BYU-Brigham Young University

Department of Mathematics

Dynamical Systems Seminar

Organizers(s): Ben Webb

Schedule of Talks:

Fall 2015

Thursday, December 10, 2015

Title of Talk: Uniqueness of equilibrium states for geodesic flows in manifolds of nonpositive curvature
Thurs, 12/10/2015-10:00am, TMCB 292
Dan Thompson, Ohio State University, Department of Mathematics

Abstract: We establish results on uniqueness of equilibrium states for geodesic flows on rank one manifolds. This is an application of machinery developed by Vaughn Climenhaga and myself, which applies when systems satisfy suitably weakened versions of expansivity and the specification property. The geodesic flow on a rank one manifold is a canonical example of a non-uniformly hyperbolic flow and I'll explain why it satisfies our hypotheses. Our methods are completely different from those used by Knieper in his seminal proof that there is a unique measure of maximal entropy in this setting. This is joint work with Keith Burns (Northwestern), Vaughn Climenhaga (Houston) and Todd Fisher (Brigham Young).

Previous Talks:

Fall 2015

Thursday, September 3, 2015

Title of Talk: Unique equilibrium states for geodesic flows in nonpositive curvature
Thurs, 9/03/2015-10:00am, TMCB 292
Todd Fisher, BYU, Department of Mathematics

Abstract: The geodesic flow for a compact Riemannian manifold with negative curvature has a unique equilibrium state for every Holder continuous potential function. This is no longer true if the curvature is only nonpositive. We show that there is a large class of potentials with unique equilibrium states. Specifically, we prove that for compact rank 1 surfaces of nonpositive curvature that the a scalar times geometric potential has a unique equilibrium state for the scalar less than 1. Furthermore, if a potential satisfies a bounded range hypothesis for compact rank 1 manifolds with nonpositive curvature, then there will be a unique equilibrium state. This is joint work with Keith Burns, Vaughn Climenhaga, and Dan Thompson.

Thursday, October 1, 2015

Title of Talk: Corners, Quasirandom Groups and Ergodic Theory
Thurs, 10/01/2015-10:00am, TMCB 292
Donald Robertson, Univeristy of Utah, Department of Mathematics

Abstract: The corners theorem, proved by Ajtai and Szemeredi in 1974, states that for any prescribed density, any dense subset of a large enough n-by-n grid contains the vertices of a right triangle. In this talk I will describe recent joint work with V. Bergelson and P. Zorin-Kranich on a version of the corners theorem for quasirandom groups (finite groups without low-dimensional complex representations) via ergodic theory.

Tuesday, October 27, 2015

Title of Talk: Lattice deformations in the Heisenberg group
Tues, 10/27/2015-10:00am, TMCB 292
Ioannis Konstantoulas, Univeristy of Utah, Department of Mathematics

Abstract: In this work, joint with J. Athreya, we study the distribution of lattice points of a Heisenberg lattice chosen uniformly on the parameter space. It is part of a broader program to understand average statistics of lattices in a wide variety of Lie groups. The non-abelian nature of the lattices and non-reductive nature of the automorphism group give rise to challenges that do not appear in the Euclidean case. To meet them we use a combination of methods from spectral theory (developed in previous work of J. Athreya with G. Margulis), geometry and number theory.

Spring 2015

Thursday, May 14, 2015

Title of Talk: Dynamics of Asynchronous Networks
Thurs, 14/05/2015-10:00am, TMCB 292
Christian Bick, University of Exeter

Abstract: Systems that have the structure of a network of interconnected nodes are abundant in nature and technology. Many mathematical models of such networks are given as ordinary differential equations defined by smooth vector fields. These traditional or “synchronous” network models, however, fail to incorporate nonsmooth features seen in real world; for example, individual nodes of a network cannot stop and restart in finite time. Asynchronous networks are an attempt to set up a mathematical framework to study systems that exhibit stopping of nodes and their bifurcations. We also consider asynchronous networks with function: given some set of initial conditions a desired final output state has to be reached. We discuss deadlocks–dynamical states that prevent a network to complete its function–and indicate how functional networks can be decomposed into simpler components.

