## Nicolas Zygouras : Pinning-depinning transition in Random Polymers

- Probability ( 206 Views )Random Polymers are modeled as a one dimensional random walk (S_n), with excursion length distribution P(S_1 = n) = \phi(n)/n^\alpha, \alpha > 1 and \phi(n) a slowly varying function. The polymer gets a random reward whenever it visits or crosses an interface. The random rewards are realised as a sequence of i.i.d. variables (\omega_n). Depending on the relation between the mean value of the disorder \omega_n and the temperature, the polymer might prefer to stick to the interface (pinnings) or undergo a long excursion away from it (depinning). In this talk we will review some aspects of random polymer models. We will also discuss in more detail the pinning-depinning transition of the `Pinning' model and prove its annealed and quenched critical points are distinct. This is joint work with Ken Alexander.

## David Andeerson : Stochastic models of biochemical reaction systems

- Probability ( 181 Views )I will present a tutorial on the mathematical models utilized in molecular biology. I will begin with an introduction to the usual stochastic and deterministic models, and then introduce terminology and results from chemical reaction network theory. I will end by presenting the deficiency zero theorem in both the deterministic and stochastic settings.

## David Herzog : Hypocoercivity for Langevin dynamics

- Probability ( 167 Views )This will be the last in his sequence of an introductory lecture on Hypocoercivity for Langevin dynamics. For those who have not attended the previous lectures and are familiar with Langevin dynamics, the talk should be accessible. We will continue our discussion on convergence to equilibrium for second-order Langevin dynamics using the Poincare approach. We'll recap convergence in H^1(\mu) and then we'll talk about the direct L^2(\mu) method of Dolbeault, Mouhot, and Schmeiser, also called the DMS approach.

## Leonid Koralov : An Inverse Problem for Gibbs Fields

- Probability ( 167 Views )It is well known that for a regular stable potential of pair interaction and a small value of activity one can define the corresponding Gibbs field (a measure on the space of configurations of points in $\mathbb{Z}^d$ or $\mathbb{R}^d$). We consider a converse problem. Namely, we show that for a sufficiently small constant $\overline{\rho}_1$ and a sufficiently small function $\overline{\rho}_2(x)$, $x \in \mathbb{Z}^d$ or $\mathbb{R}^d$, there exist a hard core pair potential, and a value of activity, such that $\overline{\rho}_1$ is the density and $\overline{\rho}_2$ is the pair correlation function of the corresponding Gibbs field.

## Mohammad Ghomi : Topology of Locally convex hypersurfaces with prescribed boundary

- Probability ( 166 Views )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.

## Louigi Addario-Berry : Probabilistic aspects of minimum spanning trees

- Probability ( 160 Views )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.

## Ronnie Sircar : Games with Exhaustible Resources

- Probability ( 160 Views )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).

## Santosh Vempala : Logconcave Random Graphs

- Probability ( 156 Views )We propose the following model of a random graph on $n$ vertices. Let F be a distribution in R_+^{n(n-1)/2} with a coordinate for every pair ij with 1 \le i,j \le n. Then G_{F,p} is the distribution on graphs with n vertices obtained by picking a random point X from F and defining a graph on n vertices whose edges are pairs ij for which X_{ij} \le p. The standard Erd\H{o}s-R\'{e}nyi model is the special case when F is uniform on the 0-1 unit cube. We determine basic properties such as the connectivity threshold for quite general distributions. We also consider cases where the X_{ij} are the edge weights in some random instance of a combinatorial optimization problem. By choosing suitable distributions, we can capture random graphs with interesting properties such as triangle-free random graphs and weighted random graphs with bounded total weight. This is joint work with Alan Frieze (CMU) and Juan Vera (Waterloo). The talk will be self-contained and no prior knowledge of random graphs is assumed.

## Jeremy Quastel : The effect of noise on KPP traveling fronts

- Probability ( 155 Views )It was noticed experimentally in the late 90's that the speeds of traveling fronts in microscopic systems approximating the KPP equation converge unusually slowly to their continuum values. Brunet and Derrida made a very precise conjecture for the basic model equation, which is the KPP equation perturbed by white noise. We will explain the conjecture and sketch the main ideas of the proof. This is joint work with Carl Mueller and Leonid Mytnik.

