to a collection of interface curves that interact with each other, the so-called multiple SLEs. In recent years much attention has been given to the characterization
of all possible systems of multiple SLEs and deriving their properties. In this talk I will explain recent joint work with Nam-Gyu Kang (KIAS) and Nikolai Makarov (Caltech) that uses a Gaussian Free
Field based Conformal Field Theory to analyze systems of multiple SLEs. In particular, we re-derive Dubedat's commutation relations, generate an infinite family of martingale observables for the processes,
and re-interpret Dubedat's method of screening all within the framework of our CFT. Our approach also motivates a construction (independent of our CFT) of the so-called pure partition functions using a
contour integral method.
Friday, October 11. No seminar due to fall break. Friday, October 18. Stochastics Seminar. 3-4 PM. LCB 215
Alex Hening, Tufts University
" Stochastic persistence and extinction "
A key question in population biology is understanding the conditions under which the species of an ecosystem persist or go extinct. Theoretical and empirical studies have shown that persistence can be facilitated or negated by both biotic interactions and environmental fluctuations. We study the dynamics of n interacting species that live in a stochastic environment. Our models are described by n dimensional piecewise deterministic Markov processes. These are processes (X(t), r(t)) where the vector X denotes the density of the n species and r(t) is a finite state space process which keeps track of the environment. In any fixed environment the process follows the flow given by a system of ordinary differential equations. The randomness comes from the changes or switches in the environment, which happen at random times. We give sharp conditions under which the populations persist as well as conditions under which some populations go extinct exponentially fast. As an example we look at the competitive exclusion principle from ecology and show how the random switching can `rescue' species from extinction.
Xuan Wu, Columbia University
" Gibbsian line ensembles and the inverse gamma polymer "
In this talk we will construct the discrete log-gamma line
ensemble, which is associated with the inverse gamma polymer model. This
log-gamma line ensemble enjoys a random walk Gibbs resampling
invariance that follows from the integrable nature of the inverse
gamma polymer model via the geometric RSK correspondence. By exploiting
such resampling invariance, we show the tightness of this log-gamma
line ensemble under the weak noise scaling. Furthermore, a Gibbs property,
as enjoyed by the KPZ line ensemble, holds for all subsequential limits.
Promit Ghosal, Columbia University
" Time correlation and tail probabilities of the KPZ equation "
The KPZ equation is a fundamental stochastic PDE related to modeling random growth processes, Burgers turbulence, interacting particle system, random polymers etc. In this talk, we focus on the time correlation and the tail probabilities of the solution of the KPZ equation. We investigate the correlation function of the KPZ equation at two different times. This will be based on a recent joint work with Prof. Alan Hammond from UC Berkeley and my advisor Prof. Ivan Corwin. One of the key inputs to the time correlation project is an estimate on the tail probabilities of the KPZ equation which we also describe. The discussion on the tail probabilities will be based on a separate joint work with Prof. Ivan Corwin. Our analysis is based on an exact identity between the KPZ equation and the Airy point process (which arises at the edge of the spectrum of the random Hermitian matrices) and the Brownian Gibbs property of the KPZ line ensemble.
Mireille Boutin, Purdue University
" Highly likely clusterable data with no cluster "
Data generated as part of a real-life experiment is often quite organized. So much so that, in many cases, projecting the data onto a random line has a high probability of uncovering a division of the data into two well-separated groups. In other words, the data can be clustered with a high probability of success using a hyperplane whose normal vector direction is picked at random. We call such data ``highly likely clusterable.” The clusters obtained in this fashion often do not seem compatible with a cluster structure in the original space. In fact, the data in the original space may not contain any cluster at all. This talk is about the geometry of ``real” data, with a focus on geometries that give rise to this surprising phenomenon. We will also discuss how to exploit this phenomenon to cluster datasets, especially datasets consisting of a small number of points in a high-dimensional space. An application to a qualitative data analysis problem in engineering education will be presented.
Mireille Boutin, Purdue University
" Reconstructing a room from echoes and other unlabeled distance geometry problems "
Suppose that some microphones are placed on a drone inside a room with planar walls/floors/ceilings. A loudspeaker emits a sound impulse and the microphones receive several delayed responses corresponding to the sound bouncing back from each planar surface. These are the first-order echoes. We are interested in reconstructing the shape of the room from the first-order echoes. The time delay for each echo determines the distance from the microphone to a mirror image of the source reflected across a wall. Since we do not know which echo corresponds to which wall, the distances are unlabeled. The problem is to figure out under which circumstances, and how, one can find out the correct distance-wall assignments and reconstruct the wall positions. This is one example of an unlabeled distance geometry problem. Another example is the problem of reconstructing a point configuration, up to a rigid motion, from the multi-set of its pairwise (unlabeled) distances. We show that under some mild genericity assumptions, these problems are well-posed. Extensions to the case where the measurements are noisy will also be discussed. This is joint work with Gregor Kemper (TU Munich).
Jeremiah Birrell, University of Massachusetts at Amherst
" Information-Theoretic Approaches to Distributional Robustness "
Probabilistic models themselves carry uncertainty, due to imperfect
knowledge of the system description and/or dynamics. This leads to
uncertainty in quantities computed from the model, e.g., expected
values. I will discuss recent progress in developing
information-theoretic tools for bounding quantities-of-interest over
infinite-dimensional model-neighborhoods. Different classes of
quantities-of-interest require different approaches; I will focus on
results for rare events and stochastic processes in the long-time
regime. These results will be illustrated by applications to diffusion
processes, option pricing, and large-deviations rate functions.
Friday, November 29. No seminar due to Thanksgiving break.