Dynamics Seminar Spring 2021

The Dynamics/Probability seminar will meet on Monday afternoons at 2:30 in Zoom. All are welcome. Please contact Anthony Quas for the zoom link. The list of seminars is as follows. For further information, please contact me at aquas(a)uvic.ca

Date Speaker Title

11/1/2021 Tyler Helmuth (Durham) Random spanning forests and hyperbolic symmetry

18/1/2021 No seminar

25/1/2021 No seminar

1/2/2021 Florian Richter (Northwestern) Dynamical generalizations of the Prime Number Theorem and disjointness of additive and multiplicative actions

8/2/2021 Bhaswar Bhattacharya (U. Penn) Birthday Paradox, Monochromatic Subgraphs, and Applications in Statistical Inference

15/2/2021 READING BREAK.

22/2/2021 Mark Piraino (Northwestern) The Central Limit Theorem for Typical Cocylces

1/3/2021 Kitty Yang (Memphis) Mapping class group of low complexity subshifts

8/3/2021 Sourav Sarkar (Toronto) Universality in Random Growth Processes

15/3/2021 Jacqueline Warren (UCSD) Effective equidistribution of horocycle orbits

22/3/2021 Kevin McGoff (UNC Charlotte) Subsystem entropies of SFTs and sofic shifts on countable amenable groups

29/3/2021 Tamara Kucherenko (CUNY) Nonlinear Thermodynamic Formalism

5/4/2021 No seminar: Easter Monday

12/4/2021 Jacopo de Simoi (Toronto) Mostly expanding slow fast partially hyperbolic systems

Date: 11/1/2021
Speaker: Tyler Helmuth (Durham)
Title: Random spanning forests and hyperbolic symmetry
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

Date: 25/1/2021
Speaker:
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Date: 1/2/2021
Speaker: Florian Richter (Northwestern)
Title: Dynamical generalizations of the Prime Number Theorem and disjointness of additive and multiplicative actions
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.

Date: 8/2/2021
Speaker: Bhaswar Bhattacharya (U. Penn)
Title: Birthday Paradox, Monochromatic Subgraphs, and Applications in Statistical Inference
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.

Date: 22/2/2021
Speaker: Mark Piraino (Northwestern)
Title: The Central Limit Theorem for Typical Cocylces
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.

Date: 1/3/2021
Speaker: Kitty Yang (Memphis)
Title: Mapping class group of low complexity subshifts
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.

Date: 8/3/2021
Speaker: Sourav Sarkar (Toronto)
Title: Universality in Random Growth Processes
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

Date: 15/3/2021
Speaker: Jacqueline Warren (UCSD)
Title: Effective equidistribution of horocycle orbits
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.

Date: 22/3/2021
Speaker: Kevin McGoff (UNC Charlotte)
Title: Subsystem entropies of SFTs and sofic shifts on countable amenable groups
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.

Date: 29/3/2021
Speaker: Tamara Kucherenko (CUNY)
Title: Nonlinear Thermodynamic Formalism
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.

Date: 12/4/2021
Speaker: Jacopo de Simoi (Toronto)
Title: Mostly expanding slow fast partially hyperbolic systems
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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