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Abstract: In this talk, I will be discussing joint work with Nathaniel Butler, Gourab Ray and Yinon Spinka on the behavior of uniformly chosen integer valued 1-lipschitz functions on regular trees, with prescribed boundary conditions on the nth generation. This falls under a much larger umbrella of the study of various gradient/height function models on lattices on which there is a vast collection of literature. In the context of regular trees, it was known that the height function is localized as n becomes large, that is the law at any given vertex is tight. Moreover, the heights have double exponential tail. We provide alternative proofs of this fact, and go further to prove that the heights locally converge in distribution if and only if the degree of the tree is less than or equal to 7. For larger degree, we establish an alternating pattern of the law on even and odd generations. Finally, with certain special boundary conditions, local convergence holds for all degrees.

Abstract: We will discuss extensions of random transpositions to other conjugacy classes. Our main probabilistic result is the total variation cutoff profile for these random walks, under a mild assumption on the number of fixed points. The proof is based on a new method to estimate the characters of symmetric groups, that makes use of exicited diagrams. During the talk we will present the main combinatorial formulas that allow to compute the eigenvalues of such chains, and give ideas on how to estimate them asymptotically. Based on joint work with Sam Olesker-Taylor and Paul Thévenin.

Abstract: we consider the model in which uniform random points are added to the unit interval at a constant intensity and independently vanish each at rate 1. The stationary distribution is a Poisson point process. Our goal is to investigate the time until an atypically large gap appears, in the high-intensity limit. To do so we develop some theory that allows us to compute the hitting time of a rare set in a family of Markov chains in terms of the restriction of the stationary distribution to that set. By studying the stationary distribution, as well as sample paths of the counting processes that describe particle numbers on fixed intervals, we obtain an asymptotic formula for the expected time until the appearance of a large gap. A component of the theory involves generalizing the exponential limit of scaled geometric random variables to the case where the relevant Bernoulli sequence is one-dependent.

Abstract: The mixing time and spectral gap of a random walk on the symmetric group can sometimes be understood in terms of its low dimensional representations (e.g., Aldous' spectral gap conjecture). It turns out that under a mild degree condition involving the step of the group, the same holds for nilpotent groups w.r.t. their one dimensional representations: the spectral gap and the epsilon total variation mixing time of the walk on G are determined by those of the projection of the walk to the abelianization G/[G,G]. We'll discuss some applications concerning the cutoff phenomenon (= abrupt convergence to equilibrium) and the dependence (or lack of!) of the spectral gap and the mixing time on the choice of generators.

As time permits we shall discuss a related result, confirming in the nilpotent setup a conjecture of Aldous and Diaconis concerning the occurrence of cutoff when a diverging number of generators are picked uniformly at random. Joint work with Zoe Huang.

Abstract: A phase transition is a phenomenon in physics, chemistry, and other related fields where the macroscopic behavior of a system changes qualitatively when the tuning parameter (such as temperature) that governs local interactions is varied by a small amount through a critical value. A familiar example of a phase transition is the evaporation of water into steam when the temperature reaches the boiling point.

In statistical mechanics, percolation is a simple model that undergoes a phase transition. At the critical threshold, percolation exhibits fractal properties that prove to be a rich area of research. In this talk, I will give a high-level introduction to the dimension-dependence of critical phenomena for percolation on Euclidean lattices.

Abstract:
The stabilized automorphism group of a dynamical system (X,T) is the group of all self-homeomorphisms of X that commute with some power of T. In this talk, we will describe the stabilized automorphism group of minimal systems. The main result we will prove is that if two minimal systems have isomorphic stabilized automorphism groups and each has at least one non-trivial rational eigenvalue, then the systems have the same rational eigenvalues.

Location: Zoom (link
Meeting ID: 893 7295 9263 Password: 033409)

(Zoom link Meeting ID: 893 7295 9263 Password: 033409)

Abstract: An highly active research area is concerned with finding conditions on sparse sets that ensure the existence of many geometric patterns. I will present some results in this direction connecting Newhouse thickness and its generalizations to higher dimensions, games of Schmidt type and the existence of an abundance of homothetic copies of small sets.

I will discuss a problem that appeared on MathOverflow and its solution by the user “Fedja”. The result we will demonstrate is that if T is subset of the reals with Hausdorff dimension exceeding ½, then there is a positive probability that a standard Brownian Motion has a zero at some time t belonging to T.

The beautiful proof relies on the second moment method, which I will discuss along the way.

Location: Zoom (https://uvic.zoom.us/j/89372959263?pwd=q7IPZKueYMnaAxLaVmlwVqHfW0H0AR.1 Meeting ID: 893 7295 9263 Password: 033409)

Abstract :
In this talk, we consider a notion of stability for solutions of random optimization problems based on small perturbations of the input data and inspired by the technique for proving CLT using Stein's method as illustrated with examples by Chatterjee (2008). This notion of stability is closely related with the size of near-optimal solution sets for those optimization problems. We establish this notion of stability for a number of settings, such as branching random walk, the Sherrington--Kirkpatrick model of mean-field spin glasses, the Edwards--Anderson model of short-range spin glasses, the Wigner and Wishart ensemble of random matrices and combinatorial optimization problems like TSP / MST / MMP on weighted complete graphs and Euclidean spaces.

Location: Zoom (https://uvic.zoom.us/j/89372959263?pwd=q7IPZKueYMnaAxLaVmlwVqHfW0H0AR.1
Meeting ID: 893 7295 9263
Password: 033409)

Abstract: Bounded density shifts are examples of hereditary subshifts. Bounded density shifts are defined by disallowing words whose sum of entries exceeds a value depending on the length of the word. After presenting some examples and reviewing the concepts of topological entropy and measure-theoretic entropy, we will provide sufficient conditions for bounded density shifts to have a unique measure of maximal entropy.

This is a joint work with Felipe García-Ramos and Ronnie Pavlov.

We thank PIMS for their generous contribution which makes this in person event possible.

Abstract: Recent interactions between probability and number theory proved fruitful. In particular, random multiplicative functions (RMFs) are random models for partial sums of the Möbius function and Dirichlet characters, which are related to L-functions like the Riemann zeta function. In this talk, I will present the past, present, and future of RMFs: their history, known results, and current work on the moments and distribution of these partial sums of RMFs, as well as a look at the connections with random matrices that are likely to lead to major breakthroughs in the upcoming years.

We thank PIMS for their generous contribution which makes this in person event possible.

Abstract: This talk will present a novel discretization scheme for Wasserstein gradient flows (i.e. curves of steepest descent for functions of probability measures) based on the successive computation of Schrödinger bridges with equal marginals. In doing so, we develop a small-time approximation of same-marginal Schrödinger bridges using Langevin diffusion. This talk will begin with an introduction to entropic regularized optimal transport, introducing all the notions essential for understanding this scheme. Based on joint work with Medha Agarwal, Zaid Harchaoui, and Soumik Pal.

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Abstract: The classical Peierls argument establishes that percolation on a graph G has a non-trivial (uniformly) percolating phase if G has “not too many small cutsets”. Severo, Tassion, and I have recently proved the converse. Our argument is inspired by an idea from computer science and fits on one page.

Our new approach resolves a conjecture of Babson of Benjamini from 1999 and provides a simpler proof of the celebrated result of Duminil-Copin, Goswami, Raoufi, Severo, and Yadin that percolation on any non-one-dimensional transitive graph undergoes a non-trivial phase transition.

Abstract: The notions of noise sensitivity and stability were recently extended for the voter model, a well-known and studied interactive particle system. In this model, vertices of a graph have opinions that are updated by uniformly selecting edges. We further extend sensitivity and stability results to a different class of perturbations when the nearest neighbor exclusion process is performed in the collection of edge selections. Under a rate depending on the underlying graph, we prove that the consensus opinion of the voter model is “exclusion stable” when the dynamics above run for a short amount of time. This is done by analyzing the expected size of the pivotal set.

We thank PIMS for their generous contribution which makes this in person event possible.

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Abstract: I will present a coupling between a massive planar Gaussian free field (GFF) and a random curve in which the curve can be interpreted as the level of the field. This coupling is constructed by reweighting the law of the standard GFF-SLE_4 coupling. I will then show that in this coupling, the marginal law of the curve is that of a massive version of SLE_4, called massive SLE_4. This law on curves was orignally introduced by Makarov and Smirnov to describe the scaling limit of a massive version of the harmonic explorer.

https://uvic.zoom.us/j/89372959263?pwd=q7IPZKueYMnaAxLaVmlwVqHfW0H0AR.1
Meeting ID: 893 7295 9263
Password: 033409

Abstract: Spin glasses are disordered statistical physics system with both ferromagnetic and anti-ferromagnetic spin interactions. The Gibbs measure belongs to the exponential family with parameters, such as inverse temperature $\beta>0$ and external field $h\in R$. Given a sample from the Gibbs measure of a spin glass model, we study the problem of estimating system parameters. In 2007, Chatterjee first proved that under reasonable conditions, for spin glass models with $h=0$, the maximum pseudo-likelihood estimator for $\beta$ is $\sqrt{N}$-consistent. However, the approach has been restricted to the single parameter estimation setting. The joint estimation of $(\beta,h)$ for spin glasses has remained open. In this talk, I will present a joint work with Wei-Kuo Chen, Arnab Sen, which shows that under some easily verifiable conditions, the bi-variate maximum pseudo-likelihood estimator is indeed jointly $\sqrt{N}$-consistent for a large collection of spin glasses, including the Sherrington-Kirkpatrick model and its diluted variants.

