This workshop is cosponsored by the dynamical systems groups at Maryland and Penn State, and has been held twice each year, beginning in Fall 1991. The Fall meetings are held at Penn State and the Spring meetings are held at the Department of Mathematics of the University of Maryland, College Park. Our spring meetings have been partially supported by the National Science Foundation from their beginning in 1992 through the present.

Links to previous conference programs and upcoming conference:

Jairo Bochi

Title: Angles between Oseledets spaces

Abstract: This talk is based on joint work with Pablo Lessa. We provide an example of a probability distribution on the group GL(2,R) with finite first moment such that the corresponding random product of i.i.d. matrices has two distinct Lyapunov exponents, but the angle between the Oseledets directions is not log-integrable. We prove that, on the other hand, if the second moment is finite, then this angle, if defined, is log-integrable. Next, we turn our attention to general GL(2,R)-cocycles over ergodic automorphisms, and ask ourselves if there is any criterion for log-integrability of the angle between the Oseledets directions in terms of a suitable integrability condition. The answer is negative. In fact, we show the following flexibility result: given any ergodic automorphism T of a non-atomic Lebesgue probability space, we can find a GL(2,R)-cocycle over T whose Lyapunov exponents and joint distribution of Oseledets spaces are prescribed a priori, and meeting any prescribed integrability condition. 

 

Aaron Brown

Title: Regularity of stationary measures 

Abstract: I will discuss some recent results about stationary measures for random dynamical systems.  In particular, I'll discuss questions about classification and absolute continuity of stationary measures.  I'll mostly focus on random dynamics on surfaces with some remarks on higher-dimensions.  

 

Emilio Corso

Title: Closed geodesics, periodic torus orbits and the distribution of shapes of unit lattices in number fields.

 Abstract: The quest to understand the asymptotic behaviour of periodic orbits of subgroup actions on homogeneous spaces bears substantial geometric and number-theoretic significance. The talk is concerned with one of the main strands of research in this direction, which focuses on compact orbits of diagonalizable subgroups on finite-volume quotients of Lie or algebraic groups. We will survey aspects of the interplay between geometry, dynamics and number theory, starting with the classical rank-one results on equidistribution of closed geodesics on hyperbolic manifolds; subsequently, we will turn our focus to the higher-rank case, mentioning the relevant generalizations in the works of Einsiedler-Lindenstrauss-

Michel-Venkatesh and Dang-Li, and discussing the arithmetic interpretation of periodic torus orbits in terms of orders, and their unit groups, in number fields. Finally, we address long-standing questions of Margulis and Gromov on the asymptotic distribution of shapes of periodic orbits, presenting recent joint work with F. Rodriguez Hertz, improved upon by Dang, Li and Gargava, on the topology of the set of shapes in the case of cubic fields.

 

David Fisher

Title: Finiteness of totally geodesic submanifolds.

Abstract:  I will give a survey of recent results concerning finiteness of totally geodesic manifolds in negativelycurved manifolds. 

 

Basak Gürel

Title: Invariant Sets and Hyperbolic Periodic Orbits

Abstract: The presence of hyperbolic periodic orbits or invariant sets often has an effect on the global behavior of a symplectic dynamical system. In this talk we discuss two theorems along the lines of this phenomenon, extending some properties of Hamiltonian diffeomorphisms to dynamically convex Reeb flows on the sphere in all dimensions. The first one, complementing other multiplicity results for Reeb flows, is that the existence of a hyperbolic periodic orbit forces the flow to have infinitely many periodic orbits. This result can be thought of as a step towards a higher-dimensional Franks’ theorem or forced existence of periodic orbits for Reeb flows. The second result is a contact analogue of the higher-dimensional Le Calvez-Yoccoz theorem proved by the speaker and Ginzburg and asserting that no periodic orbit of a Hamiltonian pseudo-rotation of a complex projective space is locally maximal. The talk is based on a joint work with Erman Cineli, Viktor Ginzburg and Marco Mazzucchelli.

 

Misha Guysinsky 

Title: Distribution of the areas of the countries.