## Krishna Athreya : Preferential attachment random graphs with general weight function

- Probability ( 155 Views )Consider a network of sites growing over time such that at step n a newcomer chooses a vertex from the existing vertices with probability proportional to a function of the degree of that vertex, i.e., the number of other vertices that this vertex is connected to. This is called a preferential attachment random graph. The objects of interest are the growth rates for the growth of the degree for each vertex with n and the behavior of the empirical distribution of the degrees. In this talk we will consider three cases: the weight function w(.) is superlinear, linear, and sublinear. Using recently obtained limit theorems for the growth rates of a pure birth continuous time Markov chains and an embedding of the discrete time graph sequence in a sequence of continuous time pure birth Markov chains, we establish a number of results for all the three cases. We show that the much discussed power law growth of the degrees and the power law decay of the limiting degree distribution hold only in the linear case, i.e., when w(.) is linear

## Mark Huber : Conditions for Parallel and Simulated Tempering to be fast or slow

- Probability ( 153 Views )In Markov chain Monte Carlo, a Markov chain is constructed whose limiting distribution is equal to some target distribution. While it is easy to build such chains, for some distributions the standard constructions can take exponentially long to come near that limit, making the chain torpidly mixing. When the limit is reached in polynomial time, the chain is rapidly mixing. Tempering is a technique designed to speed up the convergence of Markov chains by adding an extra temperature parameter that acts to smooth out the target distribution. In this talk I will present joint work with Dawn Woodard (Cornell) and Scott Schmidler (Duke) that give sufficient conditions for a tempering chain to be torpidly mixing, and a related (but different) set of conditions for the chain to be rapidly mixing.

## Jan Wehr : Entanglement percolation in quantum networks

- Probability ( 153 Views )Reliable information transmission between two sites of a network naturally leads to a percolation problem. When the information to be transmitted is quantum an exciting possibility arises: transform the network performing well chosen measurements to enhance the transmission probability. This idea, introduced recently by Acin, Cirac and Lewenstein is now systematically and successfully applied to a variety of two-dimensional networks, but open questions show that a complete theory is missing. The talk will involve some quanta, some network geometry, some percolation and, hopefully, some fun. No knowledge of quantum theory or percolation theory is assumed. Graduate students are encouraged to attend.

## James Gleeson : Determinants of meme popularity

- Probability ( 150 Views )We will describe and analyze some models of the spread of information on Twitter. The competition between memes fro the limited resource of user attention leads to critical branching processes, and resulting heavy tailed distributions for meme popularity.

## Davar Khoshnevisan : Nonlinear Stochastic Heat Equations: Existence, Growth, and Intermittency

- Probability ( 149 Views )We introduce some recent advances in the study of nonlinear stochastic heat equations, and related stochastic PDEs. Special attention will be paid to the local structure of the solution. In particular, we show that, frequently, the solution exhibits a form of intermittency. Time permitting, we discuss related connections to classical potential theory and mathematical physics as well.

## Asaf Nachmias : The Alexander-Orbach Conjecture Holds in High Dimensions

- Probability ( 137 Views )It is known that the simple random walk on the unique infinite cluster of supercritical percolation on Z^d diffuses in the same way it does on the original lattice. In critical percolation, however, the behavior of the random walk changes drastically. The infinite incipient cluster (IIC) of percolation on Z^d can be thought of as the critical percolation cluster conditioned on being infinite. Alexander and Orbach (1982) conjectured that the spectral dimension of the IIC is 4/3. This means that the probability of an n-step random walk to return to its starting point scales like n^{-2/3} (in particular, the walk is recurrent). In this work we prove this conjecture when d>18; that is, where the lace-expansion estimates hold. Joint work with Gady Kozma.

## Leonid Bogachev : Gaussian fluctuations for Plancherel partitions

- Probability ( 127 Views )The limit shape of Young diagrams under the Plancherel measure was found by Vershik & Kerov (1977) and Logan & Shepp (1977). We obtain a central limit theorem for fluctuations of Young diagrams in the bulk of the partition 'spectrum'. More specifically, under a suitable (logarithmic) normalization, the corresponding random process converges (in the FDD sense) to a Gaussian process with independent values. We also discuss a link with an earlier result by Kerov (1993) on the convergence to a generalized Gaussian process. The proof is based on poissonization of the Plancherel measure and an application of a general central limit theorem for determinantal point processes. (Joint work with Zhonggen Su.) (see more details hear.

