Dynamics Archives for Fall 2017 to Spring 2018

Introduction to geodesic flows and hyperbolicity

When: Tue, August 2, 2016 - 12:30pm
Where: Kirwan Hall 1308
Speaker: Sarah Bray (University of Michigan) - http://www-personal.umich.edu/~brays/
Abstract: In this minicourse, I'll gently introduce the Patterson-Sullivan program for
studying hyperbolic geodesic flows, and survey some modern contexts to which the program has been applied.

Ergodic geometry: Introduction to Sullivan-Patterson densities

When: Thu, August 4, 2016 - 12:30pm
Where: Kirwan Hall 1308
Speaker: Sarah Bray (University of Michigan) - http://www-personal.umich.edu/~brays/
Abstract: In this minicourse, I'll gently introduce the Patterson-Sullivan program for
studying hyperbolic geodesic flows, and survey some modern contexts to which the program has been applied.

Entropy-maximizing measures: examples beyond uniform hyperbolicity

When: Tue, August 9, 2016 - 12:30pm
Where: Kirwan Hall 1308
Speaker: Sarah Bray (University of Michigan) - http://www-personal.umich.edu/~brays/
Abstract: In this minicourse, I'll gently introduce the Patterson-Sullivan program for
studying hyperbolic geodesic flows, and survey some modern contexts to which the program has been applied. The lecture titles are:

1. Introduction to geodesic flows and hyperbolicity

2. Ergodic geometry: Introduction to Sullivan-Patterson densities

3. Entropy-maximizing measures: examples beyond uniform hyperbolicity

The moving sofa problem (joint Dynamics - Geometric Analysis seminar)

When: Thu, September 8, 2016 - 2:00pm
Where: Kirwan Hall 1311
Speaker: Dan Romik (University of California, Davis) - https://www.math.ucdavis.edu/~romik/
Abstract: As everyone knows from real-life experience, moving sofas around corners is often tricky: not every sofa shape will fit. The moving sofa problem is a mathematical question that elegantly captures the subtleties involved even in a simplified two-dimensional setting. It asks for the planar shape of maximal area that can be moved around a right-angled corner in a corridor of width 1. The problem has been open for 50 years, and has a complicated conjectured solution known as Gerver's sofa, proposed in 1992 - a shape whose boundary has 18 distinct pieces. In this talk I will explain the mathematics of this fascinating problem, and tell about a new approach to the study of the problem that I developed recently, and some additional related results. The talk will be self-contained, will require no prerequisite knowledge beyond standard calculus, and will include many entertaining animations of moving sofas.

A class of infinite translation surfaces where almost every direction is uniquely ergodic

When: Thu, September 8, 2016 - 3:00pm
Where: Kirwan Hall 1311
Speaker: Anja Randecker (University of Toronto) - http://www.math.toronto.edu/anja/
Abstract: Translation surfaces can be obtained from gluing finitely many polygons along parallel edges of the same length. In recent years, people have asked what happens when you glue infinitely instead of finitely many polygons. From that question the field of infinite translation surfaces has evolved.
It turns out that the behaviour of infinite translation surfaces is in many regards very different and more diverse than in the finite case. For instance, Kerckhoff, Masur, and Smillie showed in 1986 that on a finite translation surface the geodesic flow is uniquely ergodic in almost every direction. This is not at all true for infinite translation surfaces in general.
However, in this talk, I will introduce a class of infinite translation surfaces where the statement remains true. I will recall the original proof from Kerckhoff, Masur, and Smillie and show how the proof has to be adapted and why that class of infinite translation surfaces was chosen.
The presented work is joint with Kasra Rafi.

Coarse equivalence, topological couplings and a theorem of Gromov

When: Thu, September 22, 2016 - 2:10pm
Where: Kirwan Hall 1311
Speaker: Christian Rosendal (UI Chicago and UMD) - http://homepages.math.uic.edu/~rosendal/
Abstract: A seminal theorem of M. Gromov states that two finitely generated groups are quasi-isometric if and only if they admit a topological coupling, thus establishing a link between the geometry and topological dynamics of groups. Much work has been done recently on expanding the tools and results of geometric group theory to locally compact groups and beyond and we shall explain how Gromov's theorem admits generalisations to all locally compact groups and even a special class of non-locally compact topological transformation groups. (The initial part of the talk will be based on joint work with U. Bader.)

