## Measure-Theoretic Dvoretzky Theorem and Applications to Data Science

- Probability,Uploaded Videos ( 1451 Views )SEPC 2021 in honor of Elizabeth Meckes. Slides from the talks and more information are available <a href="https://services.math.duke.edu/~rtd/SEPC2021/SEPC2021.html">at this link (here).</a>

## Oliver Tough : The Fleming-Viot Particle System with McKean-Vlasov dynamics

- Probability,Uploaded Videos ( 1332 Views )Quasi-Stationary Distributions (QSDs) describe the long-time behaviour of killed Markov processes. The Fleming-Viot particle system provides a particle representation for the QSD of a Markov process killed upon contact with the boundary of its domain. Whereas previous work has dealt with killed Markov processes, we consider killed McKean-Vlasov processes. We show that the Fleming-Viot particle system with McKean-Vlasov dynamics provides a particle representation for the corresponding QSDs. Joint work with James Nolen.

## Zoe Huang : Motion by mean curvature in interacting particle systems

- Probability,Uploaded Videos ( 1254 Views )There are a number of situations in which rescaled interacting particle systems have been shown to converge to a reaction diffusion equation (RDE) with a bistable reaction term. These RDEs have traveling wave solutions. When the speed of the wave is nonzero, block constructions have been used to prove the existence or nonexistence of nontrivial stationary distributions. Here, we follow the approach in a paper by Etheridge, Freeman, and Pennington to show that in a wide variety of examples when the RDE limit has a bistable reaction term and traveling waves have speed 0, one can run time faster and further rescale space to obtain convergence to motion by mean curvature. This opens up the possibility of proving that the sexual reproduction model with fast stirring has a discontinuous phase transition, and that in Region 2 of the phase diagram for the nonlinear voter model studied by Molofsky et al there were two nontrivial stationary distributions.

## Alex Hening : Stochastic persistence and extinction

- Probability,Uploaded Videos ( 1224 Views )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, which says in its simplest form that two species competing for one resource cannot coexist, and show how the random switching can facilitate coexistence.

## Roman Vershynin : Mathematics of synthetic data and privacy

- Probability,Uploaded Videos ( 1110 Views )An emerging way to protect privacy is to replace true data by synthetic data. Medical records of artificial patients, for example, could retain meaningful statistical information while preserving privacy of the true patients. But what is synthetic data, and what is privacy? How do we define these concepts mathematically? Is it possible to make synthetic data that is both useful and private? I will tie these questions to a simple-looking problem in probability theory: how much information about a random vector X is lost when we take conditional expectation of X with respect to some sigma-algebra? This talk is based on a series of papers with March Boedihardjo and Thomas Strohmer.

## David Aldous: Probability Seminar

- Probability,Uploaded Videos ( 649 Views )David Aldous, Probability Seminar Sept 30, 2021 TITLE: Can one prove existence of an infectiousness threshold (for a pandemic) in very general models of disease spread? ABSTRACT: Intuitively, in any kind of disease transmission model with an infectiousness parameter, there should exist a critical value of the parameter separating a very likely from a very unlikely resulting pandemic. But even formulating a general conjecture is challenging. In the most simplistic model (SI) of transmission, one can prove this for an essentially arbitrary large weighted contact network. The proof for SI depends on a simple lemma concerning hitting times for increasing set-valued Markov processes. Can one extend to SIR or SIS models over similarly general networks, where the lemma is no longer applicable?

## Max Xu : Random multiplicative functions and applications

- Probability ( 262 Views )Random multiplicative functions are probabilistic models for multiplicative arithmetic functions, such as Dirichlet characters or the Liouville function. In this talk, I will first quickly give an overview of the area, and then focus on some of the recent works on proving central limit theorems, connections to additive combinatorics, as well as some other deterministic applications. Part of the talk is based on joint work with Soundararajan, with Harper and Soundararajan (in progress) and with Angelo and Soundararajan (in progress).

