We study evolutionary games on the torus with N points in dimensions \(d\ge 3\) with matrices of the form \(\bar G = {\bf 1} + w G\), where 1 is a matrix that consists of all 1's, and w is small. We show that there are three weak selection regimes (i) \(w \gg N^{-2/d}\), (ii) \(N^{-2/d} \gg w \gg N^{-1}\), and (iii) there is a mutation rate \(\mu\) so that \(\mu \gg N^{-1}\) and \(\mu \gg w\) where in the last case we have introduced a mutation rate \(\mu\) to make it nontrivial. In the first and second regimes the rescaled process converges to a PDE and an ODE respectively. In the third, which is the classical weak selection regime of population genetics, we give a new derivation of Tarnita's formula which describes how the equilibrium frequencies are shifted away from uniform due to the spatial structure.

The sandpile group of a finite graph is an abelian group that is defined using the graph Laplacian. I will describe a natural random walk on this group. The main questions are: what is the mixing time of the sandpile random walk, and how is it affected by the geometry of the underlying graph? These questions can sometimes be answered even if the actual group is unknown. I will present an explicit characterization of the eigenvalues and eigenfunctions of the sandpile walk, and demonstrate an inverse relationship between the spectral gaps of the sandpile walk and the simple random walk on the underlying graph. This is joint work with Lionel Levine and John Pike.

Gaussian processes are employed to model spatially varying errors in various stochastic systems. In this talk, we consider the analysis of the extreme behaviors and the rare-event simulation problems for such systems. In particular, the topic covers various nonlinear functionals of Gaussian processes including the supremum norm and integral of convex functions. We present the asymptotic results and the efficient simulation algorithms for the associated rare-event probabilities.

This is joint work with David Herzog, Kevin McGoff, David Sivakoff and Evan Myers. We develop a dynamic random graph model to capture the heterogeneous structure of adolescent sexual networks. Coupled to a stochastic model of infection with the human papillomavirus (HPV), the network model is used to investigate the effect of different HPV vaccination strategies. The results obtained with the stochastic agent-based model are confirmed and extended by means of a deterministic mean-field model amenable to rigorous analysis. Using parameter values reflecting the current situation in the US, we show that for a large class of cost-benefit measures it is more effective to start implementing male-vaccination than to extend female vaccination further. In view of the stagnating female and low male coverage in the US, our results demonstrate the necessity for empirical assessment of coverage-dependent marginal administration costs of the vaccine.

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 Borel’ s school), but the main subject will be the random zeta – function (which was introduced implicitly by Cramer) and its generalizations. We will show that “true primes are not truly random “, since zeta – functions for the random “pseudo-primes” (in the spirit of Cramer) have no analytic continuation through the critical line Re (z) = 1/2.

In this talk I will revisit the idea of viewing the Bayesian update as a variational problem. I will show how the variational interpretation is helpful in establishing the convergence of Bayesian models, and in defining and analysing diffusion processes that have the posterior as invariant measure. I will illustrate the former by proving a consistency result for graph-based Bayesian semi-supervised learning in the large unlabelled data-set regime, and the latter by suggesting new optimality criteria for the choice of metric in Riemannian MCMC.

The efficient delivery of proteins and other molecular products to their correct location within a cell (intracellular transport) is of fundamental importance to normal cellular function and development. Moreover, the breakdown of intracellular transport is a major contributing factor to many degenerative diseases. There are two major types of transport. (I) Passive diffusion within the cytosol or the surrounding plasma membrane of the cell. Since the aqueous environment (cytosol) of a cell is highly viscous at the length and velocity scales of macromolecules (low Reynolds number), a diffusing particle can be treated as an overdamped Brownian particle where inertial effects are ignored. (II) Active motor-driven transport along polymerized filaments such as microtubules and F-actin that comprise the cytoskeleton. At appropriate length and time scales, active transport can either be modeled as a velocity-jump process or as an advection-diffusion process. In this talk I present various PDE models of active and passive transport within cells. The bulk of the talk will focus on three examples: synaptic democracy and vesicular transport in axons and dendrites; stochastically gated diffusion in bounded domains; cytoneme-based transport of morphogens during embryogenesis. (A cytoneme is a thin actin-rich filament that forms direct contacts between cells and is thought to provide an alternative to diffusion-based morphogen gradient formation.) Other applications include cellular length control, cell polarization, and synaptogenesis in C. elegans.

We will explore the role of demographic stochasticity in triggering extinction events in models of large finite populations. While prior works have focused on large fluctuations from quasi-stationary distributions, we instead consider extinction events occurring before entering a metastable state. Since such extinction events require only slight deviations from the mean-field trajectories, we can derive the approximating extinction probability PDE with a modified Robin-type boundary condition. We then investigate the utility of this approximation by comparing to the Lotka-Volterra model as well as the Lotka-Volterra model with logistic growth.

