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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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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.

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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.

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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.

Abstract: Variational integrators for numerical simulations of Lagrangian systems have the advantage of conserving the momenta up to machine precision, independent of the time step. While the theory of variational integrators for mechanical systems is well developed, there are obstacles in direct applications of these integrators to systems involving fluid-structure interactions. In this talk, we derive a variational integrator for a particular type of fluid-structure interactions, namely, simulating the dynamics of a bendable tube conveying ideal fluid that can change its cross-section (collapsible tube). We start by deriving a fully three-dimensional, geometrically exact theory for flexible tubes conveying fluid. Our approach is based on the symmetry-reduced, exact geometric description for elastic rods, coupled with the fluid transport and subject to the volume conservation constraint for the fluid. Using these methods, we obtain the fully three dimensional equations of motion. We then proceed to the linear stability analysis and show that our theory introduces important corrections to previously derived results, both in the consistency at all wavelength and in the effects arising from the dynamical change of the cross-section. Based on this theory, we derive a variational discretization of the dynamics based on the appropriate discretization of the fluid’s back-to-labels map, coupled with a variational discretization of elastic part of the Lagrangian. We also show the results of simulations of the system with the discretization using a very small number of spatial points demonstrating a non-trivial and interesting behavior.  Time permitting, we shall also discuss some fully nonlinear solutions and the results of experiments.  

Joint work with F. Gay-Balmaz (ENS and LMD, Paris). The work was partially supported by NSERC and the University of Alberta Centennial Fund.

Abstract: The goal of this talk is to explain some of the number theory leading up to the conjecture named in the title, and show how a simple fact from ergodic theory -- that integer multiples of an irrational number are dense modulo 1 -- is the key to the proof of the conjecture.

Abstract: Limit theorems describe fine properties of the long-term statistical behaviourof dynamical systems, including expected averages, large deviations and fluctuations. In this talk, we discuss a new approach to establish limit theorems for non-autonomous dynamical systems, based on spectral methods and modern ergodic theory. We present an application in the context of piecewise expanding maps. This is based on joint work with D. Dragicevic, G. Froyland and S. Vaienti.

In these two talks, we will  show the construction and continuity of SRB measures for the Lorenz flow.

In these two talks, we will  show the construction and continuity of SRB measures for the Lorenz flow.

Instead of working directly with a (measurable) dynamical system (X,T), it is often easier to study the Perron-Frobenius (or transfer) operator, and obtain information about the transformation from the operator. There are some more general operator theoretic notions and techniques that allow for effective study of this operator. This will be a set of two talks; in the first talk, we will give requisite definitions, some examples, and some motivation for why we want to study operators to learn about dynamical systems, and in the second talk, we will look at a result by Hennion (1993, in French) that gives us a way to show that an operator is quasi-compact, which is a desired property for the transfer operator.

Instead of working directly with a (measurable) dynamical system (X,T), it is often easier to study the Perron-Frobenius (or transfer) operator, and obtain information about the transformation from the operator. There are some more general operator theoretic notions and techniques that allow for effective study of this operator. This will be a set of two talks; in the first talk, we will give requisite definitions, some examples, and some motivation for why we want to study operators to learn about dynamical systems, and in the second talk, we will look at a result by Hennion (1993, in French) that gives us a way to show that an operator is quasi-compact, which is a desired property for the transfer operator.

I’ll discuss the notion (and use) of distortion estimates for
one-dimensional maps, in general, and intermittent maps and families of intermittent maps in particular.

ABSTRACT:  Rudolph’s theorem states that the only x2 and x3 (mod 1) invariant ergodic probability measure that gives either map positive entropy is Lebesgue measure. The goal of this series of seminars is to present Rudolph’s original proof of this theorem.

This week: Cadenza.

ABSTRACT:  Rudolph’s theorem states that the only x2 and x3 (mod 1) invariant ergodic probability measure that gives either map positive entropy is Lebesgue measure. The goal of this series of seminars is to
present Rudolph’s original proof of this theorem.

Using the previously introduced symbolic representation, week’s talk will get to the meat of the proof where the notion of a symmetric point is defined, leading to a dichotomy between uniform measure (Lebesgue) and
zero entropy. 