Winter 2014

Thursday, February 26, 2015

Title of Talk: Periodic Path on non-autonomous Graphs
Thurs, 26/02/2015-10:00am, TMCB 292
Pedro Martins Rodrigues, Mathematics Department, Doutoramento em Matemática, Instituto Superior Técnico, Portugal.

Abstract: We discuss non-autonomous graphs, i.e. directed graphs where the set of edges depends on time, and the associated symbolic dynamical systems. The main subject of the talk will be the zeta function of time periodic non-autonomous graphs.

Thursday, January 22, 2015

Title of Talk: A Survey on Ergodic Stability of Partial Hyperbolic Systems
Thurs, 22/01/2015-10:00am, TMCB 292
Davi Obata, Department of Mathematics, UFRJ

Abstract: In this survey I will introduce the notion of ergodic stability for diffeomorphisms, with that I will discuss two good strategies to prove this property: The Hopf Argument (and its generalizations) and the usage of SRB measures to obtain thesame property. With that we will see some of the philosophy behind each method and its importance.

Tuesday, January 27, 2015

Title of Talk: Ambiguous symbolic dynamics and the Conley index
Tues, 27/01/2015-10:00am, TMCB 292
Jim Wiseman, Department of Mathematics, Agnes Scott College

Abstract: Consider a dynamical system given by a continuous map f:X-->X. We cover X with compact sets N_1,...,N_n whose interiors may intersect nontrivially (an index system), and look at the symbolic dynamics generated by the itineraries of orbits. Because of the nontrivial intersection, a given orbit may have more than one itinerary. We’ll talk about ways to extract meaningful dynamical information from these ambiguous symbolic dynamics (in particular, estimates for the topological entropy), and about how to generate index systems using the Conley index.

Thursday, January 29, 2015

Title of Talk: Lyapunov exponents, multiplicative ergodic theorem, invariant cones and invariant splitting for random linear system in a separable Banach space.
Thurs, 29/01/2015-10:00am, TMCB 292
Zeng Lian, Loughborough University

Abstract: Lyapunov exponents is an important tool in studying asymptotical behavior of dynamical system. Multiplicative ergodic theorem provides theoretical fundation of Lyapunov exponents and a sufficient condition of existence of Lyapunov exponents and coresponding invariant splittings. Another way to investigate invariant splitting is to study invariant cones. In this talk, I will report some recent work (joint with Yi Wang, USTC, China) which investigate the connections between invariant cones, invariant splitting and Lyapunov exponents.

Thursday, February 5, 2015

Title of Talk: TBA
Thurs, 5/02/2015-10:00am, TMCB 292
Jeff Xia, Department of Mathematics, Northwestern University

Abstract: TBA

Thursday, February 12, 2015

Title of Talk: Some uniformly ergodic results for discontinuous skew-products
Thurs, 12/02/2015-10:00am, TMCB 292
Zhe Zhou, Institute of Applied Mathematics Academy of Mathematics and Systems Science, Chinese Academy of Sciences

Abstract: We will push the boundary of the uniform ergodic theorem to discontinuous dynamical systems, including the additive case and subadditive case, and then give an application for the linear Schrodinger equation. This is a joint work with Prof. M. Zhang and Prof. Z. Zheng.

Thursday, March 5, 2015

Title of Talk: Follower sets for symbolic dynamics
Thurs, 5/03/2015-10:00am, TMCB 292
Nic Ormes, Department of Mathematics, University of Denver

Abstract: Suppose that (X,T) is a symbolic dynamical system, i.e. a shift map on a closed subset X of a A^Z where A is finite. For any word w that appears in X, we may consider the follower set of w, that is, the set of words u that wu appears in X. In this talk we will discuss several results about follower sets, including the following:
1) the a classical result that X is sofic if and only if there are finitely many sets F_1, F_2, …,F_n such that for all words w in X, F(w)=F_j for some j.
2) an example of M. Delacourt and more general result of T. French that exhibit surprising behavior in the sequence of the number of follower sets for words of length n for sofic systems.
3) work of R. Pavlov and myself in which we generalize the notion of follower sets for Z-actions to “extender sets” for Z^d actions and show that if a Z^d system has very few extender sets then it has a finite number of extender sets and is therefore sofic.