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.

We will show that the solutions to a large family of stochastic PDEs that behave as the linear heat equation develop large-scale space-time peaks on infinitely-many different scales. We formalize this assertion by appealing to the Barlow-Taylor theory of macroscopic fractals. We will also present some earlier work on fixed-time results for comparison purposes. This talk is based on a paper and a work in progress with Kunwoo Kim (Technion) and Yimin Xiao (Michigan State University).

An open problem in Classical Differential Geometry, posed by S. T. Yau, asks when does a simple closed curve in Euclidean 3-space bound a surface of positive curvature? We will give a survey of recent results related to this problem, including connections with the h-principle, Monge-Ampere equations, and Alexandrov spaces with curvature bounded below. In particular we will discuss joint work with Stephanie Alexander and Jeremy Wong on Topological finiteness theorems for nonnegatively curved surfaces filling a prescribed boundary, which use in part the finiteness and stability theorems of Gromov and Perelman.

Practice talk for Northeast Probability Symposium. This is joint work with Zoe Huang. We will also mention two results from the DOMath summer project.

Let \(S_n\) denote the set of permutations of length \(n\). For a permutation \(\tau \in S_n\) we say \(\tau\) contains a pattern \(\sigma\in S_k\) if there is a subsequence \(i_1 < \cdots < i_k\) such that \(\tau_{i_1} \cdots \tau_{i_k}\) has the the same relative order of \(\sigma\). If \(\tau\) contains no pattern \(\sigma\), we say that \(\tau\) avoids \(\sigma\). We denote the set of \(\sigma\)-avoiding permutations of length \(n\) by \(S_n(\sigma)\). Recently, there has been a number of results that help describe the geometric properties of a uniformly random element in \(S_n(\sigma)\). Many of these geometric properties are related to well-studied random objects that appear in other settings. For example, if \(\sigma \in S_3\), we have that a permutation chosen uniformly in \(S_n(\sigma)\) converges, in some appropriate sense, to Brownian excursion. Furthermore for \(\sigma = 123,312\) or\( 231\), we can describe properties like the number and location of fixed points in terms of Brownian excursion. Larger patterns are much more difficult to understand. Currently even the simplest question, enumeration, is unknown for the pattern \(\sigma = 4231\). However, for the monotone decreasing pattern \(\sigma= (d+1)d\cdots 21\), \(S_n(\sigma)\) can be coupled with a random walk in a cone that, in some appropriate sense, converges to a traceless Dyson Brownian motion.

The goal of this talk is to further a joint project involving Carla Staver, Simon Levin, Rick Durrett, and Ruibo Ma. The puzzle is: why can savannah and forest display stable coexistence when this is not possible in a spatially homogeneous system.

It is a well known fact that velocity grandients in a flow change the dispersion of a passive tracer. One clear manifestation of this phenomenon is that in systems with homogenization type diffusive long time/large scale behavior, the effective diffusivity often differs greatly from the molecular one. An important aspect of these well known result is that they are only valid on timescales much longer than the inverse diffusivity. We are interested in what happens on shorter timescales (subhomogenization regimes) in a family of two-dimensional incompressible periodic flows that consists only of pockets of recirculations essentially acting as traps and infinite flowlines separating these where significant transport is possible. Our approach is to follow the random motion of a tracer particle and show that under certain scaling it resembles time-changed Brownian motions. This shows that while the trajectories are still diffusive, the variance grows differently than linear.

The KPZ equation was introduced in 1986, and has become the default model in physics for random interface growth. It is a member of a large universality class with non-standard fluctuations, including directed random polymers. Even in one dimension, it turned out to be difficult to interpret and analyze mathematically, but at the same time to have a large degree of exact solvability. We will survey the history and recent progress.

Abstract: One of the most dynamic areas of probability theory is the study of the behaviour of discrete optimization problems on random inputs. My talk will focus on the probabilistic analysis of one of the first and foundational combinatorial optimization problems: the minimum spanning tree problem. The structure of a random minimum spanning tree (MST) of a graph G turns out to be intimately linked to the behaviour of critical and near-critical percolation on G. I will describe this connection and some of my results, alone and with coauthors, on the structure, scaling limits, and volume growth of random MSTs. It turns out that, on high-dimensional graphs, random minimum spanning trees are expected to be three-dimensional when viewed intrinsically, and six-dimensional when viewed as embedded objects.

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.