## Eric Foxall : The compulsive gambler with allowances

- Probability ( 126 Views )We consider a process in which a finite set of n agents continually receive a 1 dollar allowance and gamble their fortunes, all in, with one another at a constant rate. This is a variation on the existing compulsive gambler process; in that process, initial fortunes are prescribed and no further allowances are given out. For our process, we find that after some time the distribution of wealth settles into a pattern in which most people have only a few dollars, a few are very wealthy, and a single person possesses most of the cash currently present in the population. In addition, eventually the only way to attain first rank is by winning a bet against the current champion. Moreover, if agents play a fair game, i.e., the probability of winning a bet is proportional to the players' fortunes, the title of champion is assumed by every player infinitely often, although it changes less and less frequently as time goes on. Finally, by examining the process from both the perspective of typical fortune, and that of large fortune, we can go one step further and obtain two distinct limiting processes as n --> infty, with each one admitting a detailed description of its dynamics.

## Davar Khoshnevisan : A macroscopic multifractal analysis of parabolic stochastic PDEs

- Probability ( 122 Views )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).

## Elena Kosygina : Excited random walks

- Probability ( 119 Views )The idea behind excited random walks (ERWs), roughly speaking, is to take a well-known underlying process (such as, for example, simple symmetric random walk (SSRW)) and modify its transition probabilities for the "first few" visits to every site of the state space. These modifications can be deterministic or random. The resulting process is not markovian, and its properties can be very different from those of the underlying process. I shall give a short review of some of the known results for ERW (with SSRW as underlying process) on the d-dimensional integer lattice and then concentrate on a specific model for d=1. For this model we can give a complete picture including functional limit theorems.

## Ross Pinsky : Transience, Recurrence and the Speed of a Random Walk in a Site-Based Feedback Environment

- Probability ( 119 Views )We study a random walk on the integers Z which evolves in a dynamic environment determined by its own trajectory. Sites flip back and forth between two modes, p and q. R consecutive right jumps from a site in the q-mode are required to switch it to the p-mode, and L consecutive left jumps from a site in the p-mode are required to switch it to the q-mode. From a site in the p-mode the walk jumps right with probability p and left with probability (1-p), while from a site in the q-mode these probabilities are q and (1-q). We prove a sharp cutoff for right/left transience of the random walk in terms of an explicit function of the parameters $\alpha = \alpha(p,q,R,L)$. For $\alpha > 1/2$ the walk is transient to $+\infty$ for any initial environment, whereas for $\alpha < 1/2$ the walk is transient to $-\infty$ for any initial environment. In the critical case, $\alpha = 1/2$, the situation is more complicated and the behavior of the walk depends on the initial environment. We are able to give a characterization of transience/recurrence in many instances, including when either R=1 or L=1 and when R=L=2. In the noncritical case, we also show that the walk has positive speed, and in some situations are able to give an explicit formula for this speed. This is joint work with my former post-doc, Nick Travers, now at Indiana University.

## Sourav Chatterjee : Superconcentration

- Probability ( 119 Views )We introduce the term `superconcentration' to describe the phenomenon when a function of a Gaussian random field exhibits a far stronger concentration than predicted by classical concentration of measure. We show that when superconcentration happens, the field becomes chaotic under small perturbations and a `multiple valley picture' emerges. Conversely, chaos implies superconcentration. While a few notable examples of superconcentrated functions already exist, e.g. the largest eigenvalue of a GUE matrix, we show that the phenomenon is widespread in physical models; for example, superconcentration is present in the Sherrington-Kirkpatrick model of spin glasses, directed polymers in random environment, the Gaussian free field and the Kauffman-Levin model of evolutionary biology. As a consequence we resolve the long-standing physics conjectures of disorder-chaos and multiple valleys in the Sherrington-Kirkpatrick model, which is one of the focal points of this talk.

## Peter Bubenik : Multivariate topological data analysis

- Probability ( 119 Views )I will present results on constructing an estimator of a function on a compact manifold for the purpose of recovering its "topology". What this means will be explained in detail. The talk will conclude with an application to brain imaging.