ABSTRACT:  Rudolph’s theorem states that the only x2 and x3 (mod 1) invariant ergodic probability measure that gives either map positive entropy is Lebesgue measure. The goal of this series of seminars is to present Rudolph’s original proof of this theorem.

This week’s talk (third in the series) will continue with the Z^2 symbolic representation of the x2 x3 system introduced last week.

ABSTRACT:  Rudolph’s theorem states that the only x2 and x3 (mod 1)
invariant ergodic probability measure that gives either map positive
entropy is Lebesgue measure. The goal of this series of seminars is to
present Rudolph’s original proof of this theorem.

The first lecture (last week) reviewed basics of measure theoretic
entropy. In this lecture we
will start to look at Rudolph’s argument in detail.

ABSTRACT:  Rudolph’s theorem states that the only x2 and x3 (mod 1) invariant ergodic probability measure that gives either map positive entropy is Lebesgue measure. The goal of this series of seminars is to present Rudolph’s original proof of this theorem. In the first lecture I will develop the theory of measure theoretic entropy, and the following two lectures will be devoted to proving the theorem.

Abstract: We study the mixing property for the skew product F: T^d x T^l \to T^d x T^l given by F(x, y)= (Tx; y + \tau(x)), where T:  T^d \to T^d is a C^\infty uniformly expanding endomorphism, and the fiber map \tau: T^d \to T^l is a C^\infty map. We apply the semiclassical approach to show the dichotomy: either F mixes exponentially fast or \tau is an essential coboundary. This is joint work with Jianyu Chien.

A Gibbs measure in thermodynamic formalism is an invariant measure on a sub-shift of finite type which describes systems at equilibrium for some temperature T and some interacting potential or Hamiltonian H. When the temperature goes to zero these measures tend to converge to special invariant measures called minimizing measures or ground states. Although the Gibbs measure at finite temperature is ergodic, the different limits, by taking sub-sequences as T->0, may be supported on disjoint invariant components. For finite-range potentials, it is known that the limit exists as convex combination of these components. We show that, for a class of long-range potentials, by freezing the system, the Gibbs measure oscillates between the two only possible ground states.

NOTE  we have returned to the standard START TIME.

Abstract: Now that we have seen what a cocycle is (and we have a good linear algebra background), we are in a position to state the Multiplicative Ergodic Theorem. This will be done, and we will see examples of what it can do for us, in terms of the examples we looked at last time; we will also see a familiar result appear in a special case.

NOTE THE EARLY START TIME PLEASE

Abstract: The idea of a cocycle is a natural one; we provide some motivating situations before proceeding to the definition of a cocycle. After some theoretical comments, we will explore how cocycles aid us in studying dynamical systems, in particular by forming natural skew products. The goal of the next talk is to describe the Multiplicative Ergodic Theorem, so we intend on covering the necessary background in this talk.

This week we will conclude with a definition of anisotropic spaces of densities adapted to GBTs and some key steps in the proof of a Lasota-Yorke inequality.

Abstract: This week we will see what happens when we try to apply the methods developed for expanding interval maps to study generalized baker's transformations. These are piecewise hyperbolic maps of the unit square. We will see that the presence of both expanding and contracting coordinates precludes the use of standard spaces of functions to study the Frobenius-Perron operator. We will introduce anisotropic Banach space norms and see that we can recover the usual Lasota-Yorke inequality for the Frobinius-Perron operator acting on these spaces of anisotropic functions.

Abstract: This week we will look at what can be done when an interval map has an indifferent fixed point (T(x) =x and DT(x)=1). We will investigate some simple examples of generating function techniques and their application to renewal processes in probability theory. We will then discuss results of Sarig and Gouezel that extend this method to analyse Frobenius-Perron operators. We will try to see how this method works for a simple two branched map with onto branches and an indifferent fixed point at 0.

Last time Chris showed us that for piecewise Lasota-York maps the associated Frobenius-Perron (FP) operator can be viewed as an operator on C^1 functions and that this restriction is quasi-compact. Today we will investigate the effect of allowing the map T to have branches that are not onto. This small change in the map requires a substantial change in the analysis. We will see that the FP operator no longer restricts to C^1. Instead we will consider the restriction to functions of bounded variation (BV) and deduce quasi-compactness of the FP operator viewed as an operator on BV.

