## 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?

## Sayan Banerjee : Singular Reflected Diffusions

- Probability ( 384 Views )I will talk about some models coming from Physics and Queueing Theory that give rise to singular reflected processes in their diffusion limit. Such diffusions are characterized by non-elliptic generators (which are not even hypoelliptic) in the interior, and ergodicity arises from non-trivial interactions between the diffusion, drift and reflection. I will introduce a regenerative process approach which identifies renewal times in diffusion paths and analyzes excursions between successive renewal times. This provides a detailed description of the stationary distribution even when closed form expressions are unavailable. Based on joint works with Chris Burdzy, Brendan Brown, Mauricio Duarte and Debankur Mukherjee.

## 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)

## John McSweeney : A Nonuniform Stochastic Coalescent Process with applications to Biology and Computer Science

- Probability ( 225 Views )Viewed forwards in time, a population reproducing according to some random mechanism can be thought of as a branching process. What if it is viewed backwards? We can take a sample of individuals from the current generation and trace their genealogy backwards, and for instance find their most recent common ancestor; this is known as a coalescent process. If we know a population's random mating process, but have no actual data as to what the phylogenetic tree looks like, how do we derive the distribution of the time until its most recent common ancestor? I will discuss a variant on the classical Wright-Fisher reproductive model and deduce some parameter thresholds for emergence of different qualitative features of the tree. An isomorphic problem may also be useful in computer science for bounding the running time of certain random sampling algorithms.

## David Herzog : Supports of Degenerate Diffusion Processes: The Case of Polynomial Drift and Additive Noise

- Probability ( 223 Views )We discuss methods for computing supports of degenerate diffusion processes. We assume throughout that the diffusion satisfies a stochastic differential equation on R^{d} whose drift vector field X_{0} is ``polynomial'' and whose noise coefficients are constant. The case when each component of X_{0} is of odd degree is well understood. Hence we focus our efforts on X_{0} having at least one or more components of even degree. After developing methods to handle such cases, we shall apply them to specific examples, e.g. the Galerkin truncations of the Stochastic Navier-Stokes equation, to help establish ergodic properties of the resulting diffusion. One benefit to our approach is that, to prove such consequences, all we must do is compute certain
Lie brackets.

## 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.

## Ofer Zeitouni : Slowdown in Branching random walks and the inhomogeneous KPP equation

- Probability ( 207 Views )The classical result of Bramson gives a precise logarithmic correction to the speed of front propagation in one dimensional branching random walks and Brownian motions. I will discuss several variants of this model where the slowdown term is not classical.

## Scott Schmidler : Mixing times for non-stationary processes

- Probability ( 204 Views )Markov chain methods for Monte Carlo simulation of complex physical or statistical models often require significant tuning. Recent theoretical progress has renewed interest in "adaptive" Markov chain algorithms which learn from their sample history. However, these algorithms produce non-Markovian, time-inhomogeneous, irreversible stochastic processes, making rigorous analysis challenging. We show that lower bounds on the mixing times of these processes can be obtained using familiar ideas of hitting times and conductance from the theory of reversible Markov chains. The bounds obtained are sufficient to demonstrate slow mixing of several recently proposed algorithms including adaptive Metropolis kernels and the equi-energy sampler on some multimodal target distributions. These results provide the first non-trivial bounds on the mixing times of adaptive MCMC samplers, and suggest a way of classifying adaptive schemes that leads to new hybrid algorithms. Many open problems remain.

## David Sivakoff : Nucleation scaling in jigsaw percolation

- Probability ( 203 Views )Jigsaw percolation is a nonlocal process that iteratively merges elements of a partition of the vertices in a deterministic puzzle graph according to the connectivity properties of a random collaboration graph. We assume the collaboration graph is an Erdos-Renyi graph with edge probability p, and investigate the probability that the puzzle graph is solved, that is, that the process eventually produces the partition {V}. In some generality, for puzzle graphs with N vertices of degrees about D, this probability is close to 1 or 0 depending on whether pD(log N) is large or small. We give more detailed results for the one dimensional cycle and two dimensional torus puzzle graphs, where in many instances we can prove sharp phase transitions.