Abstract: In the talk exchangeability appears in two different meanings. In the first part, the determination of the phase diagram of the Curie-Weiss model relies on De Finetti’s Theorem. The Curie-Weiss distribution will be expressed as a random mixture of Bernoulli distributions. The competition between the Gaussian randomness in the CLT of Bernoulli’s and the randomness in their mixture replaces the standard energy-entropy competition. In the second part, we study a mean-field spin model with three- and two-body interactions. In a recent paper by Contucci, Mingione and Osabutey, the equilibrium measure for large volumes was shown to have three pure states, two with opposite magnetization and an unpolarized one with zero magnetization, merging at the critical point.

The aim is to apply the exchangeable pair approach due to Stein to prove (non-uniform) Berry-Esseen bounds, a concentration inequality, Cramér-type moderate deviations and a moderate deviations principle for the suitably rescaled magnetization.

Abstract: Double dimers are superimposition of two perfect matchings. Such superimpositions can be decomposed into disjoint simple loops. The question we address is: as the graphs become large, in a 'typical' double dimer sample, do some of the loops diverge to an infinite loop (or a bi-infinite path)? It is known that in the square lattice, this is not the case. In this talk, I will address the same question, but on planar hyperbolic graphs. (I will not spoil the answer in the abstract! )

The answer involves elements from the theory of hyperbolic geometry, circle packings, spanning trees, random walks and unimodular random graphs.

Click https://uvic.zoom.us/j/83154536954 to start or join a scheduled Zoom meeting.

Abstract: A spanning tree of a finite connected graph G is a connected subgraph of G that includes every vertex and contains no cycles. In this talk we will consider uniformly drawn spanning trees of high-dimensional graphs, and explain why, under appropriate rescaling, they converge in distribution as metric-measure spaces to Aldous’ Brownian CRT. This extends an earlier result of Peres and Revelle (2004) who previously showed a form of finite-dimensional convergence. If time permits, we may also discuss scaling limits of random spanning trees with non-uniform laws. Based on joint works with Asaf Nachmias and Matan Shalev.

Abstract: The directed landscape is a random directed metric on the plane that arises as a scaling limit of a classical metric models in the KPZ universality class. In this talk, we will discuss a functional large deviation principle (LDP) for the entire random metric. Applying the contraction principle, our result yields an LDP for the geodesics in the directed landscape. If time permits, we will also mention certain interesting features of the rate function for the geodesic LDP. Based on a joint work with Duncan Dauvergne and Balint Virag.

Abstract: We analyse the metastable behaviour of the disordered Curie-Weiss-Potts (DCWP) model subject to a Glauber dynamics. The model is a randomly disordered version of the mean-field q-spin Potts model (CWP), where the interaction coefficients between spins are general independent random variables. This model comprises also, e.g., the CWP model on inhomogeneous dense random graphs. We are interested in a comparison of the metastable behaviour of the CWP and the DCWP models, for fixed temperature and the infinite volume limit. We prove the CWP model is metastable and through this prove metastability for the DCWP model.

Then we identify the ratio of the (random) mean time the DCWP model takes to reach the stable phase when it starts from a certain probability distribution on the metastable state (called the last-exit biased distribution) and the (non-random) corresponding quantity for the CWP model. In particular, we obtain the asymptotic tail behaviour and the moments of the ratio of the metastable hitting times of the disordered and annealed systems. Our proof is based on a combination of the potential theoretic approach to metastability and concentration of measure techniques, the later adapted to our particular setup.

Abstract: Feynman’s path integral expresses the probability amplitude of a quantum mechanical system as a “sum of trajectories” of the classical system. Since this integral has not been given a satisfactory mathematical meaning, a widely accepted treatment is M. Kac’s method which starts with the idea of rotating “real time” to “imaginary time.” The corresponding path integrals are stochastic, given by exponential functionals of Brownian motion.

This talk will introduce a Feynman–Kac-type formula given by a non-exponential multiplicative functional of a non-Gaussian process. The formula represents the many-body delta-Bose gas in two dimensions, extending technically the two-body case obtained earlier. To contrast the classical and new, for this seminar, a significant part of the talk will discuss the classical Feynman–Kac formula.

Abstract: The Lorentz gas was originally introduced as a model for the movement of electrons in metals. It consists of a massless point particle (electron) moving through Euclidean space bouncing off a given set of scatterers $\mathcal{S}$ (atoms of the metal) with elastic collisions at the boundaries $\partial \mathcal{S}$. If the set of scatterers is periodic in space, then the quotient system, which is compact, is known as the Sinai billiard. There is a great body of work devoted to Sinai billiards and in many ways their dynamics is well understood. In contrast, very little is known about the behavior of the Lorentz gases with aperiodic configurations of scatterers which model quasicrystals and other low-complexity aperiodic sets. This case is the focus of our joint work with Rodrigo Treviño. We establish some dynamical properties which are common for the periodic and quasiperiodic billiard. We also point out some significant differences between the two. The novelty of our approach is the use of tiling spaces to obtain a compact model of the aperiodic Lorentz gas on the plane.

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Abstract: We discuss ergodic properties of coded shift spaces. A coded shift is defined as a closure of all bi-infinite concatenations of words from a fixed countable generating set. It turns out that many well-known classes of shifts are coded including transitive subshifts of finite type, S-gap shifts, generalized gap shifts, transitive Sofic shifts, Beta shifts, and many more. We derive sufficient conditions for the uniqueness of measures of maximal entropy based on the partition of the coded shift into its sequential set (sequences that are concatenations of generating words) and its residual set (sequences added under the closure). Under these conditions we provide a simple explicit description of the measure. (Joint work with M. Schmoll and C. Wolf)

Abstract: In this talk, I will introduce my recent work on the asymptotic behaviors of the stochastic heat equation (SHE) and the Kardar-Parisi-Zhang (KPZ) equation for spatial dimensions d >== 3. The former describes the dynamics of non-equilibrium growth processes arising in statistical physics, and the latter is connected with the directed polymer model, which describes the evolution of a hydrophilic polymer chain wafting in water. As an application, the asymptotic structures of the partition function and the free energy of the continuous directed polymer will be clarified.

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Abstract: In this talk, I will present a recent work in which two co-authors and myself studied the behaviour of a local algorithm optimizing the weight of a graph. More precisely, the process starts with a given subgraph H of the complete graph with uniform weights and a maximal weight W, and inductively replaces a subgraph of H and of weight less than W by the minimum spanning tree on the corresponding set of vertices. Our main result shows that there is a sharp threshold for W regarding the asymptotic behaviour of this algorithm (i.e. with high probability): if W is less than 1, it is impossible to reach the global minimum spanning tree, whereas it is possible when W is larger than 1. Since this work introduces a new type of local algorithm, I will also present some related open problems. In particular, our results prove when such algorithms can reach the global minimum spanning tree and it is then only natural to ask how fast they can do so when possible. The answer to this question actually relates to efficiently packing sets of uniforms into a special type of partition and leads to a surprisingly difficult open question.

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Abstract: We consider a class of disordered mean-field combinatorial optimization problems, focusing on the Gibbs measure, where the inverse temperature does not vary with the size of the graph and the edge weights are sampled from a general distribution. We prove Central Limit Theorems for the log-partition function, the weight of a typical configuration, and the Gibbs average in both quenched and annealed forms. We also derive quenched Poisson convergence for the size of the intersection of two independent samples, yielding replica symmetry of the model. Applications cover popular models from the literature, such as the Minimal Matching Problem, Traveling Salesman Problem, and Minimal Spanning Tree Problem, on a sequence of deterministic and random dense block graphs of increasing size. Joint work with Grigory Terlov.

Abstract: For a population model that encodes parent-child relations, an ordered representation is a partial or complete labelling of individuals, in order of their descendants’ long-term success in some sense, with respect to which the ancestral structure is more tractable. The two most common types are the lookdown and the spinal decomposition(s), used respectively to study exchangeable models and Markov branching processes. We study the lookdown for an exchangeable model with a fixed, arbitrary sequence of natural numbers, describing population size over time. We give a simple and intuitive construction of the lookdown via the complementary notions of forward and backward neutrality. We discuss its connection to the spinal decomposition in the setting of Galton-Watson trees. We then use the lookdown to give sufficient conditions on the population sequence for the existence of a unique infinite line of descent. For a related but slightly weaker property, takeover, the necessary and sufficient conditions are more easily expressed: infinite time passes on the coalescent time scale. The latter property is also related to the following question of identifiability: under what conditions can some or all of the lookdown labelling be determined by the unlabelled lineages? A reasonably good answer can be obtained by comparing extinction times and relative sizes of lineages.