The classical CLT can be generalized to sequences of RV generated by a single observable and iterates of a measure preserving transformation. In particular, no independence is assumed, although some control on decay of correlation (for a class of functions containing the observable) is necessary.  The method of proof involves the use of functional analytic versions of the classical characteristic functions from probability, something which seems to be interesting in its own right. 

We are following a paper of Sebastien Gouezel  which can be downloaded as item 5 from  ‘other publications’  on the website

https://perso.univ-rennes1.fr/sebastien.gouezel/publis.html

Abstract: Actions of groups like Z or Z^d using automorphisms of compact abelian groups have served as a source of inspiration and rich examples in dynamics for over fifty years. More recently, the study of similar actions of groups such as the discrete Heisenberg group have revealed a host of new phenomena and connections to areas such as von Neumann algebras. Using concrete examples that anyone can understand, I will sketch some of the previous theory, and indicate some of the many fascinating open problems remaining. This is necessarily just a sampler, in the spirit of Halmos's dictum for math talks "to attract and inform”.

Abstract: Piecewise rotations in the complex plane divide the space into two half-planes, let's say upper and lower, sending points in the upper (lower) half-plane to a right (respectively, left) translation, followed by a common rotation. The orbits of such maps lead to some beautiful pictures and itineraries of points lead to some interesting symbolic dynamics. Early researchers include Boshernitzan, Cheung, Goetz, and Quas. Curiously, I had been interested in a (superficially, at least) similar map over finite fields. The motivation comes from a problem in combinatorics and extremal graph theory. A one-factorization of a graph is a colouring or partition of its edges into perfect matchings (also called one-factors). The union of two perfect matchings is a disjoint union of even-length cycles, and it is of interest to study the possible cycle lengths arising from such unions in a one-factorization. For instance, a one-factorization is called "perfect" if the union of any two colour classes is a Hamiltonian cycle. At the other extreme, if the union of any two colour classes always shatters into small components, this answers a question of Häggkvist. I will discuss how a class of one-factorizations of complete graphs arising from finite fields can give some neat results on cycle structures. The idea is essentially a study of the dynamics of my finite analog of piecewise rotations. This is based on joint work with Jeff Dinitz. As fair warning, I admit that the connection with dynamics is a bit of a stretch; however, the parallels with the planar case may be amusing.

Abstract: In their 1973 paper, Lasota and Yorke observed that that the transfer operator associated to a piecewise expanding map of the interval acting on functions of bounded variation has a regularizing effect. This property allowed them to prove that the map possesses absolutely continuous invariant measures. The main step in this proof is the calculation of the so called Lasota-Yorke inequality. The classical definition of variation is not easily generalized to higher dimensions. Many variants of multidimensional variation were introduced in the literature. The widely accepted modern definition of variation is due to Ennio De Giorgi and is formulated in terms of C^1 test functions. The modern definition generalizes nicely to higher dimensions. In this talk we will prove a Lasota-Yorke inequality for a class of expanding interval maps using the modern definition of variation. The formulation allows for the main technical issue encountered in proving the inequality to be treated in a particularly transparent and geometric manner.

Abstract: Fix a graph G and place some number (random or otherwise) of sleeping frogs at each site, as well as one awake frog at the root. Set things in motion by having awake frogs perform independent simple random walk, waking any "sleepers" they encounter. Say the model is recurrent if the root is a.s. visited by infinitely many frogs and otherwise transient. When G is the rooted d-ary tree with one-frog-per-site we prove a phase transition from recurrence to transience as d increases. Alternatively, for fixed d with Poi(m)-frogs-per-site we prove a phase transition from transience to recurrence as m increases. The proofs use two different recursions and two different versions of stochastic domination. Several open problems will be discussed. Joint with Christopher Hoffman and Tobias Johnson.

Abstract: A central concept of probability and ergodic theory is mixing in its various forms. The strongest and simplest mixing condition is finite dependence, which states that variables at sufficiently well separated locations are independent. A 50-year old conundrum is to understand the relationship between finitely dependent processes and block factors (a block factor is a finite-range function of an independent family). The issue takes a surprising new turn if we in addition impose a local constraint (such as proper coloring) on the process. In particular, this has led to the discovery of a beautiful yet mysterious stochastic process that seemingly has no right to exist.