## Nayantara Bhatnagar : Subsequence Statistics in Random Mallows Permutations

- Probability ( 258 Views )The longest increasing subsequence (LIS) of a uniformly random permutation is a well studied problem. Vershik-Kerov and Logan-Shepp first showed that asymptotically the typical length of the LIS is 2sqrt(n). This line of research culminated in the work of Baik-Deift-Johansson who related this length to the GUE Tracy-Widom distribution. We study the length of the LIS of random permutations drawn from the Mallows measure, introduced by Mallows in connection with ranking problems in statistics. We prove limit theorems for the LIS for different regimes of the parameter of the distribution. I will also describe some recent results on the longest common subsequence of independent Mallows permutations. Relevant background for the talk will be introduced as needed. Based on work with Ron Peled, Riddhi Basu and Ke Jin.

## Lisa Hartung : Extreme Level Sets of Branching Brownian Motion

- Probability ( 253 Views )Branching Brownian motion is a classical process in probability theory belonging to the class of Â?Log-correlated random fieldsÂ?. We study the structure of extreme level sets of this process, namely the sets of particles whose height is within a fixed distance from the order of the global maximum. It is well known that such particles congregate at large times in clusters of order-one genealogical diameter around local maxima which form a Cox process in the limit. We add to these results by finding the asymptotic size of extreme level sets and the typical height and shape of those clusters which carry such level sets. We also find the right tail decay of the distribution of the distance between the two highest particles. These results confirm two conjectures of Brunet and Derrida.(joint work with A. Cortines, O Louidor)

## Didong Li : Learning & Exploiting Low-Dimensional Structure in High-Dimensional Data

- Probability ( 240 Views )Data lying in a high dimensional ambient space are commonly thought to have a much lower intrinsic dimension. In particular, the data may be concentrated near a lower-dimensional subspace or manifold. There is an immense literature focused on approximating the unknown subspace and the unknown density, and exploiting such approximations in clustering, data compression, and building of predictive models. Most of the literature relies on approximating subspaces and densities using a locally linear, and potentially multiscale, dictionary with Gaussian kernels. In this talk, we propose a simple and general alternative, which instead uses pieces of spheres, or spherelets, to locally approximate the unknown subspace. I will also introduce a curved kernel called the Fisherâ??Gaussian (FG) kernel which outperforms multivariate Gaussians in many cases. Theory is developed showing that spherelets can produce lower covering numbers and mean square errors for many manifolds, as well as the posterior consistency of the Dirichlet process mixture of the FG kernels. Time permitting, I will also talk about an ongoing project about stochastic differential geometry.

## Lea Popovic : Genealogy of Catalytic Populations

- Probability ( 230 Views )For neutral branching models of two types of populations there are three universality classes of behavior: independent branching, (one-sided) catalytic branching and mutually catalytic branching. Loss of independence in the two latter models generates many new features in the way that the populations evolve. In this talk I will focus on describing the genealogy of a catalytic branching diffusion. This is the many individual fast branching limit of an interacting branching particle model involving two populations, in which one population, the "catalyst", evolves autonomously according to a Galton-Watson process while the other population, the "reactant", evolves according to a branching dynamics that is dependent on the number of catalyst particles. We show that the sequence of suitably rescaled family forests for the catalyst and reactant populations converge in Gromov-Hausdorff topology to limiting real forests. We characterize their distribution via a reflecting diffusion and a collection of point-processes. We compare geometric properties and statistics of the catalytic branching forests with those of the "classical" (independent branching) forest. This is joint work with Andreas Greven and Anita Winter.

## Stanislav Molchanov : On the random analytic functions

- Probability ( 230 Views )The talk will contain a review of several recent results on the analytic continuation of the random analytic functions. We will start from the classical theorem on the random Taylor series (going to BorelÂ? s school), but the main subject will be the random zeta Â? function (which was introduced implicitly by Cramer) and its generalizations. We will show that Â?true primes are not truly random Â?, since zeta Â? functions for the random Â?pseudo-primesÂ? (in the spirit of Cramer) have no analytic continuation through the critical line Re (z) = 1/2.