Abstract: Lifting theorems have played a key role in many recent breakthrough results in diverse areas of theoretical cs and mathematics via simple yet profound connections between these areas and communication complexity. A typical lifting theorem translates the query complexity (a very simple model of computation) of a Boolean function to the communication complexity (a more powerful model) of an associated function obtained by a central operation of Boolean functions known as block-composition by composing with another function known as inner function. Most lifting theorems work for any Boolean function f and depend upon the pseudo-random properties of the inner function g known as the gadget. The main parameter of efficiency in lifting theorems is the input size of the inner function g. Obtaining lifting theorems for constant-sized gadgets would give us breakthrough results and a nearly complete understanding of lifting phenomena; current techniques are far from achieving this goal. The main barrier is the existence of “nice” pseudo-random properties for well-known gadgets when their input length is relatively small compared to the outer function. Recent results have shown that understanding the pseudo-random properties is inherently connected to interesting questions in combinatorics (like the sunflower lemma) and in Boolean function analysis. In this talk, we will go through a high level overview of lifting theorems and the necessary condition that enable lifting phenomenon. We will also see some applications of lifting theorems in diverse areas of theoretical cs and mathematics, achieved via surprising yet simple connections. We will also go over recent advances in understanding the pseudo-random properties that drive the lifting phenomena and important barriers and open problems related to this.

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Abstract: The word complexity function p(n) of a subshift X measures the number of n-letter words appearing in sequences in X, and X is said to have linear complexity if p(n)/n is bounded. It's been known since work of Ferenczi that linear word complexity highly constrains the dynamical behavior of a subshift. In recent work with Darren Creutz, we show that if X is a transitive subshift with limsup p(n)/n < 3/2, then X is measure-theoretically isomorphic to a compact abelian group rotation. On the other hand, limsup p(n)/n = 3/2 can occur even for X measurably weak mixing. Our proofs rely on a substitutive/S-adic decomposition for such subshifts. I'll give some background/history on linear complexity, discuss our results, and will describe several ways in which 3/2 turns out to be a key threshold (for limsup p(n)/n) for several different types of dynamical behavior.

Abstract: I will talk about the `minimal spanning arborescence', a directed version of the Minimal spanning tree. I will explain how this naturally leads to a new type of stochastic process which we call `loop contracting random walk'. I will show how this can be analyzed in the setting of trees. I will finish with some simulations and open questions. Joint work with Arnab Sen.

Abstract: We consider an arbitrary sequence of d*d matrices (A_n) of norm at most 1 and subject them to noiselike perturbations of size epsilon. We show that the perturbed system has simple Lyapunov spectrum (no repeated exponents) and discuss universal bounds on gaps between exponents as a function of d and epsilon.

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Abstract: The random interlacement point process (introduced by Sznitman, generalized by Teixeira) is a Poisson point process on the space of labeled doubly infinite nearest neighbour trajectories modulo time-shift on a transient graph G. We show that the random interlacement point process on any transient transitive graph G is a factor of i.i.d., i.e., it can be constructed from a family of i.i.d. random variables indexed by vertices of the graph via an equivariant measurable map. Our proof uses a variant of the soft local time method (introduced by Popov and Teixeira) to construct the interlacement point process as the almost sure limit of a sequence of finite-length variants of the model with increasing length. We also discuss a more direct method of proving that the interlacement point process is a factor of i.i.d. which works if and only if G is non-unimodular. Based on joint work with Márton Borbényi and Sándor Rokob.

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Abstract. We study a mean-field spin glass model whose coupling distribution has a power-law tail with exponent \alpha \in (0, 2). This is known as Levy spin glasses in literature. Though it is a fully connected model, many of its important characteristics are driven by the presence of strong bonds that have a sparse structure. In this sense, the Levy model sits between the widely studied Sherrington-Kirkpatrick model (with Gaussian couplings) and diluted spin glass models, which are more realistic but harder to understand. In this talk, I will report a number of rigorous results on the Levy model. For example, when 1< \alpha < 2, in the high-temperature regime, we obtain the limit and the fluctuation of the free energy. Also, we can determine the behaviors of the site and bond overlaps at the high temperature. Furthermore, we establish a variational formula of the limiting free energy that holds at any temperature. Interestingly, when 0< \alpha < 1, the effect of the strong bonds becomes more pronounced, which significantly changes the behavior of the model. For example, the free energy requires a different normalization (N^{1/\alpha} vs N), and its limit has a simple description via a Poisson point process at any temperature.

This is a joint work with Wei-Kuo Chen and Heejune Kim.

Abstract: In this talk I will be speaking about the dimer model sampled on a general Riemann surface. Dimer model on a graph consists of sampling a random perfect matching with the probability proportional to the product of edge weights. In the case when the graph is planar, perfect matchings are in correspondence with their height functions defined on faces of the graph. Given a sequence of graphs approximating a given planar domain in a small mesh size, one can ask whether the underlying sequence of dimer height functions has a scaling limit and how to describe it. The landmark result of Kenyon asserts that in the case of a simply-connected domain approximated by Temperley polygons on the square grid the fluctuations of the height functions around their means converge to the (properly normalized) Gaussian free field in the target domain. The same result in the case of general Temperley graphs was established by Berestycki, Laslier and Ray 15 years later.

The dimer height function can still be defined when the graph is not planar, but is embedded into a general Riemann surface. In this case the height function becomes additively multivalued and is expected to converge to the compactified free field on the surface in the scaling limit. Recently, this problem was studied by Berestycki, Laslier and Ray in the case of general Temperley graphs embedded into the Riemann surface. Using soft probabilistic methods they proved the scaling limit exists, is conformally invariant and does not depend on a particular sequence of graphs. However, its identification with the compactified free field was missing. My goal is to fill this gap by studying the same problem from the perspective of discrete complex analysis. For this I consider graphs embedded into locally flat Riemann surfaces with conical singularities and satisfying certain geometric assumptions with respect to the local Euclidean structure. Using various analytic methods (both discrete and continuous counterparts are non-trivial) I obtain convergence to the compactified free field when the Riemann surface is chosen generically. Moreover, I am able to prove that the tightness of the underlying height fluctuations always implies their convergence to the compactified free field.

Abstract: The Liouville quantum gravity (LQG) surface, formally defined as a 2-dimensional Riemannian manifold with conformal factor being the exponentiation of a Gaussian free field, is closely related to random planar geometry as well as scaling limits of models from statistical mechanics. In this talk we consider the bounded, simply connected setting, and explain the Weyl's law for the asymptotics of the eigenvalues of the (random) Laplace-Beltrami operator. We also discuss a few open problems about the spectral geometry of LQG. This is a joint work with Nathanael Berestycki.

Abstract: The homomorphism density of a graph H in a graph G is the probability that a random function from the vertex set of H to the vertex set of G is a graph homomorphism. A natural question studied by Leontovich and Sidorenko back in the 80s and 90s is: given two trees, say H and T, under what conditions does the homomorphism density function of H dominate the homomorphism density function of T? We apply a beautiful information-theoretic approach of Kopparty and Rossman to reduce this problem to solving a particular linear program. We then use this perspective to answer the question for various pairs H and T of trees. Roughly speaking, short bushy trees tend to be more numerous than tall skinny ones, but there are exceptions. Joint work with Natalie Behague, Gabriel Crudele and Lina M. Simbaqueba.

Abstract: Recent advances in single cell sequencing has led to the availability of gene expression data. We model the trajectories of cells in gene expression space using a diffusion and gradient drift process that also undergoes branching. In this talk, we show that one can recover the law of this process given time marginals using an entropy minimization problem. The optimization is over the set of all processes with the same branching mechanism and given marginals. We prove that among these the one which minimizes the relative entropy with respect to a reference branching Brownian motion is the ground truth. We conclude this talk with a general result showing that the solution to a entropy minimization problem with respect to a Markov reference measure with given marginals is also Markov.

Abstract: The random connection model (RCM) is a random graph model where the vertices are given by a Poisson point process with a given intensity, $\lambda>0$, and the edges exist independently with a probability that depends upon the relative positions of the two vertices in question. A standard example would be the Gilbert disc model. As we vary $\lambda$, we observe a percolation phase transition at a critical intensity $\lambda_c>0$. Finding $\lambda_c$ is only possible in very exceptional cases, so here we investigate a high-dimension asymptotic expansion for the critical intensity that applies for a great variety of RCMs. This is based on arXiv:2309.08830 with Markus Heydenreich (Universität Augsburg).

Abstract: I will introduce the Monomer-Dimer model, a Gibbs probability measure on the space of (not necessarily perfect) weighted matchings on a graph. I will recap some known results of Heilmann and Lieb which establish the absence of a phase transition. I will then describe how many of these results carry over when we consider iid random weights to the edges and vertices. In certain special cases, we show how laws of large numbers and central limit theorems can be established for partition function as well as statistics such as the number of unpaired vertices.

Flat Wieler solenoids are inverse limits of locally expanding affine maps. They appear in several contexts in dynamical systems, for example as tiling spaces of self-similar tilings, and they are very interesting because they are not manifolds but have an interesting local product structure. I will talk about several geometric and dynamical properties of these spaces, emphasizing the interplay between intrinsic topological properties and renormalization. This talk is for everyone as no prior knowledge of inverse limits will be assumed.

Abstract: Probabilistic zero-forcing can be thought of as a model for rumour spreading, where a person is more likely to spread a rumour if several of their friends already believe it . We start with a graph that has one infected vertex. At each time step an infected vertex infects an uninfected neighbour with probability proportional to how many of its neighbours are already infected. I will focus in this talk on probabilistic zero-forcing on hypercubes and grids, and demonstrate tight bounds on how long it takes for every vertex to be infected (asymptotically almost surely). This is joint work with Trent Marbach and Pawel Pralat.