Abstract: At the end of the 1990s, V. Arnold suggested the following model that could explain certain properties of the areas of countries. Let the areas of the countries be denoted as x_1, x_2, ... , x_N. We fix a k x k column stochastic matrix M. In each iteration, we randomly select k countries x_{i_1}, x_{i_2},..., x_{i_k}, form a vector \mathbf{x}, and calculate a new vector M \mathbf{x}. The coordinates of this new vector are then used as the new areas for the selected countries. For example, during an iteration, two countries may merge, or one country may split into two countries of equal area. Specifically, if we select three countries with areas x, y, and z, we replace them with areas x + y, z/2, and z/2. It was observed in computer experiments that, for large N, the distribution of the areas of the countries converges to the stable distribution. We will prove it rigorously. This problem is equivalent to analyzing random products of N x N stochastic matrices and proving that, as N goes to infinity, the corresponding stationary measure on (N-1)-dimensional simplices concentrates around points whose coordinates form the same distribution. This is a joint work with J. Paik.

 

Bryna Kra 

Title : The density finite sums theorem

Abstract : Since Szemeredi's Theorem and Furstenberg's proof thereof using ergodic  theory, dynamical methods have been used to show the existence of numerous configurations in sets of positive upper density. Until recently, a common feature of all of these configurations is that they were all finite. Resolving questions and conjectures of Erdos, we use dynamical methods to prove a density version of the finite sums theorem of Hindman. This is joint work with Joel Moreira, Florian Richter, and Donald Robertson.

 

Egor Shelukhin

Title: A symplectic Hilbert-Smith conjecture

Abstract: A well-known open question in group actions and topological dynamics is the Hilbert-Smith conjecture. It extends Hilbert's fifth problem and states that a locally compact group acting faithfully on a manifold must be a Lie group. We discuss a recent result regarding p-adic actions by homeomorphisms of symplectic nature, which provides new evidence to support this conjecture. 

 

Pablo Shmerkin

Title: Higher rank Furstenberg slicing

Abstract: I will discuss upper bounds for the dimensions of the affine and smooth slices of Cartesian products of Cantor sets invariant under multiplication by p_i on the circle. These upper bounds generalize the case of linear slices of products of two invariant Cantor sets, which is the original Furstenberg slicing problem. The higher rank version requires several new tools and ideas. Joint work with Emilio Corso.

   

Masato Tsuji

Title: Regularity of Stable and Unstable Foliations in Anosov Flows and Partially Hyperbolic Dynamical Systems

Abstract: This talk addresses the regularity of the stable and unstable foliations in Anosov flows and partially hyperbolic dynamical systems. Even when assuming the systems to be smooth, it is known that the holonomy maps of the stable and unstable foliations are generally not smooth, but only Hölder continuous. This irregularity presents significant challenges in the analysis of smooth dynamical systems. We introduce a new approach to this problem using "non-stationary normal trivializations" of the normal bundle of the stable and unstable manifolds. We then explore a few implications of this method for the ergodic properties of Anosov flows and partially hyperbolic dynamical systems.

 

Hao Wu

Title: A temporal central limit theorem for irrational rotations

Abstract: Dynamical systems are deterministic systems. However, in chaotic systems where the entropy is positive, the ergodic sums often behave similarly to the sums of independent random variables and satisfy the spatial central limit theorems (CLT) according. In zero-entropy systems, the spatial CLT often fails due to the lack of (fast) mixing properties. But in some cases, such as irrational rotations of bounded type, one can retrieve the central limit theorem if we study single orbit statistics, where we fix the starting point and randomise time, hence the word “temporal”. In this talk, I will present an ongoing joint work with Bromberg and Ulcigrai, where we generalise their previous method of coding and Markov chains to obtain a temporal central limit theorem for a broader class of observables over bounded irrational rotations.

 

SHORT TALKS:

Jason Day

Title: Topologically mixing suspension flows over shift spaces

Abstract: We show the necessary and sufficient conditions for suspension flows over certain families of shift spaces to be topologically mixing.  We also show that the set of roof functions that produce non-mixing suspension flows is dense in the set of all continuous roof functions.  This is joint work with the late Todd Fisher.

 

Gregory Hemenway

Title: Conjugacies of Expanding Skew Products

Abstract: Historically, the Ruelle-Perron-Frobenius (RPF) theorem has provided a fundamental framework for understanding the long-term behavior of dynamical systems and their associated equilibrium states. We will discuss how the Hilbert metric, a powerful tool for analyzing convex cones in Banach spaces, can be employed to construct a family of non-stationary measures which satisfy a non-stationary RPF theorem. In particular, we will see how these measures can be used to build a C^{1+\alpha} conjugacy of the model expanding map on T^2 that preserves Lebesgue measure.
 