It is an old question in dynamical systems: which compact metric spaces admit minimal (uniquely ergodic) homeomorphisms? The circle obviously has irrational rotations, but for higher dimensional spheres, this question becomes quite interesting. One of the major positive results is by Fathi and Herman which asserts that odd dimensional spheres admit minimal, uniquely ergodic homeomorphisms. On the other hand, the most famous negative result is the Hopf-Lefschetz theorem which, in particular, shows that even dimensional spheres do not. After discussing these, I will give an example of another space which admits a minimal, uniquely ergodic homeomorphism, managing in a crafty manner to avoid the Hopf-Lefschetz theorem, which says that it shouldn't. This has interesting consequences for George Elliott's classification program for C*-algebras. I'll discuss these, without assuming any prior knowledge of C*-algebras.

Inspired by topological notions coming from symbolic dynamics, we introduce a new combinatorial condition (TSSM) on the support of Z^d Markov random fields, especially useful when dealing with hard constrained systems. We establish some of its properties and different characterizations. We also show how TSSM is related with the absence of phase transitions in the context of nearest-neighbour Gibbs measures.

When dealing with matrices, putting them into a triangular form often yields meaningful information about their structure. In the case of invertible real matrix cocycles over an invertible, ergodic map, the Multiplicative Ergodic Theorem provides a subspace decomposition of $\mathbb{R}^n$, which may be interpreted as a statement about block diagonalizing the cocycle. One can then ask if cocycles may always be block triangularized, not just block diagonalized; we answer this question in the negative, and therefore show that the MET is in this sense optimal.

The set of 13 Kari Culik tiles is currently the smallest know set of Wang tiles (square tiles with colored edges and the rule that two tiles may lie adjacent iff their common edges share the same color) that tile the plane only in an aperiodic way. They do so for fundamentally different reasons than previously known tilings. A subset of Kari Culik tilings have rows which may be interpreted as Sturmian sequences. This talk will show how this Sturmian-like subset can be thought of as a generalization of rotation sequences and how to get explicit waiting time bounds for n*m configurations.

The systems we consider are homeomorphisms of compact metric spaces, shifts of finite type, countable state Markov shifts, surface diffeomorphisms which are C^{1+} (have Holder continuous derivative), and isomorphisms of standard Borel spaces.

The "almost-Borel" viewpoint is to consider a system with respect to all its invariant Borel probabilities, neglecting sets which have measure zero for all of them. We consider issues around the classification of systems in this category, and give complete (or close to complete) results for the Markov shifts and C^{1+} surface diffeomorphisms.

The basis for this comes from Mike Hochman, who showed that various familiar systems have a universality property: all Borel systems of smaller entropy can be embedded into them (modulo those universal null sets). Such a system has a universal part (for this category), supporting its nonatomic ergodic measures of less than maximum entropy, and a top part supporting the measures of maximum entropy. The strictly universal part is uniquely determined up to Borel isomorphism (modulo the universal null sets). Thus the entropy and the almost-Borel structure of the top part determine the system up to almost-Borel isomorphism. The universal part is fantastically complicated, but it is homogeneous. The idiosyncratic part at the top is sometimes easily classified.

We extend Hochman's results to more general systems. This requires considering universality and entropy relative to periods. From Hochman's previous work and proper definitions, one gets a complete almost-Borel classification of finite entropy countable state Markov shifts (NOT necessarily irreducible).

The main payoff: up to a set (often empty) supporting some zero entropy measures, every C^{1+} surface diffeomorphism is almost Borel isomorphic to a countable state Markov shift. This result rests on the remarkable symbolic dynamics of Omri Sarig, and an analysis of their factors under this symbolic dynamics with respect to measures with entropy "maximal at a period".

One of the basic open questions about generalizing these results arises from the work of Quas and Soo.

Many constructions in ergodic theory involve skew products. The use of skew products are much more common for constructing actions of Z than for actions of other groups. In this talk I will describe how to make the Z^2 analogs of some common skew products. Along the way we will find connections to several models statistical mechanics as well as questions in computability.