Abstract: A momentous legacy of twentieth-century mathematics is the realisation that deterministically evolving systems frequently exhibit, when observed for sufficiently extended periods of time, a statistical behaviour akin to the limiting behaviour of independent random variables. We shall explore a geometric incarnation of this surprising phenomenon, overviewing various kinds of statistical limit theorems for the free motion of a particle on a negatively curved surface. In order to emphasise the richness of possible asymptotic behaviours, as well as the variety of sources of randomness, we will further compare the free-motion dynamics with a closely related evolution on the same phase space, known as the horocycle flow.

Abstract: Reproductive value is the relative expected number of off-springs produced by an individual in its remaining lifetime. It is also an invariant function for population processes with birth and death rates independent of the time except in cases when they are periodic. In this study, we prove that the limiting ratio of the survival probability of two branching processes starting from distinct state-time positions is equal to the relative reproductive value. Moreover, we developed a method to obtain the reproductive value for continuous time general branching population models with state-dependent rates and a renewal state. We are using size-biased birth rates for constructing the spinal representation of the branching process, which establishes the relation between survival probability and the Martingale of the process. Further, we provide sufficient conditions for a successful coupling to compare the survival probabilities of two branching population models.

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joint work with E. Lesigne, A. Quas, J. Rosenblatt.

Abstract: Let S= (s_1<s_2<\dots) be a strictly increasing sequence of positive integers. We say S is good if for every real \alpha the sequence

\left(\frac1N\sum_{n\le N}e^{2\pi is_n\alpha}\right)_N

of complex numbers is convergent. Equivalently, the sequence S is good if for every real \alpha the sequence (s_n\alpha) has asymptotic distribution modulo 1. We are interested in finding out what the limit probability measure

\mu_{S,\alpha}=\lim_N \frac1N\sum_{n\le N}\delta_{s_n\alpha}

can be. It turns out that for an irrational \alpha the limit measure must be continuous. So now the main question is can it be any continuous measure? An affirmative answer would also affirm Furstenberg's x2x3 conjecture. What we can prove is that the limit measure can be any absolutely continuous measure.

Abstract: I will describe the large-scale behavior of a random growth model (Internal DLA) on random planar maps which approximate a random fractal surface embedded in the plane (Liouville quantum gravity, LQG). No prior knowledge of these objects will be assumed. Joint work with Ewain Gwynne.

Abstract: We introduce a model of a controlled random process. In this model, the vertices of a hypergraph are ordered randomly and then revealed, one by one, to an algorithm. The algorithm must decide, immediately and irrevocably, whether to keep each observed vertex. Given the total number of observed vertices ("time"), the algorithm's goal is to succeed - asymptotically almost surely - in completing a hyperedge by keeping ("purchasing") the smallest possible number of vertices.

We analyse this model in the context of random graph processes, where the corresponding hypergraph defines a natural graph property, such as minimum degree, connectivity, Hamiltonicity and the containment of fixed-size subgraphs.

Joint work with Alan Frieze and Michael Krivelevich.

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Abstract: The Perceptron model was proposed as early as the 1950's as a toy model of a one-layer neural network. The basic model consists of a set of solutions (either the Hamming cube or the sphere of dimension n) and a set of constraints given by independent n-dimensional Gaussian vectors. The constraints are that the inner product of a solution vector with each constraint vector scaled by sqrt{n} must lie in some interval on the real line. Probabilistic questions about the model include the satisfiability threshold (or the "storage capacity") and questions about the typical structure of the solution space. Algorithmic questions include the tractability of finding a solution (the learning problem in the neural network interpretation). I will describe the model, the main problems, and recent progress.

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Abstract: It is common in probability theory and statistics to study distributional convergences of sums of random variables conditioned on another such sum. In this talk I will present a novel approach using Stein’s method for exchangeable pairs that allows to derive a conditional central limit theorem of the form $(X_n|Y_n = k)$ with explicit rate of convergence as well as its extensions to a multidimensional setting. We will apply these results to particular models including pattern counts in a random binary sequence and subgraph counts in Erdös-Rényi random graph. This talk is based on joint work with Partha S. Dey.

Abstract: Take an Ising model with very low temperature. What is the largest p such that the Ising model dominates Bernoulli percolation with parameter p ? We will show that the answer to this question depends drastically on the geometry of the graph. We also obtain similar results for for two Ising models at very low, but close temperatures. A process is a finitary factor of iid if it can be written as a measurable and equivariant function of an iid process. As an application of the domination results, we show that the very low temperature Ising model on a nonamenable graph is a finitary factor of iid. This is in stark contrast with the amenable setting, where it is known through a celebrated result of Van Den Berg and Steif that the low temperature Ising model is not a finitary factor of iid. Joint work with Yinon Spinka.

We provide a brief motivational overview of recent developments of extensions of stochastics tools to deal with uncertainty. These are: Peng's nonlinear expectations and Ito's calculus without probabilities. We then describe a non-probabilistic version of a supermartingale theory closely motivated by financial considerations of no-arbitrage. The basic object replacing the classical filtered probability space is a structured trajectory set which allows the definition of conditional outer integrals as well as null sets. The conditional outer integrals are non linear functionals that allow to circumvent the linearity of the classical conditional expectations in proofs and definitions. Integrability notions emerge in our setting through non-classical conditional integral operators that lead to the special case of non-probabilistic martingales.

One can define non-probabilistic supermartingales and prove analogous of classical results like: Doob's optional sampling theorem, Dubin's upcrossing inequalities and Doob's a.e. convergence for non-negative supermartingales. All constructions and results have a hedging and superhedging interpretation and there is a direct way in which the new results generalize the classical case. Null sets appearing in the results have a financial interpretation and are handled in a more concrete way than in the classical theory.

Abstract: Many systems whose physics is generally well understood are modeled with differential equations. However, many of these differential equations have the property that they are sensitive to the choice of initial conditions. If one instead has snapshots of a system, i.e. data, one can make a more educated guess at the true state by incorporating the data via data assimilation. Many of the most popular data assimilation methods were developed for general physical systems. However, in the context of fluids, data assimilation works better than would be anticipated for a general physical system. In particular, turbulent fluid flows have been proven to have the property that, given enough perfect observations, one can recover the full state irrespective of the choice of initial condition. This property is surprisingly unique to turbulent fluid flows, a consequence of their finite dimensionality. Unfortunately, the development of data assimilation methods has not actively take this finite dimensionality into account, as it has been developed independent of a considered system. This will be an informal presentation of what I currently understand with regards to data assimilation, both statistical and continuous methods. In particular, I will highlight the difficulties in equating the existing methods, and where these difficulties are presenting themselves in current research directions.

Abstract: Schrödinger operators with delta potentials are of longstanding interest for admitting solutions expressible in closed analytic forms. By some duality, these operators also receive new interest for solving moments of the continuum directed random polymers and the Kardar–Parisi–Zhang equation for interface growth. Along with a review of the above background of up to 3D, this talk will discuss a Feynman–Kac-type path integral representation for a standard model of Schrödinger operator in 2D with delta potential. In particular, it will be explained why this representation falls outside the scope of standard Feynman–Kac formulas and is probabilistically of “two-minus dimensions” by construction and nature.

Abstract: A fundamental problem in statistical mechanics is the description of phase transitions. A useful method to show phase transitions in specific models is known as the Peierls argument, where it is shown that the energetic cost to create certain objects called contours is large and, in a certain regime of temperature, has a small probability of appearing, forcing the system to pass from a disordered to an ordered phase. We will review some results of phase transitions in deterministic as well as recent advances in random systems and, after that, will explain how a notion of multiscaled contours, inspired by previous articles of Fröhlich and Spencer, allowed us to show not only phase transition in a direct way for deterministic systems but also to study models where the usual argument for phase transition via correlation inequalities is not available. The talk will be based on recent works joint with Rodrigo Bissacot, Eric O. Endo, Satoshi Handa, and João Vitor Maia.

Abstract: I'll begin by describing shifts of finite: topological dynamical systems based on some simple combinatorial data. These have played a fundamental role in hyperbolic dynamics. I will then describe a new construction of factors (quotients) of such systems, also based on simple combinatorial data along with the idea of binary expansion of real numbers. This produces surprisingly elaborate geometric objects. I'll also briefly discuss applications to operator algebras.

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Abstract: The Non Linear Schrodinger Equation is a canonical example of a dispersive PDE that can also display stable, spatially localized solutions called solitons. Invariant measures for the flow of the equation have been used to study not only the well-posedness of the equation, but also the typicality of the long term behavior, whether dispersive or solitonic. In this talk, I will review some of the existing results and then describe joint work with Partha Dey and Kay Kirkpatrick on the corresponding Discrete PDE in dimension 3 and higher. In particular, I will define a family of invariant Gibbs measures for the discrete equations where the key parameter is the strength of the non linearity. We prove convergence of the associated free energy, and as the strength of the non linearity is varied, we establish existence of a phase transition. In the supercritical regime the support of the measure lies on very sharply peaked functions corresponding to a soliton phase, and resembles the Gaussian free field conditioned to have given L^2- norm in the subcritical regime.

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Abstract: In this talk we will give an exact characterization for the asymptotic distribution of quadratic forms in IID random variables with finite second moment, where the underlying matrix is the adjacency matrix of a graph. In particular we will show that the limit distribution of such a quadratic form can always be expressed as the sum of three independent components: a Gaussian, a (possibly) infinite sum of centered chi-squares, and a Gaussian with a random variance. As a consequence, we derive necessary and sufficient conditions for asymptotic normality, and universality of the limiting distribution.