 

Homin Lee

Title: Partially hyperbolic actions on nilmanifolds by higher rank groups

Abstract: In this talk, we will discuss the rigidity of actions on nilmanifolds with a partially hyperbolic diffeomorphism by higher rank abelian groups or lattices. I will discuss motivations, known results, and work with Sven Sandfeldt as well as ongoing works with Sven Sandfeldt and Kurt Vinhage in this direction.

 

Thomas O’Hare 

Title: Finite Periodic Data Rigidity For Two-Dimensional Area-Preserving Anosov Diffeomorphisms

Abstract: Let f,g be C^2 area-preserving Anosov diffeomorphisms on T^2 which are topologically conjugated by a homeomorphism h. It was proved by de la Llave in 1992 that the conjugacy h is automatically C^{1+} if and only if the Jacobian periodic data of f and g are matched by h for all periodic orbits. We prove that if the Jacobian periodic data of f and g are matched by h for all points of some large period N, then f and g are ``approximately smoothly conjugate." That is, there exists a a C^{1+\alpha} diffeomorphism h_N that is exponentially close to h in the C^0 norm, and such that f and f_N:=h_N^{-1}\circ g\circ h_N is exponentially close to f in the C^1 norm.

 

Abror Pirnapasov

Title: C^{0} Stability of Topological Entropy for Reeb and Geodesic Flows

Abstract: In this talk, we establish the lower semi-continuity of topological entropy for closed contact 3-manifolds in the C^0topology for C^\infty-generic contact forms. By applying these techniques to geodesic flows on surfaces, we show that non-degenerate metrics are among the points of lower semi-continuity. In particular, for a geodesic flow with positive topological entropy, this property persists under sufficiently small C^0-perturbations of the metric. This work is joint with Marcelo Alves, Lucas Dahinden, and Matthias Meiwes

 

Chengyang Shao

Title: Paradifferential calculus and dynamical conjugacy problems

Abstract: In this talk, I will introduce how paradifferential calculus can be applied to dynamical conjugacy problems involving "loss of regularity". A powerful tool in modern harmonic analysis, paradifferential operators share the algebraic structure of usual differential and Fourier integral operators, while on the other hand provide a scheme in which a nonlinearity can be manipulated as if it were linear, with delicate control of regularity. Therefore, paradifferential calculus not only serves as a promising alternative to the usual KAM/Nash-Moser method, but also applies to problems where the "loss of regularity" disables the latter. The talk is based on joint works with Thomas Alazard, together with a recent preprint of Giovanni Forni. 

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 Spring 2025 Maryland-Penn State Semiannual Workshop on Dynamical Systems and Related Topics

SCHEDULE

Schedule 2

Beginning in Fall 1991, the dynamics groups of the University of Maryland and Pennsylvania State University have jointly sponsored two annual three to four day meetings in dynamical systems and related topics: a spring meeting here in College Park, and a fall meeting in State College. These meetings, though primarily regional, also attract participants outside the region and the country; the conference archives indicate their scope.

All the talks will take place in the Colloquium Room (3rd floor) in the Math Department (Kirwan Hall) at UMD.

Confirmed speakers:

Jairo Bochi (PSU) 

Aaron Brown (Northwestern)

Emilio Corso (PSU)

David Fisher (Rice)

Basak Gurel (UCF, Orlando)

Misha Guysinsky (PSU)

Bryna Kra (Northwestern) 

Egor Shelukhin (Montreal) 

Pablo Shmerkin (British Columbia)

Masato Tsujii (Kyoto) 

Hao Wu (Zurich) 

There will be also 8 talks by students and post-docs.

 

2014 12 11 15.07.42 1

Dynamics at the University of Maryland is distinguished by its quality, breadth and activity. The Chaos Group spans several departments and units on campus. In this website you will find information regarding the Mathematics Department.