Thursday, March 12, 2015

Title of Talk: TBA
Thurs, 5/03/2015-10:00am, TMCB 292
Kamlesh Parwani, Department of Mathematics, Eastern Illinios University

Abstract: TBA

Thursday, March 26, 2015

Title of Talk: Study of ion channel problems via analysis of Poisson-Nernst-Planck systems
Thurs, 5/02/2015-10:00am, TMCB 292
Weishi Liu, Department of Mathematics, University of Kansas

Abstract: In this talk, we will report our work on Poisson-Nernst-Planck (PNP) type systems, a class of primitive continuum models for electrodiffusion, mainly in the content of ionic flow through membrane channels. We will begin with a brief background of ion channel problems and a description of PNP models. A dynamical system framework for analyzing PNP will be presented, which relies on a combination of a general theory of geometric singular perturbations and of specific structures of PNP models. Application of the analysis will be demonstrated by several concrete results that involve specifics of ion channel problems such as ion sizes, permanent charges, channel geometry, and boundary conditions.

Thursday, April 2, 2015

Title of Talk: Equilibrium states for certain non-uniformly hyperbolic diffeomorphisms.
Thurs, 2/04/2015-10:00am, TMCB 292
Todd Fisher, Mathematics Department, Brigham Young University.

Abstract: We will prove the existence and uniquencess of equilibrium states for a large class of potentials associated with certain robustly transitive diffeomorphisms. These are some of the first results for equilibrium states in the partially hyperbolic setting.

 

Fall 2014

Thursday, September 11, 2014

Title of Talk: Ergodicity of the Weil-Petersson geodesic flow
Thurs, 09/011/2014-10:00am, TMCB 292
Keith Burns, Mathematics Department, Northwestern University.

Abstract: The Weil-Petersson is a naturally defined Riemannian metric on the Teichmueller space of a surface. It descends to the moduli space which then has finite volume. All sectional curvatures of the metric are negative. It is natural to hope that the  geodesic flow for the metric will be ergodic, and this has now been proved. The proof of is greatly complicated by the fact that the metric is incomplete. The talk will outline ideas involved in the proof. This is joint work with Howie Masure and Amie Wilkinson.

Thursday, September 18, 2014

Title of Talk: Self-Avoiding Modes of Motion in a Deterministic Lorentz Lattice Gas
Thurs, 09/18/2014-10:00am, TMCB 292
Ben Webb, Department of Mathematics, BYU.

Abstract: We study the motion of a particle on the two-dimensional honeycomb lattice whose sites are occupied by flipping rotators, which scatter the particle according to a deterministic rule. What we find is that the particle's trajectory is a self-avoiding walk between returns to its initial position. We show that this behavior is a consequence of the deterministic scattering rule and the particular class of initial scatterer configurations we consider. This result allows us to deterministically generate a self-avoiding walk, in contrast to the standard way of using random motion to generate this type of motion. This is joint work with E.G.D. Cohen.

Thursday, September 25, 2014

Title of Talk: A Separating Surface for Sitnikov-like Problems
Thurs, 09/25/2014-10:00am, TMCB 292
Skyler Simmons , Department of Mathematics, BYU.

Abstract: We consider the a generalization of the Sitnikov problem of Newtonian mechanics.  For periodic, planar configurations of n bodies which are symmetric under rotation by a fixed angle, the z-axis is invariant. We consider the effect of placing a massless particle on the z-axis. The study of the motion of this particle can then be modeled as a time-dependent Hamiltonian System. We give a geometric construction of a surface in phase space separating orbits for which the massless particle escapes to infinity from those for which it does not. The construction is demonstrated numerically in a few examples. This is joint work with Lennard Bakker.

Thursday, October 9, 2014

Title of Talk: On the perfect reconstruction of the topology of dynamic networks
Thurs, 10/09/2014-10:00am, TMCB 292
Alan Veliz-Cuba, Department of Mathematics, University of Houston & Department of Biochemistry and Cell Biology, Rice University.