See the abstract (PDF file).

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See abstract (PDF)

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Abstract: I will present our recent results on estimating the Lyapunov exponents of weakly-damped, weakly-dissipated stochastic differential equations. Our primary tool is a new, mildly-quantitative version of Furstenberg’s criterion.

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Abstract: Co-evolving network models, wherein dynamics such as random walks on the network influence the evolution of the network structure, which in turn influences the dynamics, are of interest in a range of domains. While much of the literature in this area is currently supported by numerics, providing evidence for fascinating conjectures and phase transitions, proving rigorous results has been quite challenging. We propose a general class of co-evolving tree network models driven by local exploration, started from a single vertex called the root. New vertices attach to the current network via randomly sampling a vertex and then exploring the graph for a random number of steps in the direction of the root, connecting to the terminal vertex. Specific choices of the exploration step distribution lead to the well-studied affine preferential attachment and uniform attachment models, as well as less well understood dynamic network models with global attachment functionals such as PageRank scores. We obtain local weak limits for such networks and use them to derive asymptotics for the limiting empirical degree and PageRank distribution. We also quantify asymptotics for the degree and PageRank of fixed vertices, including the root, and the height of the network. Two distinct regimes are seen to emerge, based on the expected exploration distance of incoming vertices, which we call the `fringe' and `non-fringe' regimes. These regimes are shown to exhibit different qualitative and quantitative properties. In particular, networks in the non-fringe regime undergo `condensation' where the root degree grows at the same rate as the network size. Networks in the fringe regime do not exhibit condensation. A non-trivial phase transition phenomenon is also displayed for the PageRank distribution, which connects to the well known power-law hypothesis. In the process, we develop a general set of techniques involving local limits, infinite-dimensional urn models, related multitype branching processes and corresponding Perron-Frobenius theory, branching random walks, and in particular relating tail exponents of various functionals to the scaling exponents of quasi-stationary distributions of associated random walks. These techniques are expected to shed light on a variety of other co-evolving network models.

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Abstract:
We prove universality of spin correlations in the scaling limit of the planar Ising model on isoradial graphs and Z–invariant weights. Specifically, we show that in the massive scaling limit, (i. e. as the mesh size tends to zero at the same rate as the inverse temperature goes to the critical one) the two-point spin correlations converges to a rotationally invariant function, which is universal among isoradial graphs and independant of the local geometry. We also give a simple proof of the fact that the infinite-volume sub-critical magnetization is independent of the site and the local geometry of the lattice. Finally, we provide a geometrical interpretation of the correlation length using the formalism of s-embeddings introduced recently by Chelkak. Based on a joint work (arXiv:2104.12858) with Dmitry Chelkak (ENS), Konstantin Izyurov (Helsinki).

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Abstract: In this talk, we will discuss the relation between two types of two-dimensional lattice models: on one hand, we will consider the spin models with an O(2)-invariant interaction, such as the famous XY and Villain models. On the other, we study integer-valued height function models, where the interaction depends on the discrete gradient. We show that delocalization of a height function model implies that an associated O(2)-invariant spin model has a power-law decay phase. Motivated by this observation, we also extend the recent work of Lammers to show that a certain class of integer-valued height functions delocalize for all doubly periodic graphs (in particular, on the square lattice). Together, these results give a new perspective on the celebrated Berezinksii-Kosterlitz-Thouless phase transition for two-dimensional O(2)-invariant lattice models. This is joint work with Michael Aizenman, Ron Peled, and Jacob Shapiro.

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Abstract: A process Y is a factor of a process X if it can be written as Y=F(X) for some function F which commutes with translations. The factor is finitary if Y_0 is almost surely determined by some finite portion of the input X. Given a process Y, the question of whether Y is a (finitary) factor of an IID process is fundamental in ergodic theory and has received much attention in probability as well. As it turns out, contrary to the prevailing belief, some classical results about factors do not have finitary counterparts, as was recently shown by Gabor. We will present a complementary result that any process Y which is a finitary factor of an IID process furthermore admits an entropy-efficient finitary coding by an IID process. Here entropy-efficient means that the IID process has entropy arbitrarily close to that of Y. As an application we give an affirmative answer to an old question of van den Berg and Steif about the critical Ising model. Joint work with Tom Meyerovitch.

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Abstract: In this talk, I will describe some progress towards the construction of the 3D Yang-Mills (YM) measure. In particular, I will introduce a state space of “distributional gauge orbits” which may possibly support the 3D YM measure. Then, I will describe a result which says that assuming that 3D YM theories exhibit short distance behavior similar to the 3D Gaussian free field (which is the expected behavior), then the 3D YM measure may be constructed as a probability measure on the state space. The underlying technical details involve analyzing the YM heat flow (which is a certain PDE) with random distributional initial data. This is based on joint work with Sourav Chatterjee.

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Abstract: For negatively curved Riemannian manifolds, various natural geometric quantities grow exponentially quickly: the volume of a ball in the universal cover; the number of "distinguishable" geodesics of a given length; the number of closed geodesics with length below a given threshold. Margulis gave very precise asymptotic estimates in this setting. After surveying the general background and history of Margulis-type results, I will describe joint work with Gerhard Knieper and Khadim War in which we obtain Margulis asymptotics for surfaces without conjugate points.

Abstract: The Oseledets multiplicative ergodic theorem plays a fundamental role in smooth ergodic theory. Here we present a generalization where instead of multiplying matrices along an orbit, we compose linear operators. Theorems of this type play a role in the study of delay differential equations, and have applications in oceanographic measurements.

Abstract: Given square matrices A_1, ..., A_d we can consider random products and the associated (top) Lyapunov exponent. We will consider two applications where the Lyapunov exponent plays an interesting role: firstly to barycentric subdivisions of triangles in the Euclidean plane; and secondly to random walks in the hyperbolic plane.

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Abstract: In 1975, the physicists Imry and Ma predicted that the incorporation of a random field in low-dimensional spin systems leads to the rounding of first-order phase transitions. These predictions were rigorously established by Aizenman and Wehr in 1989 for general spin systems, and recently quantified in the case of the random field Ising model. While the Imry-Ma phenomenon has been mainly studied in the case of compact spin spaces, it has been observed that a similar effect occurs for other models of statistical physics such as random surfaces. In this talk, we will present how the qualitative properties of these models are affected by the additon of a random field and discuss some open questions.

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Abstract: We prove that skew systems with a sufficiently expanding base have "approximate" statistical properties similar to random ergodic Markov chains. For example, they exhibit approximate exponential decay of correlations, meaning that the exponential rate is observed modulo a controlled error. The fiber maps are only assumed to be Lipschitz regular and to depend on the base in a way that guarantees diffusive behaviour on the vertical component. The assumptions do not imply an hyperbolic picture and one cannot rely on the spectral properties of the transfer operators involved. The approximate nature of the result is the inevitable price one pays for having so mild assumptions on the dynamics on the vertical component. The error in the approximation is shown to go to zero when the expansion of the base tends to infinity.

Abstract: The stochastic heat equation (driven space-time white noise) arises from interface growth dynamics via the Kardar-Parisi-Zhang equation and from the theory of disordered systems via continuum directed random polymers. Despite these physical relations, many fundamental questions for the stochastic heat equation above one dimension remain open. In particular, two dimensions are the critical dimensions for the stochastic heat equation and are known for much more subtle properties.

In this talk, I will begin with an overview of the stochastic heat equation. Then I will turn to the setting of two dimensions and explain connections between (1) the moments of certain solutions from renormalizing the noise and (2) a solvable quantum mechanical system given by the many-body Bose gas with delta-function potentials. Accordingly, the last part of the talk will concentrate on related constructions of the Bose gas dynamics, with an emphasis on a Euclidean, probabilistic analytic view.

I will describe variational problems whose solutions imply statements about dynamical systems, as well as how relaxations of these variational problems often can be solved computationally using tools of polynomial optimization. The talk will be an informal overview, focusing on the ODE case and including both deterministic and stochastic ODEs. In addition to showing computational results for particular systems, I will describe some theoretical results and open questions.

Abstract: One of the challenges of the accurate simulation of turbulent flows is that initial data is often incomplete. Data assimilation circumvents this issue by continually incorporating the observed data into the model. A new approach to data assimilation known as the Azouani-Olson-Titi algorithm (AOT) introduced a feedback control term to the 2D incompressible Navier-Stokes equations (NSE) in order to incorporate sparse measurements. In 2014, the solution to the AOT algorithm applied to the 2D NSE was proven to converge exponentially to the true solution of the 2D NSE with respect to the given initial data. In this talk, I will present our work on parameter recovery, well-posedness for the sensitivity equations, a nonlinear version of the AOT algorithm, and real-world applications in ocean models, as well as new ideas based off of these investigations.

Abstract: In this talk I will discuss activated random walk (ARW) and the stochastic sandpile (SSM), two interacting particle systems that exhibit a phase transition on infinite domains and self-organized criticality on finite domains. ARW and SSM both consist of identical particles that perform simple random walk, initially with average density μ particles per site, but they follow different local interaction rules. Depending on the value of μ, particles may eventually stop moving, or remain active forever. Current research is focused on determining where the transition between these two phenomena occurs, and many questions still remain – even on Z. I will present some recent results and discuss the novel tools involved.