## Carl Mueller : Nonuniqueness for some stochastic PDE

- Probability ( 119 Views )The superprocess or Dawson-Watanabe process is one of the most intensively studied stochastic processes of the last quarter century. It arises as a limit of population processes, and includes information about the physical location of individuals. Usually the superprocess is measure valued, but In one dimension it has a density that satisfies a parabolic stochastic PDE. For a long time uniqueness for this equation was unknown. In joint work with Barlow, Mytnik, and Perkins, we show that nonuniquess holds for the superprocess equation and several related equations.

## Shish Luo : Multiscale evolutionary dynamics: A measure-valued process perspective

- Probability ( 117 Views )Evolution by natural selection can act at multiple biological levels, often in opposing directions. This is particularly the case for pathogen evolution, which occurs both within the host it infects and via transmission between hosts, and for the evolution of cooperative behavior, where individually advantageous strategies are disadvantageous at the group level. In mathematical terms, these are multiscale systems characterized by stochasticity at each scale. We show how a simple and natural formulation of this can be viewed as a ball-and-urn (measure-valued) process. This equivalent process has very nice mathematical properties, namely it converges weakly to either (i) the solution of an analytically tractable integro-partial differential equation or (ii) a Fleming-Viot process. We can then study properties of these limiting objects to infer general properties of multilevel selection.

## Ivana Bozic : Dynamics of cancer in response to targeted combination therapy

- Probability ( 117 Views )A link to the paper can be found on her web page. In solid tumors, targeted treatments can lead to dramatic regressions, but responses are often short-lived because resistant cancer cells arise. The major strategy proposed for overcoming resistance is combination therapy. We present a mathematical model describing the evolutionary dynamics of lesions in response to treatment. We first studied 20 melanoma patients receiving vemurafenib. We then applied our model to an independent set of pancreatic, colorectal, and melanoma cancer patients with metastatic disease. We find that dual therapy results in long-term disease control for most patients, if there are no single mutations that cause cross-resistance to both drugs; in patients with large disease burden, triple therapy is needed. We also find that simultaneous therapy with two drugs is much more effective than sequential therapy. Our results provide realistic expectations for the efficacy of new drug combinations and inform the design of trials for new cancer therapeutics.

## Tobias Johnson : Galton-Watson fixed points, tree automata, and interpretations

- Probability ( 116 Views )Consider a set of trees such that a tree belongs to the set if and only if at least two of its root child subtrees do. One example is the set of trees that contain an infinite binary tree starting at the root. Another example is the empty set. Are there any other sets satisfying this property other than trivial modifications of these? I'll demonstrate that the answer is no, in the sense that any other such set of trees differs from one of these by a negligible set under a Galton-Watson measure on trees, resolving an open question of Joel Spencer's. This follows from a theorem that allows us to answer questions of this sort in general. All of this is part of a bigger project to understand the logic of Galton-Watson trees, which I'll tell you more about. Joint work with Moumanti Podder and Fiona Skerman.

## Rick Durrett : Evolutionary Games on the Torus

- Probability ( 116 Views )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.

## Gerard Letac : Dirichlet curve of a probability in \(R^d\)

- Probability ( 115 Views )A Dirichlet random probability \(P_t\) on \(\mathbb{R}^d\) of intensity \(t\) and governed by the probability \(\alpha\) is such that for any partition \( (A_0,\ldots,A_n)\) of \(\mathbb{R}^d\) the random variable \( (P_t(A_0),\ldots,P_t(A_n))\) is Dirichlet distributed with parameters \( (t\alpha(A_0),\ldots,t\alpha(A_n).\) If \(\mu(t\alpha)\) is the distribution of \(X_t=\int xP_t(dx),\) the Dirichlet curve is the map \(t\mapsto \mu(t\alpha)\). Its study raises challenging problems and explicit computations are rare. We prove that if \(\lim_{t\to\infty}\mu(t\alpha)\) exists, it is a Cauchy or Dirac distribution on \(\mathbb{R}^d\). If \(\alpha\) has an expectation we prove that \(t\mapsto \int \psi(x)\mu(t\alpha)(dx)\) is decreasing for any positive convex function \(\psi\) on \(\mathbb{R}^d.\) In other terms the Dirichlet curve decreases in the Strassen order. This is joint work with Mauro Piccioni.