## Li-Cheng Tsai : Interacting particle systems with moving boundaries

- Probability ( 218 Views )In this talk I will go over two examples of one-dimensional interacting particle systems: Aldous' up-the-river problem, and a modified Diffusion Limited Growth. I will explain how these systems connect to certain PDE problems with boundaries. For the up-the-river problem this connection helps to solve AldousÂ? conjecture regarding an optimal strategy. For the modified DLA, this connection helps to characterize the scaling exponent and scaling limit of the boundary at the critical density. This talk is based on joint work with Amir Dembo and Wenpin Tang.

## Johan Brauer : The Stabilisation of Equilibria in Evolutionary Game Dynamics through Mutation

- Probability ( 208 Views )The multi-population replicator dynamics (RD) can be considered a dynamic approach to the study of multi-player games, where it was shown to be related to Cross-learning, as well as of systems of co-evolving populations. However, not all of its equilibria are Nash equilibria (NE) of the underlying game, and neither convergence to an NE nor convergence in general are guaranteed. Although interior equilibria are guaranteed to be NE, no interior equilibrium can be asymptotically stable in the multi-population RD, resulting, e.g., in cyclic orbits around a single interior NE. We report on our investigation of a new notion of equilibria of RD, called mutation limits, which is based on the inclusion of a naturally arising, simple form of mutation, but is invariant under the specific choice of mutation parameters. We prove the existence of such mutation limits for a large range of games, and consider an interesting subclass, that of attracting mutation limits. Attracting mutation limits are approximated by asymptotically stable equilibria of the (mutation-)perturbed RD, and hence, offer an approximate dynamic solution of the underlying game, especially if the original dynamic has no asymptotically stable equilibria. Therefore, the presence of mutation will indeed stabilise the system in certain cases and make attracting mutation limits near-attainable. Furthermore, the relevance of attracting mutation limits as a game theoretic equilibrium concept is emphasised by the relation of (mutation-)perturbed RD to the Q-learning algorithm in the context of multi-agent reinforcement learning. However, in contrast to the guaranteed existence of mutation limits, attracting mutation limits do not exist in all games, raising the question of their characterization.

## Hendrik Weber : Convergence of the two-dimensional dynamic Ising-Kac model

- Probability ( 207 Views )The Ising-Kac model is a variant of the ferromagnetic Ising model in which each spin variable interacts with all spins in a neighbourhood of radius $\ga^{-1}$ for $\ga \ll1$ around its base point. We study the Glauber dynamics for this model on a discrete two-dimensional torus $\Z^2/ (2N+1)\Z^2$, for a system size $N \gg \ga^{-1}$ and for an inverse temperature close to the critical value of the mean field model. We show that the suitably rescaled coarse-grained spin field converges in distribution to the solution of a non-linear stochastic partial differential equation. This equation is the dynamic version of the $\Phi^4_2$ quantum field theory, which is formally given by a reaction diffusion equation driven by an additive space-time white noise. It is well-known that in two spatial dimensions, such equations are distribution valued and a \textit{Wick renormalisation} has to be performed in order to define the non-linear term. Formally, this renormalisation corresponds to adding an infinite mass term to the equation. We show that this need for renormalisation for the limiting equation is reflected in the discrete system by a shift of the critical temperature away from its mean field value. This is a joint work with J.C. Mourrat (Lyon).

## Hao Shen : Stochastic PDEs and regularity structures

- Probability ( 207 Views )In this talk I will review the basic ideas of the regularity structure theory developed by Martin Hairer, as well as its applications to stochastic PDE problems. I will then discuss my joint work with Hairer on the sine-Gordon equation and central limit theorems for stochastic PDEs.

## Robin PEMANTLE : Zeros of random analytic functions and their derivatives

- Probability ( 207 Views )I will discuss a series of results concerning the effect of the derivative operator on the locations of the zeros of a random analytic function. Two models are considered. In the first, the zeros are chosen IID from some measure on the complex plane. In the second, the zeros are chosen to be a Poisson point process on the real line. Repeated differentiation results in a nearly deterministic zero set.