I will present new, non-asymptotic bounds on the heights of random combinatorial trees and conditioned Bienaymé trees, as well as stochastic inequalities relating the heights of combinatorial trees with different degree sequences. The tool for most of the proofs is a new approach to coding rooted trees by sequences, based on a new proof of Cayley's formula.

Abstract: Joint work with: Omer Angel, Yinon Spinka

We prove that a well-known graphical expansion of the Ising model, known as the Loop O(1) model, is a factor of i.i.d. on unimodular random rooted graphs under various boundary conditions and in the presence of a non-negative external field. As an application we show that the gradient of the free Ising model is a factor of i.i.d. on unimodular planar maps. The key idea is to develop an appropriate theory of local limits of uniform even subgraphs with various boundary conditions and prove that they can be sampled as a factor of i.i.d. Another key tool we prove and exploit is that the wired uniform spanning trees on unimodular transient graphs are factors of i.i.d. This partially answers some questions posed by Tom Hutchcroft.

Abstract: Joint work with Christophe Garban. The discrete Coulomb gas is a model where an integer amount of charged particles are put on the $d$-dimensional grid. In this talk, I will discuss the basic properties of the Coulomb gas, its connection with other statistical physics models and the scaling limit of the potential of the discrete Coulomb gas at high enough temperature.

If you are local, please use your UVic SSO credentials so we don’t have to hand-admit you. Zoom link: https://uvic.zoom.us/j/98731132367?pwd=SzJKOWlKanZNaEVYMXRQQ0RpbGxJdz09

Abstract: Closely motivated by financial considerations, we develop an integration theory which is not classical i.e. it is not necessarily associated with a measure. The base space, denoted by $\mathcal{S}$ and called a trajectory space, substitutes the set $\Omega$ in probability theory and provides a fundamental structure via conditional subsets $\mathcal{S}_{(S,j)}$ that allows the definition of conditional integrals. The setting is a natural by-product of no arbitrage assumptions that are used to model financial markets and games of chance (in a discrete infinite time framework). The constructed conditional integrals can be interpreted as required investments, at the conditioning node, for hedging an integrable function, the latter characterized a.e. and in the limit as we increase the number of portfolios used. The integral is not classical due to the fact that the original vector space of portfolio payoffs is not a vector lattice. In contrast to a classical stochastic setting, where price processes are associated with conditional expectations (with respect to risk neutral measures), we uncover a theory where prices are naturally given by conditional non-lattice integrals. One could then study analogues of classical probabilistic notions in such a non-classical setting, we enumerate some of the possible results such as Doob's martingale convergence theorem and indicate an analogy with a dynamical system setting and the ergodic theorem.

Abstract:

Interacting particle models are often employed to gain understanding of the emergence of macroscopic phenomena from microscopic laws of nature. These individual-based models capture fine details, including randomness and discreteness of individuals, that are not considered in continuum models such as partial differential equations (PDE) and integral-differential equations. The challenge is how to simultaneously retain key information in microscopic models as well as efficiency and robustness of macroscopic models.

In this talk, I will discuss how this challenge can be overcome by elucidating the probabilistic connections between particle models and PDE. These connections also explain how stochastic partial differential equations (SPDE) arise naturally under a suitable choice of level of detail in modeling complex systems. I will also present some novel scaling limits including SPDE on graphs and coupled SPDE. These SPDE not only interpolate between particle models and PDE, but also quantify the source and the order of magnitude of stochasticity. Scaling limit theorems and new duality formulas are obtained for these SPDE, which connect phenomena across scales and offer insights about the genealogies and the time-asymptotic properties of the underlying population dynamics.

Abstract (PDF)

Abstract:
Russo-Seymour-Welsh's theory is a key concept in two-dimensional statistical physics models. A version of Russo--Seymour--Welsh type assumption for random walks was exploited by Berestycki, Laslier and Ray to study the decorrelation of uniform spanning trees (UST) in a planar graph. This ultimately led to a universality result about convergence of the dimer model. I will discuss an extension of this decorrelation theorem to USTs on random planar graphs. The major example we consider is the infinite cluster of supercritical bond percolation in the square lattice. This is joint work with my supervisor Gourab Ray.

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Abstract: In an earlier work in collaboration with C. Liverani, we studied stochastic properties of slow-fast partially hyperbolic local diffeomorphisms of the 2-torus. We showed a Local Central Limit Theorem which was instrumental in our proof of existence of finitely many SRB measures and exponential decay of correlation towards SRB measures, provided that the system is mostly contracting (e.g. every SRB measure has negative center Lyapunov exponents). In this talk I will present a work in progress with K. Fernando (U. Toronto), which deals with mostly expanding systems. Such systems have paradoxical features (such as statistical sinks with positive Lyapunov exponents). Once again using the LCLT, we prove that in this situation we also have finitely many SRB measures and exponential decay of correlations with bounds similar to the mostly contracting case.

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Abstract: Leplaideur and Watbled recently applied the tools from thermodynamic formalism to the Curie-Weiss mean-field theory with a new twist: they used a variant of the pressure where the energy functional is quadratic. Inspired by this result, Buzzi and Leplaideur initiated an effort to broaden the thermodynamical approach by considering an arbitrary function of free energies. In this talk we discuss the concepts of the pressure and equilibrium states in such nonlinear settings. We establish a nonlinear variational principle and characterize the nonlinear equilibrium measures as classical equilibrium states for some multiples of the potential.​

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Abstract: When the acting group is the integers, classical results in symbolic dynamics guarantee that shifts of finite type (SFTs) and sofic shifts of positive entropy must contain rich families of subsystems. In this talk, I will discuss recent (ongoing) work with Robert Bland and Ronnie Pavlov on the subsystems of SFTs and sofic shifts on arbitrary countable amenable groups. In particular, we prove that for any countable amenable group G, any G-SFT X with positive entropy h(X) > 0 contains a family of SFT subsystems whose entropies are dense in the interval [0, h(X)]. We also establish analogous results for G-sofic shifts. Our results recover those of Desai in the case G = Z^d, and our proofs make use of recent work on exact tilings of amenable groups by Downarowicz, Huczek, and Zhang.

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Abstract: In this talk, we will discuss equidistribution of the orbits of the horocycle subgroup acting on homogeneous spaces (e.g. SL2(R)/SL2(Z)). This roughly means that orbits spend "the expected" amount of time in every set, or that it "fills out" the space. In the case that the space has finite volume, this follows from Ratner's famous equidistribution theorem. It was recently proved by Mohammadi and Oh for certain infinite volume cases as well. Such results have seen many applications, especially in number theory. However, in applications, one often needs to know more than just equidistribution: specifically, at what rate does the orbit equidistribute? We call a statement that includes a quantitative error term effective. In this talk, I will present an effective equidistribution theorem, with polynomial rate, for the horocycle subgroup acting on certain infinite volume quotients (called geometrically finite), and if time permits, an application showing a fractal distribution of certain vectors in R^n under the action of a geometrically finite subgroup of SO(n,1) (or e.g. SL2(R)). No prior knowledge of homogeneous dynamics will be assumed. This is joint work Nattalie Tamam.

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Abstract: Universality in disordered systems has always played a central role in the direction of research in Probability and Mathematical Physics, a classical example being the Gaussian universality class (the central limit theorem). In this talk, I will describe a different universality class for random growth models, called the KPZ universality class. Since Kardar, Parisi and Zhang introduced the KPZ equation in their seminal paper in 1986, the equation has made appearances everywhere from bacterial growth, fire front, coffee stain to the top edge of a randomized game of Tetris; and this field has become a subject of intense research interest in Mathematics and Physics for the last 15 to 20 years. The random growth processes that are expected to have the same scaling and asymptotic fluctuations as the KPZ equation and converge to the universal limiting object called the KPZ fixed point, are said to lie in the KPZ universality class, though this KPZ universality conjecture has been rigorously proved for only a handful of models till now. Here, I will talk about some recent results on universal geometric properties of the KPZ fixed point and the underlying landscape and show that the KPZ equation and exclusion processes converge to the KPZ fixed point under the 1:2:3 scaling, establishing the KPZ universality conjecture for these models, which were long-standing open problems in this field. The talk is based on joint works with Jeremy Quastel, Balint Virag and Duncan Dauvergne.

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Abstract: Let (X,sigma) be a subshift. A flow equivalence of two dynamical systems is an orientation-preserving homeomorphism of the suspensions of the systems. The mapping class group of a subshift is the group of self-flow equivalences up to isotopy. We compute the mapping class group for various classes of minimal low complexity subshifts.

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Abstract: We establish a central limit theorem for the maximal Lyapunov exponent of typical cocycles (in the sense of Bonatti and Viana) over irreducible subshifts of finite type with respect to the unique equilibrium state for a Hölder potential. We also establish other related results such as the analytic dependence of the top Lyapunov exponent on the underlying equilibrium state and a large deviation principle. The transfer operator and its spectral properties play key roles in establishing these limit laws. This is joint work with Kiho Park.

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Abstract: What is the chance that among a group of n friends, there are s friends all of whom have the same birthday? This is the celebrated birthday problem which can be formulated as the existence of a monochromatic s-clique (s-matching birthdays) in the complete graph K_n, where every vertex of K_n is uniformly colored with 365 colors (corresponding to birthdays). More generally, for a connected graph H, let T(H, G_n) be the number of monochromatic copies of H in a uniformly random coloring of the vertices of the graph G_n with c_n colors. In this talk, characterization theorems for the limiting distribution of this quantity, and related random multilinear polynomials, will be derived. Applications of these results in testing high-dimensional discrete distributions, motif estimation in large networks, and the discrete logarithm problem will also be discussed.