We consider an ensemble of N interacting particles modeled by a system of N stochastic differential equations (SDEs). The coefficients of the SDEs are taken to be such that as N approaches infinity, the system undergoes Kac’s propagation of chaos, and is well-approximated by the solution to a McKean-Vlasov Equation. Rare but possible deviations of the behavior of the particles from this limit may reflect a catastrophe, and computing the probability of such rare events is of high interest in many applications. In this talk, we design an importance sampling scheme which allows us to numerically compute statistics related to these rare events with high accuracy and efficiency for any N. Standard Monte Carlo methods behave exponentially poorly as N increases for such problems. Our scheme is based on subsolutions of a Hamilton-Jacobi-Bellman (HJB) Equation on Wasserstein Space which arises in the theory of mean-field control. This HJB Equation is seen to be connected to the large deviations rate function for the empirical measure on the ensemble of particles. We identify conditions under which our scheme is provably asymptotically optimal in N in the sense of log-efficiency. We also provide evidence, both analytical and numerical, that with sufficient regularity of the solution to the HJB Equation, our scheme can have vanishingly small relative error as N increases.

Self-exciting point processes are simple point processes that have been widely used in neuroscience, sociology, finance and many other fields. In many contexts, self-exciting point processes can model the complex systems in the real world better than the standard Poisson processes. We will discuss the Hawkes process, the most studied self-exciting point process in the literature. We will talk about the limit theorems and asymptotics in different regimes. Extensions to Hawkes processes and other self-exciting point processes will also be discussed.

Abstract:One way to convince ourselves that no cooperation can evolve among defectors is via a simple yet one of the most famous games in all of game theory - the Prisoner’s dilemma (PD) game. The players of this game adopt one of the two strategies: a) a cooperator who pays a cost so that another individual can receive a benefit, or b) a defector who can receive benefits, but it has no cost as it does not deal out any benefits at all. As seen from this formulation, no rational individual would opt to be a cooperator. Yet, we can see cooperation everywhere around us and thus (assuming defectors were here first) there must exist at least one mechanism for its evolution. Nowak (2006, 2012) discusses several of such mechanisms, including the kin selection by which cooperation can spread if the benefits go primarily to genetic relatives. In this talk we will introduce a simple PD-like asymmetric matrix game and show how Hamilton’s rule can easily be recovered. We will also introduce a simple PD-like symmetric matrix game to model the evolution of cooperation via greenbeard mechanism, which can be seen as a special case of kin selection.

The analytic framework for estimating coefficients of a generating function is the same in many variables as in one variable: evaluate Cauchy's integral by manipulating the contour into a "standard" position. That being said, the geometry when dealing with several complex variables can be much more complicated. This talk, drawing on the recent book (with Mark Wilson) of the same title, surveys analytic methods for extracting asymptotics from multivariate generating functions. I will try to give an idea of the main pieces of the puzzle. In particular, I will try to explain in pictures the roles of Morse theory, complex algebraic geometry and hyperbolicity in the asymptotic evaluation of integrals.

We study N-player repeated Cournot competitions that model the determination of price in an oligopoly where firms choose quantities. These are nonzero-sum (ordinary and stochastic) differential games, whose value functions may be characterized by systems of nonlinear Hamilton-Jacobi-Bellman partial differential equations. When the quantity being produced is in finite supply, such as oil, exhaustibility enters as boundary conditions for the PDEs. We analyze the problem when there is an alternative, but expensive, resource (for example solar technology for energy production), and give an asymptotic approximation in the limit of small exhaustibility. We illustrate the two-player problem by numerical solutions, and discuss the impact of limited oil reserves on production and oil prices in the dupoly case. Joint work with Chris Harris (Cambridge University) and Sam Howison (Oxford University).

I will discuss a version of stochastic Loewner evolution with branching introduced in my student Vivian Olsiewski Healey's 2017 thesis. Our main motivation was to find natural conformal processes that embed Aldous' continuum random tree in the upper half plane. Unlike previous attempts that rely on lattice models or conformal welding, our model relies on a careful choice of driving measure in the Loewner evolution and the theory of continuous state branching processes. The most important feature of our model is that it has a very nice scaling limit, where the driving measure is a superprocess.

In this talk, I will discuss a generalized Moran process from the evolutionary game theory. The generalization incorporates arrangement of by graphs and games among individuals. For these additional features, there has been consistent interest in using general spatial structure as a way to explain the ubiquitous game behavior in biological evolutions; the introduction of games leads to technical complications as basic as nonlinearity and asymmetry in the model. The talk will be centered around a seminal finding in the evolutionary game theory that was obtained more than a decade ago. By an advanced mean-field method, it reduces the infinite-dimensional problem of solving for the game fixation probabilities to a one-dimensional diffusion problem in the limit of a large population. The recent mathematical results and some related mathematical methods will be explained.