Entropy minimality was introduced by Coven and Smítal (1992) as a property of dynamical systems which is stronger than topological transitivity and weaker than minimality. A topological dynamical system (X,T) is said to be entropy minimal if all closed T-invariant subsets of X have entropy strictly less than (X,T). In the sphere of symbolic dynamics this translates to the following: a shift-space is entropy minimal if by exclusion of any globally allowed pattern we incur a drop in the topological entropy. In one dimension it is well known that a nearest neighbour shift of finite type is entropy minimal if it is irreducible; the same is known in higher dimensions under strong irreducibility. In this talk we will discuss a class of nearest neighbour shift of finite type which appear as the space of graph homomorphisms from Z^d to graphs without four cycles; for instance, we will see why the space of 3-colourings of Z^d is entropy minimal even though it does not have any of the nice mixing properties.

If f is a real-valued function on the unit interval, we can extend its domain to matrices that are Hermitian with eigenvalues in the unit interval. Now assume f is continuous. It is uniformly continuous, on its original domain. Is f also uniformly continuous for matrices? That is, can we find eta tending to zero at zero so that, for 0<=X<=I and 0<=Y<=I, ||f(X)-f(Y)| <= eta(||X-Y||) holds for the spectral norm? Ando showed this is false for f(t)=|t-1/2| but true when f is operator monotone. A related question is uniformly bounding ||f(X)A-Af(X)|| in terms of ||XA-AX|| whenever ||A||<=1. The conjecture ||sqrt(X)A-A\sqrt(X)||<= sqrt(||XA-AX||) has been around for decades. We are able to prove this in an important special case. I will discuss the Monte Carlo methods used when computing billions of examples and present some conjectures based on the data we collected. This is joint work with my graduate student, Fredy Vides. 

I will talk about one-dimensional solenoids and their homology, as defined for Smale spaces by Putnam. First I will start with definition of Smale spaces and show how shifts of finite type spaces are examples, which play an important role in this homology. Then I will go into solenoids by giving some examples. We will show how there are natural shifts of finite type which factor onto them, which is used in computing their homology. Since the process of computing this homology is complicated, I will not enter into the details of the computations and just give the results. This is joint work with M.Amini and I.F Putnam

Abstract: I will describe some interesting dynamical systems of an algebraic nature. In many ways, they are similar to the well-known hyperbolic automorphisms of tori, such as Arnold's cat map. The new ingredient is the use of the field of p-adic numbers in place of the real numbers. My guess here is that most people may not be very familiar with p-adic numbers so I will give a short introduction to them, their wonderful properties and what about them might appeal to people in dynamical systems. After describing the systems, I will discuss the problem of finding convenient symbolic codings for them (i.e. Markov partitions) which was done for hyperbolic toral automorphisms by Adler and Weiss. This is joint work with Nigel Burke (UVic/ Cambridge). From these codings, we are able to compute the homology, as I defined in earlier work, but this will be a brief footnote to the rest of the talk.

Abstract: Heat flow is one of the most pervasive concepts in mathematics. Understood from antiquity as a solution to a PDE, the mid-Twentieth Century development of stochastic processes gave a new understanding, rigorously completing Einstein's picture of heat flowing through random collisions of particles with aggregate behavior described by (a mean value of) Brownian motion. This picture makes sense on Riemannian manifolds, and led to a revolution in understanding geometry through heat flow.

Abstract: The two-stage contact process, first introduced by Steve Krone, is a stochastic growth model in which each individual must reach a mature stage before reproducing, and is a natural generalization of the contact process. We first give some background on the contact process, outlining some important results of the 80's and 90's including complete convergence, We then introduce the two-stage process, for which the same results hold. We indicate why this should be so, by showing that the two-stage process possesses the same useful properties as the contact process. Finally, we describe an important difference in the two-stage process, namely the existence of a critical value of the maturation rate below which survival is not possible. This fact is in qualitative contrast to the corresponding two-stage branching process. 

Particle interacting systems on a lattice are widely used to model complex physical processes that occur on much smaller scales than the observed phenomenon one wishes to model. However, their full