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Abstract: One of the fundamental challenges in number theory is to understand the intricate way in which the additive and multiplicative structures in the integers intertwine. We will explore a dynamical approach to this topic. After introducing a new dynamical framework for treating questions in multiplicative number theory, we will present an ergodic theorem which contains various classical number-theoretic results, such as the Prime Number Theorem, as special cases. This naturally leads to a formulation of an extended form of Sarnak's Mobius randomness conjecture, which deals with the disjointness of actions of (N,+) and (N,*). This talk is based on joint work with Vitaly Bergelson.

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Abstract: In this talk I will describe recent joint work with Kevin McGoff. We consider topological dynamical systems (X,T) defined by a compact metrizable space X and action T of a countable amenable group G on X by homeomorphisms.

For two such systems (X,T) and (Y,S) and a factor map pi from X to Y, an intermediate factor is a topological dynamical system (Z,R) for which pi can be written as a composition of factor maps from X to Z and from Z to Y. Our main result establishes that if G is countable amenable and X and Y are zero-dimensional, then every number between h(X,S) and h(Y,T) can be realized as the entropy of an intermediate zero-dimensional factor.

I'll describe some connections to work of Shub, Weiss, and Lindenstrauss on "lowerability of entropy," and will give a vague outline of some parts of the proof.

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Abstract: In Bernoulli(p) bond percolation, each edge of a given graph is declared open with probability p. The set of open edges is a random subgraph. The arboreal gas is the probability measure that arises from conditioning the random subgraph to be a spanning forest, i.e., to contain no cycles. In the special case p=1/2 the arboreal gas is the uniform measure on spanning forests. What are the percolative properties of these forests? This turns out to be a surprisingly rich question, and I will discuss what is known and conjectured. I will also describe a magic formula for the connection probabilities of the arboreal gas. This formula, analogous to the magic formula for reinforced random walks, arises due to an important connection between the arboreal gas and spin systems with hyperbolic symmetry. Based on joint work with Roland Bauerschmidt, Nick Crawford, and Andrew Swan.

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Abstract: Recently Athreya, Lalley, Sapir and Wroten have been interested in the tangle of geodesics in a compact riemaniann surface of negative curvature. One question is to understand locally the picture of a geodesic segment of length T, in a vicinity of any point given on the surface. With Françoise Pène we recover their main result, applying our work on spatio-temporal point processes for visits to small sets.

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Abstract: Consider a diffusion spreading through a network. Given a snapshot of the history, can the starting point be determined? I will discuss the ideas and problems surrounding this question in two contexts: for simple random walk / Brownian motion, and rumour spread in social networks. For random walks, I will sketch how the theory of self-intersections lends a hand; for rumour spread, I will present the state-of-the-art, an algorithm called adaptive diffusion, discuss its shortfalls, and suggest a path forward.​

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Abstract: We obtain large and moderate deviations for both sequential and random compositions of slowly mixing intermittent type maps. We also address the question of whether or not centering is needed for quenched central limit theorems. This is joint work with Felipe Perez Pereira and Andrew Torok.

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Abstract: Pressure plays a critical role in the thermodynamic formalism, giving a unifying description of many invariant measures of dynamical importance, as well as giving rise to a procedure for computing the Hausdorff dimension of some dynamically-defined sets. In particular, for a dynamical system T:X->X and for a potential function f:X->R, one studies the function g(t)=Pressure(t*f). One can show that provided T has finite topological entropy, the function g(t) is convex, and the set of intercepts with the y-axis forms a bounded sub-interval of [0,infty).

We will recall the definition and basic properties of pressure, and show that the constraint above is essentially the only restriction on g. (Joint work with Tamara Kucherenko)

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For chaotic dynamical systems, it is unfeasible to compute long-term orbits precisely. Nevertheless, we may be able to describe the statistics of orbits, that is, to compute how often an orbit will visit a prescribed region of the phase space. Different orbits may or may not follow different statistics. I will explain how to measure the statistical diversity of a dynamical system. This diversity is called emergence, is independent of the traditional notions of chaos.

I will begin by discussing classic problems of discretization of metric spaces and measures. Then I will apply these ideas to dynamics and define two forms of emergence. I will present several examples, culminating with new dynamical systems for which emergence is as large as we could possibly hope for.

This talk is based on joint work with Pierre Berger (Paris).

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Abstract: Superprocesses are measure-valued Markov processes modelling spatial branching behaviour in continuous space and time. In this talk I will consider the superprocess associated to an α-stable spatial motion and a (1+β)-stable branching mechanism in the parameter regime in which it has a density. After introducing this process and some classical results, I will discuss some newly proven path properties of the density. These include (i) strict positivity of the density at a fixed time (for certain parameters) and (ii) a classification of the measures which the density charges almost surely when conditioned on survival. The duality between the superprocess and a fractional parabolic PDE is central to our method, and I will discuss how the probabilistic statements above correspond to new results about singular solutions to the PDE.

Abstract: We consider the unique measure of maximal entropy for proper 3-colorings of Z2, or equivalently, the so-called zero-slope Gibbs measure. We will prove that this measure is Bernoulli, or equivalently, that it can be expressed as the image of a translation- equivariant function of independent and identically distributed random variables placed on Z2. Joint with Y. Spinka (UBC).

This is part II of a pair of presentations. Part I that was given in pre-covid era. I will recall all the relevant information, and everything will be self-contained.

Please contact the organizer if you require the Zoom link.

Abstract: We explore a novel percolation-type model that was proposed as a crude model of lightning propagation. In short, each cell in a network is randomly assigned a potential (a uniform [0,1] random variable). The lightning then propagates from a cell i to a neighbouring cell j if the potential at j is less than i's potential plus a transfer threshold, t. We found that for small values of t, there is no long-range propagation of the "lightning", while for larger values of t, the lightning almost certainly propagates. This model is interesting for a number of reasons: the directionality of the potential bonds (for positive values of t); and the fact that conditioned on the existence of a long path to a node j, the conditional distribution of its potential is strongly affected.

Abstract: We consider the unique measure of maximal entropy for proper 3-colorings of Z2, or equivalently, the so-called zero-slope Gibbs measure. We will prove that this measure is Bernoulli, or equivalently, that it can be expressed as the image of a translation- equivariant function of independent and identically distributed random variables placed on Z2.​ Joint with Y. Spinka (UBC).

Abstract: We consider a sequence of matrices that are associated to Markov dynamical systems and use determinant-free linear algebra techniques (as well as some algebra, complex analysis, and perhaps surprisingly, algebraic geometry) to rigorously estimate the eigenvalues of every matrix simultaneously without doing any calculations on the matrices themselves. These estimates give rise to mixing rates for the dynamical systems and provide an asymptotic sharpness result in the study of random compositions of the maps in these dynamical systems.

Abstract: We study the κ-color cyclic particle system on the one-dimensional integer lattice, first introduced by Bramson and Griffeath. In their original article they show that almost surely, every site changes its color infinitely often if κ ∈ {3, 4} and only finitely many times if κ ≥ 5. In addition, they conjecture that for κ ∈ {3, 4} the system clusters, that is, for any pair of sites x, y, with probability tending to 1 as t → ∞, x and y have the same color at time t. Here we prove that conjecture.

(joint work with Hanbaek Lyu)

Abstract: The Lyapunov spectrum of Perron-Frobenius operator cocycles contains relevant information about dynamical properties of time-dependent (non-autonomous, random) dynamical systems. In this talk we characterize the Lyapunov spectrum of a class of analytic expanding maps of the circle, and discuss stability and instability properties of this spectrum under perturbations. (Joint work with Anthony Quas.)

Abstract: Dynamical entropy is an important tool in classifying measure-preserving or topological dynamical systems up to measure or topological conjugacy. Classical dynamical entropy theory, of an action of a single transformation, has been studied since the 50s and 60s. Recently L. Bowen and Kerr-Li have introduced entropy theory for actions of sofic groups. Although a conjugacy invariant, sofic entropy in general appears to be less well-behaved than classical entropy. In particular, sofic entropy may depend on the choice of sofic approximation, although only degenerate examples have been known until now.

We present an example, inspired by hypergraph 2-colorings from statistical physics literature, of a mixing subshift of finite type with two different positive topological sofic entropies corresponding to different sofic approximations. The measure-theoretic case remains open. This is joint work with Lewis Bowen and Dylan Airey.

Abstract: A subset of the integers P is called predictive if for all zero-entropy processes X_i; i in Z, X_0 can be determined by X_i; i in P. The classical formula for entropy shows that the set of natural numbers forms a predictive set. In joint work with Benjamin Weiss, we will explore some necessary and some sufficient conditions for a set to be predictive. These sets are related to Riesz sets (as defined by Y. Meyer) which arise in the study of singular measures. This and several questions will be discussed during the talk.

Abstract: In this talk, I will discuss stochastic interface growth dynamics from the viewpoint of the Kardar–Parisi–Zhang equation. After a brief overview, the discussion will turn to the case of two spatial dimensions. I will explain some aspects of recent results that prove a conjecture on the irrelevance of nonlinearity in the equation.