## Laurie Field : Relating variants of SLE using the Brownian loop measure

- Probability ( 205 Views )In this talk I will discuss a framework for transforming one variant of the SchrammÂ?Loewner evolution (SLE) into another. The main tool in this approach is the Brownian loop measure. A simple case is to relate the reversal of radial SLE to whole-plane SLE, which looks the same locally. Writing the formula one might naÃ¯vely expect fails, because the loop measure term is infinite. In joint work with Greg Lawler, we show that there is a finite normalized version of the loop measure term, and that with this change, the naÃ¯ve formula relating the two SLEs becomes correct.

## Rick Durrett : Voter Model Perturbations

- Probability ( 202 Views )We consider particle systems that are perturbations of the voter model and show that when space and time are rescaled the system converges to a solution of a reaction diffusion equation in dimensions $d \ge 3$. Combining this result with properties of the PDE and a block construction, we give general, and often asymptotically sharp, conditions for the existence of non-trivial stationary distributions, and for extinction of one type. As applications, we describe the phase diagrams of three systems when the parameters are close to the voter model: (i) a stochastic spatial Lotka-Volterra model of Neuhauser and Pacala, (ii) a model of the evolution of cooperation of Ohtsuki, Hauert, Lieberman, and Nowak, and (iii) a continuous time version of the non-linear voter model of Molofsky, Durrett, Dushoff, Griffeath, and Levin. The first two applications confirm conjectures of Cox and Perkins and Ohtsuki et al.

## Brian Rider : Log-gases and Tracy-Widom laws

- Probability ( 201 Views )The now ubiquitous Tracy-Widom laws were first discovered in the context of the Gaussian Orthogonal, Unitary, and Symplectic Ensembles (G{O/U/S}E) of random matrix theory. The latter may be viewed as logarithmic gases with quadratic (Gaussian) potential at three special inverses temperatures (beta=1,2,4). A few years back, Jose Ramirez, Balint Virag, and I showed that that one obtains generalizations of the Tracy-Widom laws at all inverse temperatures (beta>0), though still for quadratic potentials. I'll explain how similar ideas (and considerably more labor) extends the result to general potential, general temperature log-gases. This is joint work with Manjunath Krishnapur and Balint Virag.

## Erika Berenice Roldan Roa : Asymptotic behavior of the homology of random polyominoes

- Probability ( 197 Views )In this talk we study the rate of growth of the expectation of the number of holes (the rank of the first homology group) in a polyomino with uniform and percolation distributions. We prove the existence of linear bounds for the expected number of holes of a polyomino with respect to both the uniform and percolation distributions. Furthermore, we exhibit particular constants for the upper and lower bounds in the uniform distribution case. This results can be extend, using the same techniques, to other polyforms and higher dimensions.

## Elizabeth Meckes : Projections of probability distributions: a measure-theoretic Dvoretzky theorem

- Probability ( 196 Views )Dvoretzky's theorem tells us that if we put an arbitrary norm on n-dimensional Euclidean space, no matter what that normed space is like, if we pass to subspaces of dimension about log(n), the space looks pretty much Euclidean. A related measure-theoretic phenomenon has long been observed: the (one-dimensional) marginals of many natural high-dimensional probability distributions look about Gaussian. A question which had received little attention until recently is whether this phenomenon persists for k-dimensional marginals for k growing with n, and if so, for how large a k? In this talk I will discuss recent work showing that the phenomenon does indeed persist if k less than 2log(n)/log(log(n)), and that this bound is sharp (even the 2!).

## Swee Hong Chan : Random walks with local memory

- Probability ( 195 Views )In this talk we consider this question for a family of random walks on the square lattice. When the randomness is turned to the maximum, we have the symmetric random walk, which is known to scale to a planar Brownian motion. When the randomness is turned to zero, we have the rotor walk, for which its scaling limit is an open problem. This talk is about random walks that lie in between these two extreme cases and for which we can prove their scaling limit. This is a joint work with Lila Greco, Lionel Levine, and Boyao Li.