applicability is hindered by the curse of dimensionality so that in most practical applications a mean field equation is derived and used. Unfortunately, the mean field limit does not retain the inherent variability of the microscopic model. Recently, a systematic methodology is developed and used to derive stochastic coarse-grained birth-death processes which are intermediate between the microscopic model and the mean field limit, for the case of the one-type particle-Ising system. Here we consider a stochastic multicloud model for organized tropical convection introduced recently to improve the variability in climate models. Each lattice site is either clear sky of occupied by one of three cloud types. In earlier work, local interactions between lattice sites were ignored in order to simplify the coarse graining procedure that leads to a multi-dimensional birth-death process; Changes in probability transitions depend only on changes in the large-scale atmospheric variables. Here the coarse-graining methodology is extended to the case of multi-type particle systems with nearest neighbour interactions and the multi-dimensional birth-death process is derived for this general case. The derivation is carried under the assumption of uniform redistribution of particles within each coarse grained cell given the coarse grained values. Numerical tests show that despite the coarse graining the birth-death process preserves the variability of the microscopic model. Moreover, while the local interactions do not increase considerably the overall variability of the system, they induce a significant shift in the climatology and at the same time boost its intermittency from the build up of coherent patches of cloud clusters that induce long time excursions from the equilibrium state.

Abstract: The concept of a compact, open subgroup being tidy for an element of a totally disconnected, locally compact group first appeared in the solution of concentration functions problem for random walks on these groups. The first part of the talk will describe the background to this work, which was done when I first moved to Newcastle 21 years ago. Since then, the concept has been considerably extended, most recently to endomorphisms of totally disconnected, locally compact groups, and has been used to solve other problems in ergodic theory and concerning rigidity of arithmetic groups. The second part of the talk will describe some of these developments.

Abstract:
In this talk, I will discuss the problem of studying  billiard
trajectories in a polygon whose interior angles are  rational
multiples of pi.  This leads naturally to the study of
translation surfaces, which is another term for the notion  of a
holomorphic 1-form on a (closed) Riemann surface.   The moduli
space of such objects admits an action of  SL(2,R) with a finite
invariant measure that can be used  to give information about the
dynamics of billiard trajectories  in the original polygon; for
example, a theorem of  Kerckhoff-Masur-Smillie tells us that for
almost every  direction theta, every trajectory with initial
direction theta  will be uniformly distributed.  In this talk, I
will describe some  refinements of the general technique of
exploiting dynamics  on the moduli space to obtain information
about dynamics of  billiard trajectories. This talk is intended
for a general audience.

Abstract:

The main objects of interest of this talk are topological
dynamical systems on measure metric spaces that are
equicontinuous with respect to a measure. We will talk about
different notions of continuity and equicontinuity with respect
to a measure and characterize them. We will see how to check for
weak convergence of sequences of measures of a certain family.

Abstract: Recently, Peter Dukes alerted me to a way of computing
the digits of Pi via a "billiard" game. I will use this
remarkable process to introduce the collision transformation,
discuss its properties and provide a geometric proof that the
number of collisions in a system of N hard spheres (any N,
arbitrary diameters and masses) in all space is always finite.
This answers a question originally asked by Sinai.

Abstract: The Krieger generator theorem says that every
invertible measure- preserving system with finite
measure-theoretic entropy can be embedded into a full shift with
strictly greater topological entropy. In joint work with Anthony
Quas, we extend Krieger’s theorem to include toral automorphisms
(which are not necessarily hyperbolic) to give a positive answer
to a question of Lind and Thouvenot.

Abstract: In the 1960's, Stephen Smale initiated an ambitious program to understand the dynamics of a certain class maps on smooth manifolds, which he called Axiom A. David Ruelle gave a definition of a Smale space to describe, in purely topological terms, the dynamics of an Axiom A system on its non-wandering set. I will discuss these definitions and give several concrete examples. Later, Anthony Manning proved that the zeta function for such systems was rational and Rufus Bowen conjectured that this was due to the existence of an underlying homology theory. A partial solution to this was given by Krieger and also Bowen and Franks when they constructed a very beautiful invariant for shifts of finite type. I will discuss this invariant and then show how it can be extended to all Smale spaces as a homology theory, as predicted by Bowen.

An open dynamical system is a measure preserving transformation
on a state space with a 'hole'. As orbits enter the hole, they
are lost from the system. Example: a billiard table.  Typically,
Lebesgue almost every orbit enters the hole eventually, so there
can be no absolutely continuous invariant measure. Instead we
look for a quasi-invariant measure, meaning, a measure stationary
after conditioning on the state of not being in the hole.  In
this setting one has a Perron-Frobenius operator and P-F equation
whose solution gives the density of a quasi-invariant measure.

Borrowing shamelessly from Anthony's careful setup last week in
the closed dynamics setting, I hope to describe the basic
functional analysis setup and a fundamentally different method
for calculating (= approximating) the quasi-invariant density
using optimization instead of discretization and projection.