## 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.

## F. Baudoin : Functional Inequalities: Probability and geometry in interaction

- Probability ( 202 Views )The talk will be an introduction to the world of functional inequalities with a geometric content. I will in particular focus on the family of log-Sobolev and Sobolev inequalities and show how these inequalities are closely connected to the geometry of the ambient space. I will mainly follow the Bakry-Ledoux approach to these inequalities which is is based on the notion of intrinsic curvature of a diffusion operator and at the end of the presentation will explain how these ideas have recently been used in sub-Riemannian geometry.

## 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.

## Maria Gordina : Gaussian type analysis on infinite-dimensional Heisenberg groups

- Probability ( 183 Views )This is a joint work with B.Driver. The groups in question are modeled on an abstract Wiener space. Then a group Brownian motion is defined, and its properties are studied in connection with the geometry of this group. The main results include quasi-invariance of the heat kernel measure, log Sobolev inequality (following a bound on the Ricci curvature), and the Taylor isomorphism to the corresponding Fock space. The latter is a version of the Ito-Wiener expansion in the non-commutative setting.

## Yu Gu : Scaling limits of random fluctuations in homogenization of divergence form operators

- Probability ( 182 Views )Recently, quantitative stochastic homogenization of operators in divergence form has witnessed important progress. Our goal is to go beyond the error bound to analyze statistical fluctuations around the homogenized limit. We prove a pointwise two-scale expansion and a large scale central limit theorem for the solution. The approach is probabilistic. The main ingredients include the Kipnis-Varadhan method applied to symmetric diffusion in random environment, a quantitative martingale central limit theorem, the Helffer-Sjostrand covariance representation and Stein's method. This is joint work with Jean-Christophe Mourrat.

## Gerandy Brito : Alons conjecture in random bipartite biregular graphs with applications.

- Probability ( 177 Views )This talk concerns to spectral gap in random regular graphs. We prove that almost all bipartite biregular graphs are almost Ramanujan by providing a tight upper bound for the second eigenvalue of its adjacency operator. The proof relies on a technique introduced recently by Massoullie, which we developed for random regular graphs. The same analysis allow us to recover hidden communities in random networks via spectral algorithms.

## Ivan Corwin : Brownian Gibbs line ensembles.

- Probability ( 167 Views )The Airy line ensemble arises in scaling limits of growth models, directed polymers, random matrix theory, tiling problems and non-intersecting line ensembles. This talk will mainly focus on the "non-intersecting Brownian Gibbs property" for this infinite ensemble of lines. Roughly speaking, the measure on lines is invariant under resampling a given curve on an interval according to a Brownian Bridge conditioned to not intersect the above of below labeled curves. This property leads to the proof of a number of previously conjectured results about the top line of this ensemble. We will also briefly touch on the KPZ line ensemble, which arises as the scaling limit of a diffusion defined by the Doob-h transform of the quantum Toda lattice Hamiltonian. The top labeled curve of this KPZ ensemble is the fixed time solution to the famous Kardar-Parisi-Zhang stochastic PDE. This line ensemble has a "softer" Brownian Gibbs property in which resampled Brownian Bridges may cross the lines above and below, but at exponential energetic cost. This is based on joint work with Alan Hammond.

## Lee Deville : Stochastic dynamics on networks. Emergence of collective behaviors

- Probability ( 164 Views )Dynamical systems defined on networks have applications in many fields in science and engineering. In particular, it is important to understand when networks exhibit synchronous or other types of coherent collective behaviors. Other questions include whether such coherent behavior is stable with respect to random perturbation, or how to described the detailed structure of this behavior during its evolution. We will examine the mathematical challenges of understanding these questions for networked dynamical systems with a particular focus on the dynamics of the Kuramoto oscillator model.