Abstract: The network inference problem consists in reconstructing the topology or wiring diagram of a dynamic network from time-series data. Solving this problem is particularly important for gene regulatory networks, because in many cases regulation mechanisms are unknown or cannot be detected directly. Even though this problem has been studied in the past, there is no algorithm that guarantees perfect reconstruction of the topology of a dynamic network. In this talk I will present a framework and algorithm to solve the network inference problem for discrete-time networks that, given enough data, is guaranteed to reconstruct the topology of a dynamic network perfectly. The framework uses tools from algebraic geometry

Tuesday, October 14, 2014

Title of Talk: Random Z^d-shifts of finite type
Tues, 10/14/2014-10:00am, TMCB 292
Kevin McGoff, Department of Mathematics, Duke University.

Abstract: Let A be a finite set, and let d>=1. For n in N and p in [0,1], define a random subset w of A^[1,n]^d by independently including each pattern in A^[1,n]^d with probability p. Let X_w be the (random) Z^d-SFT built from the set w. For each p in [0,1] and n tending to infinity, we compute the limit of the probability that X_w is empty, as well as the limiting distribution of entropy of X_w. Furthermore, we show that the probability of obtaining a nonempty system without periodic points tends to zero.

For d>1, the class of Z^d-SFTs is known to contain strikingly different behavior than is possible within the class of Z-SFTs. Nonetheless, the results of this work suggest a new heuristic: typical Z^d-SFTs have similar properties to their Z-SFT counterparts.

Thursday, October 23, 2014

Title of Talk: Hyperbolicity of some asymmetric lemon tables
Tues, 10/23/2014-10:00am, TMCB 292
Pengfei Zhang, Department of Mathematics, University of Houston.

Abstract: Bunimovich stadium is a famous billiard system on a convex domain while being completely hyperbolic. In fact, Bunimovich discovered a new mechanism--defocusing, that generates hyperbolic billiards with focusing boundary components. A widely accepted ingredient of this mechanism, after generalizations by several experts (Wojtkowski, Markarian, Donnay and Bunimovich himself), is the sufficient separation of focusing components. We proved that some lemon billiards, which strongly violate the separation condition of focusing arcs, are completely hyperbolic. Our work shows that the defocusing mechanism, after 40 years of its discovery, has a broader application than one ever expected. This is a joint work with L. Bunimovich and H.-K. Zhang.

Tuesday, December 2, 2014

Title of Talk: A characterization of non-planar solutions of the 3-Body problem with one body moving on a line
Tues, 12/02/2014-10:00am, TMCB 292
Oscar Perdomo, Department of Mathematical Sciences, Central Conneticut State University.

Abstract: In this talk we characterize all non-planar solutions of the 3-Body Problem with one body moving on a line and the other two bodies having the same mass. This characterization establishes that the only solutions of the 3-Body Problem with these properties are those introduced in 1993 by Kennett Meyer with the name of (n+1)-non-alternating problem. As the name suggests, the (n+1)-non-alternating solutions are define for any number of bodies. We continue the talk by showing that for any n and any pair of masses m1 and m2, there exists an open set of initial conditions such that every solution of the (n+1)-non-alternating problem with initial conditions on this open set is bounded. We end the talk by showing a method to numerically find periodic solutions of the (n+1)-non-alternating problem. These periodic solutions look like the ones shown in the YouTube video http://youtu.be/PtEMb6Rvflg elaborated by the Speaker.

Tuesday, December 9, 2014

Title of Talk: Applications of polynomial dynamics to spectral theory of aperiodic infinite Jacobi matrices.
Tues, 12/09/2014-10:00am, TMCB 292
William Yessen, Department of Mathematics, Rice University.

Abstract: Aperiodic Jacobi operators, arising in the study of the physics of quasicrystals, have been widely studied for the past
thirty years. A well-developed technique for studying the (topological structure of the) spectrum of such operators relies
on dynamical properties (Axiom A, partial hyperbolicity, and other) of a certain class of polynomial maps, called the
trace maps. We shall present this technique in a general context, as well as some classical and recent results obtained by
application of this technique. We shall also state a few open problems of modern interest that relate not only to spectral
theory of the aforementioned operators, but also to some questions in holomorphic dynamics