Keywords:  Open dynamics, Perron-Frobenius operator, density,
maximum entropy principle, moment constraints.

Abstract: Ulam's method is a very successful and practical method
for computing densities of absolutely continuous invariant
measures of transformations. In this talk, we recall our recent
work on Multiplicative ergodic theorems and indicate how this
work can be used to give a version of Ulam's method for
non-autonomous dynamical systems.  (This is joint work with
Gary Froyland and Cecilia Gonzalez-Tokman)

We consider an arbitrary directed graph on finitely many vertices, on which each edge is assigned a positive length and a positive weight.  The asymptotic growth is given for the weighted count of paths of length at most t, as t approaches infinity.

Speaker: Professor David Ruelle (Institute des Hautes Etudes Scientifiques Bures-sur-Yvette, France)

The Lee-Yang circle theorem is the prototype of a class of results giving very precise information on the locus of the zeros of certain families of (arbitrarily high degree) polynomials in one variable. The standard application of these results is to statistical mechanics, but we shall see that there are also results about graph-counting polynomials. The intriguing feature of all these results is that they appear quite inaccessible by more standard methods of study of polynomials.

Speaker: Wael Bahsoun (Loughborough)

Title: Metastability of Interval Maps

Abstract: Gonzalez-Tokman, Hunt and Wright studied a metastable
expanding system which is described by a piecewise smooth and
expanding interval map. It is assumed that the metastable map has
two invariant sub-intervals and exactly two ergodic invariant
densities. Due to small perturbations, the system starts to allow
for infrequent leakage through subsets (called holes) of the
initially invariant sub-intervals, forcing the two invariant
sub-systems to merge into one perturbed system which has exactly
one invariant density. It is proved that the unique invariant
density of the perturbed interval map can be approximated by a
particular convex combination of the two invariant densities of
the original interval map, with the weights in the combination
depending on the sizes of the holes. In this talk I will present
analogous results in two cases: 1. intermittent interval maps; 2.
Randomly perturbed expanding maps. In the random case, if time
permits, I will also present random versions of Keller-Liverani
escape rate formulae. This is joint work with Sandro Vaienti.

Speaker: Tom Meyerovitch (UBC)

Title: Stationary Markov random fields and Gibbs measures

Abstract: Markov random fields are higher-dimensional processes
with a conditional independence, which can be viewed as analogs
of Markov-chains in ine dimension. The Hammersley-Clifford
theorem states that Markov Random fields are Gibbs measures with
a nearest neighbor interaction, under an assumption on the
support: the existence of a ``safe symbol''. In this talk I will
present joint work (in progress) with Nishant Chandgotia,
Guangyue Han, Brian Marcus and Ronnie Pavlov, investigating
Markov random fields without assuming a ``safe symbol'' in the
support.

Speaker: Susie Weiler (UVic)

Title: Hyperbolic dynamical systems via inverse limits

Abstract: A Smale space is a dynamical system with canonical
coordinates of contracting and expanding directions.  The basic
sets for Smale’s Axiom A systems are a key class of examples.  We
will give a brief overview of R.F. Williams' work on expanding
attractors. In this case, the local stable sets are totally
disconnected and the local unstable sets  are Euclidean. Williams
gave a description of such a system as a stationary inverse
limit, where the space in the limit is a branched manifold. We
consider Smale spaces with totally disconnected local stable sets
and no restrictions on the unstable sets. We give criteria on a
topological stationary inverse limit which ensures the result is
a Smale space and give examples. We also prove that any such
Smale space is obtained through this construction.

Speaker: Antoine Julien (UVic)

Title: Metric properties of tiling space.

Abstract: I will describe how the usual distance on tiling space
provides a nice approach for some properties of the tiling,
namely its complexity.  In some example, the complexity and the
Hausdorff dimension of the space are closely related. I will also
describe some results on bi-Lipschitz embedding which were
obtained with Jean Savinien and Jean Bellissard.

Speaker: Siri-Malén Høynes

Title: Toeplitz dynamical systems and their K-theory

Abstract: We will show that the family of Toeplitz systems can be
associated to simple dimension groups with non-trivial rational
subdimension groups. Furthermore, we will present a class of
examples which has a particularly nice Bratteli diagram
presentation.