Abstract: Chase-escape is a competitive growth process in which red particles spread to adjacent uncoloured sites while blue particles overtake and kill adjacent red particles. We can think of this model as prey escaping from pursuing predators. If the red particles spread fast enough, both particle types occupy infinitely many sites with positive probability. Otherwise, both almost surely occupy only finitely many sites. In this talk, we introduce the modification that red particles die at some rate. When the underlying graph is a d-ary tree, chase-escape with death exhibits a new phase in which blue almost surely occupies finitely many sites, while red reaches infinity with positive probability. Moreover, the critical behaviour, which we precisely characterize, is different with the presence of death. Many of our arguments make use of novel connections to analytic combinatorics. Joint work with Erin Beckman, Keisha Cook, Nicole Eikmeier and Matthew Junge.

Abstract: The classical Perron-Frobenius theorem can be applied to Markov chains with a single primitive transition matrix to show that there is a unique stationary distribution for the chain, and that distributions relax exponentially quickly to that stationary distribution, where the rate is determined by the second-largest eigenvalue. We can generalize Markov chains: first to chains with randomly chosen transition matrices, then to cocycles of transfer operators, which describe how densities move around according to random underlying dynamics on a state space. I will describe a generalization of the classical Perron-Frobenius theorem that can be applied in this setting to give an analog of a stationary distribution and a relaxation rate, along with some of the definitions and background required to understand the theorem statement.

Abstract: Consider an ergodic Markov chain on a countable state space for which the return times have exponential tails. We show that the stationary version of any such chain is a finitary factor of an i.i.d. process. Joint work with Omer Angel.

Abstract: We consider three types of dynamical systems: shifts of finite type (SFT), substitution tilings spaces (STS), and hyperbolic toral automorphisms (HTA). A Markov partition on a HTA gives a finite-to-one factor map from a SFT onto the HTA. We investigate when such a factor map can be split as a composition of $u$-and $s$-bijective maps. It has already been shown that if the tiling system, given by the Markov partition, satisfies the forcing the border condition, then a splitting exists. We will show a partial converse, namely, if a splitting exists then the Markov partition must satisfy a certain condition. Finally, we will show that the existence of a splitting does not imply that the tiling system forces its border.

Title: The phase transition of planar random cluster model for q>4 is discontinuous: two short proofs and a new graphical representation of the six-vertex model

Abstract: It is a well-known conjecture due to Baxter that the phase transition of the critical planar random cluster and Potts model for q>4 is discontinuous, meaning that at criticality, there exist multiple Gibbs states. This conjecture was recently solved by Duminil--Copin et. al (https://arxiv.org/abs/1611.09909). using the so-called Bethe Ansatz techniques. We present two new (depending on time), short, soft and probabilistic proofs of this fact. One of the proofs introduce a new graphical representation of the six-vertex model which we believe is of independent interest. Joint with Yinon Spinka (UBC).

Abstract: A topological dynamical system $(X,S)$ is an "h-universal model for ergodic actions" if it's ergodic measures include an isomorphic copy of any free measure preserving action of entropy strictly lower than then h. In this talk, which is based on recent joint work with Nishant Chandgotia I will describe a new sufficient condition for a topological dynamical system to be a universal model in the above sense (and also in the stronger "almost Borel" since ). Our main result extends a line of previous work starting from Krieger in the 1970's to Quas and Soo from several years back, to very recently published work of David Burguet. I will explain try to outline the main ideas of the proof and also (as time premits) how we use our main result (combined with previously known results) to deduce various conclusions and answer a number of questions. Among these applications:

• A ``generic'' homeomorphisms of a compact manifold of topological dimension at least two can model any ergodic transformation,
• Non-uniform specification implies almost Borel universality (This is an independent and extended proof of Burguet's recent strengthening of the Quas-Soo result).
• $3$-colorings in $\mathbb{Z}^d$ and dimers in $\mathbb{Z}^2$ are almost Borel universal.

Abstract: Nonzero Lyapunov exponents carry crucial information on the dynamics of a system. Positive exponents are considered a hallmark of chaos, while negative exponents are often associated to regular behaviour such as synchronisation. Deciding whether a given system admits nonzero exponent is, in general, a very hard task, and providing analytic estimates is an even harder endeavour. The subtlety of this problem is attested by the fact that L. exponents are often not stable under small perturbations of the dynamics.

During the first talk, I will introduce basic notions on Lyapunov exponents and random systems, and will review some relevant literature for the topic of the second talk. This talk will target undergraduate and graduate students with basic background in ergodic theory. During the second talk, I will describe recent results on stability of Lyapunov exponents for Morse-Smale diffeomorphisms of the unit circle with a sink and a source that are perturbed by a a fixed rotation that occurs rarely, but that can be large in magnitude sending the sink in the vicinity of the source.

Abstract: Nonzero Lyapunov exponents carry crucial information on the dynamics of a system. Positive exponents are considered a hallmark of chaos, while negative exponents are often associated to regular behaviour such as synchronisation. Deciding whether a given system admits nonzero exponent is, in general, a very hard task, and providing analytic estimates is an even harder endeavour. The subtlety of this problem is attested by the fact that L. exponents are often not stable under small perturbations of the dynamics.

During the first talk, I will introduce basic notions on Lyapunov exponents and random systems, and will review some relevant literature for the topic of the second talk. This talk will target undergraduate and graduate students with basic background in ergodic theory. During the second talk, I will describe recent results on stability of Lyapunov exponents for Morse-Smale diffeomorphisms of the unit circle with a sink and a source that are perturbed by a a fixed rotation that occurs rarely, but that can be large in magnitude sending the sink in the vicinity of the source.

Abstract: We will recall the notion of Lyapunov exponents, and discuss them in a context of Perron-Frobenius for some Blaschke products. The key motivating question is to determine circumstances under which the Lyapunov exponents are stable under perturbations of the system.

Note: This will be part TWO a two part presentation.

Abstract: We will recall the notion of Lyapunov exponents, and discuss them in a context of Perron-Frobenius for some Blaschke products. The key motivating question is to determine circumstances under which the Lyapunov exponents are stable under perturbations of the system.

Note: This will be part one a two part presentation.

Title: A lower bound for the gap between the first and second Lyapunov
exponents for a class of cocycles of Perron-Frobenius operators preserving
a cone.

Abstract: Lyapunov exponents provide a generalization to cocycles of
operators (“random” compositions of operators) of the spectrum for single
operators. This spectral data allows us to understand expansion and decay
rates of parts of the space on which the operators act; when applied in
the case of Perron-Frobenius operators related to dynamics on a space, it
helps us to understand how mass evolves under the dynamics and decay rates
of transients in terms of spectral gaps.  The problem of computing
Lyapunov exponents is generally acknowledged to be a hard one.

In this series of three talks, we will see how we can find a lower bound
on the gap between the first and second Lyapunov exponents for certain
Perron-Frobenius operator cocycles: those that preserve and contract
(somehow) a nice cone of functions. The proof begins with a general
theorem about cocycles of operators and cones where the hypotheses are
checkable and the numbers are, in principle, computable. Applications to
concrete examples require specific tools if they are not “toy” problems.
We will develop one such tool, a specialized Lastota-Yorke inequality, and
use it to estimate the gap for our specific example.  We will also use
this analysis to study the situation where the system passes under
perturbation from multiple equilibria to a unique equilibrium with a small
spectral gap.

In this second talk, we will describe a new Lasota-Yorke inequality that
is 
well-adapted for cocycles of paired tent maps, and we will explain why we
need 
this new inequality, recalling some facts about cones and the Cocycle
Perron-Frobenius theorem.

Abstract: As a prelude to a reading group being organized by David Goluskin, I will introduce the subject of ergodic optimization, and talk about the questions, the paradigms, and some results in the field.

Abstract: One can construct a stationary system where the objects in
question are rooted graphs and the shift operation is a random walk step.
Another equivalent way to say this is to say that the graph satisfies an
intrinsic "mass transport principle”. I will explain why this is an
important idea and survey some recent results. Based on some results of
Benjamini and Curien and some joint work of myself with Omer Angel, Tom
Hutchcroft and Asaf Nachmias.

This will be Part II of two consecutive lectures.

I’m going to step in tomorrow and talk about something current. I’ll
finish a little bit early and we can chat about who might like to talk
this term, and when. Please give this little thought so that we can, at
least, have someone lined up for  next Thursday.  If you need more than
one session, please feel free to request.

Transport phenomena can be studied by investigating underlying dynamical
systems from a “probabilistic” or “mass transfer” point of view. Insight
can sometimes be gained from the dominant parts of the spectrum of
associated transfer operators. Modern computer power makes practical
studies of these objects feasible in a plethora of application areas; it
is natural to consider whether inferences from numerically approximated
transfer operators are mathematically plausible. These are classic
questions of “stochastic stability”.

Progress this decade has been significant, with advances in the
multiplicative ergodic theory of non-autonomous (or “driven” or “random”)
dynamical systems. This talk will report some current joint work with Gary
Froyland (UNSW) and Cecilia Gonzalez-Tokman (UQ) about non-autonomous
interval maps in which we obtain a fully quenched stochastic stability
result under an “eventually expanding on average condition” (with only
ergodicity of the driving process).

Abstract: Gibbs states in thermodynamics are expressed as Gibbs measures on shift spaces. We apply the study of the transition classes of factor maps to investigate how Gibbs properties are transformed under factor maps from shifts of finite type.