 

Faculty

  • Joe Auslander (Emeritus): topological dynamics
  • Ken Berg (Emeritus): ergodic theory, topological dynamics
  • Dmitry Dolgopyat: hyperbolic dynamics, smooth ergodic theory,  probability, mathematical physics
  • Giovanni Forni: conservative dynamics, parabolic dynamics, smooth dynamics, smooth ergodic theory, Teichmueller dynamics
  • Brian Hunt (Emeritus): chaotic dynamics, smooth dynamics
  • Michael Jakobson: one dimensional dynamics, smooth ergodic theory, stochastic dynamics
  • Vadim Kaloshin: Arnold diffusion, celestial mechanics, hamiltonian PDEs, KAM theory, smooth dynamics, stochastic dynamics
  • Adam Kanigowski: ergodic theory
  • Rodrigo Trevino : ergodic theory, smooth dynamics, geometry and mathematical physics
  • Jim Yorke (research professor): applied mathematics, chaotic dynamics, smooth dynamics 

Dynamics at UMD is intrinsically wide ranging, and even within the mathematics department the interest area goes beyond the above list, for example in the work of Mark Freidlin and Leonid Koralov (random perturbations of dynamical systems), William Goldman (ergodic theory on moduli spaces), Karin Melnick (group actions on manifolds), and Scott Wolpert (dynamics of the Weil-Petersson flow).

 

Postdoctoral fellows and visitors

  • Alex Blumenthal: smooth ergodic theory, infinite dimensional dynamics, PDEs, stochastic dynamics
  • Thibaut Castan: celestial mechanics
  • Changguang Dong: smooth ergodic theory
  • Davit Karagulyan: Mobius orthogonality, ergodic theory.
  • Peter Nandori: hyperbolic dynamical systems, probability theory and mathematical physics

Former postdoctoral fellows

  • Vaughn Climenhaga : non-uniform hyperbolicity, thermodynamic formalism, dimension theory, and multifractal analysis
  • Todd Fisher : robust transitivity, hyperbolic dynamical systems, symbolic dynamics, and topological dynamics
  • Marcel Guardia : Hamiltonian Systems, Celestial Mechanics, Hamiltonian PDE
  • Guan Huang: Hamiltonian PDEs
  • Yuri Lima : smooth ergodic theory, fractal geometry, probability and combinatorics
  • Pablo Roldan : celestial mechanics
  • Paul Wright: hyperbolic dynamcs
  • Kelly Yancey: abstract ergodic theory, topological dynamics
  • Ke Zhang : Hamiltonian Dynamics, thermodynamical formalism.

Graduate students

  • Hamid Al-Saqban
  • Keagan Callis
  • Minsung Kim
  • Jianlong Liu
  • Jenny Rustad
  • Jing Zhou

Former Graduate students

 

Activities

The Dynamics group at UMD organizes the following activities.

Dynamics seminar

The meetings occur on Thursdays from 2--4pm, and are divided into two parts: the first one is an introduction of the topic that is accessible to graduate students, and the second is a discussion/explanation of the (idea of the) proof of the main result. After the seminar there is a social hour that offers drinks and snakcs.

Student Dynamics Seminar

Student dynamics seminar meetson Tuesdays 2-3pm. It features talks by graduate students where the speakers discuss their research interests.

Applied Dynamics Seminar

The seminar meets on Thursdays at 12:30pm. It brings in a range of visitors from labs, agencies and other universities in the region, and features dynamics related talk over pizza lunch.

 

Maryland-Penn State Dynamics Conference

 

Beginning in Fall 1991, the dynamics groups of the University of Maryland and Pennsylvania State University have jointly sponsored two annual three and half day meetings in dynamical systems and related topics: a spring meeting here in College Park, and a fall meeting in State College. These meetings, though primarily regional, also attract participants outside the region and the country. Along with talks of prominent mathematicians, the conferences always include several short talks by graduate students. 

The Michael Brin Chair in Mathematics

Vadim Kaloshin occupies the Michael Brin Chair in Mathematics. 
This position controls substantial discretionary funds, and is permanent as long as the appointed person remains a professor of mathematics at Maryland. The Chair also sponsors Brin Postdoctoral Fellows. 

 

The Michael Brin Prize in Dynamical Systems

This is an international prize awarded by an international panel of dynamics luminaries. It recognizes mathematicians who have made substantial impact at an early stage of their careers. The first two prizes were awarded to Maryland faculty members Giovanni Forni and Dmitry Dolgopyat. Starting from 2017 the prize is awarded annually. During the even years the Brin Prize is awarded during Maryland Dynamics Conference.

Selected recent and upcoming events

Summer School on Dynamical Systems, August 17--25, 2014, College Park MD.

Program on Hamiltonian systems, from topology to applications through analysis August 13--December 14, 2018, MSRI, Berkeley, CA.

School on Probabilistic methods in negative curvature March 11--22, 2019, Bangalore, India.

2019 Workshop on Dynamical Systems and Related Topics, April 11--14, 2019, College Park, MD.