Ergodic Actions of Lattices in Higher-Rank Semisimple Groups

When: Thu, October 6, 2016 - 2:10pm
Where: Kirwan Hall 1311
Speaker: Darren Creutz (US Naval Academy) - http://www.dcreutz.com/mathematics
Abstract: Lattices in higher-rank semisimple groups arise naturally in many areas of mathematics, and include groups such as SL_n[Z] for n >= 3. These groups exhibit a variety of rigidity properties, most notably the results of Margulis--the Normal Subgroup Theorem that every nontrivial normal subgroup of an irreducible lattice in a center-free higher-rank semisimple group has finite index and the Superrigidity Theorem that every isomorphism of such a lattice into any algebraic group either has precompact image or extends to the ambient semisimple group.

I will present work of myself and J. Peterson generalizing both of these theorems. The main focus of the talk will be on our theorem that every ergodic action of such a lattice on a nonatomic probability space is essentially free (taking the action to be the Bernoulli shift on the lattice modulo a normal subgroup recovers the NST); the proof of which involves a careful understanding of the dynamics of the Poisson boundary and of Howe-Moore groups. I will also present (largely without proof) our operator-algebraic superrigidity theorem that any representation of such a lattice as unitary operators on a finite von Neumann algebra is either finite-dimensional (hence coming from a quotient by a finite index normal subgroup) or extends to the entire group von Neumann algebra of the lattice.

From Siegel-Veech to Eskin-Mirzakhani-Mohammadi

When: Thu, October 13, 2016 - 2:10pm
Where: Kirwan Hall 1311
Speaker: Giovanni Forni (UMD) -
Abstract: TBA

Decay of correlations for maximal measure of maps derived from Anosov.

When: Thu, October 20, 2016 - 2:10pm
Where: Kirwan Hall 1311
Speaker: Fan Yang (Universidade Federal do Rio de Janeiro, Brazil) -
Abstract: It was proven by Ures that C^1 diffeomorphism of T^3 that is derived from Anosov admits a unique measure of maximal entropy. With Jiagang Yang we show that the maximal measure has exponential decay of correlations for H\"older observables, assuming the middle eigenvalue of the linear Anosov model is contracting. I will also discuss the case when the center is expanding and how to generalize the result to other C^1 maps.

Temporal distributional limit theorems for dynamical systems (joint with D. Dolgopyat)

When: Thu, November 3, 2016 - 2:10pm
Where: Kirwan Hall 1311
Speaker: Omri Sarig (Weizmann Institute of Science ) - http://www.wisdom.weizmann.ac.il/~sarigo/
Abstract: The orbits of zero entropy uniquely ergodic map do not
always all have the same qualitative behavior, but to expose the
richness of the orbit structure one needs to look at second order
asymptotic behavior such as the error term in the ergodic theorem.
“Temporal distributional limit theorems” are a probabilistic tool for
doing this. Time permitting, we will consider the examples of the
irrational rotation and the horocycle flow.
Joint with D. Dolgopyat.

Blow-ups of partially hyperbolic diffeomorphisms

When: Thu, November 10, 2016 - 2:10pm
Where: Kirwan Hall 1311
Speaker: Andrey Gogolev (Binghamton) - http://www.math.binghamton.edu/agogolev/
Abstract: The blow-up construction replaces a submanifold of a given manifold with a space of lines
(real or complex) normal to the submanifod. We will explain how this construction can be
used to produce new examples of partially hyperbolic dynamical systems by "blowing-up"
existing examples. A modification of this technique allows, under certain assumptions, to
take a "connected sum" of two partially hyperbolic diffeomorphisms or flows. For example,
one can connect-sum the geodesic flow on a complex hyperbolic manifold and a product
flow and obtain a hybrid partially hyperbolic flow.

Smooth K non Bernoulli automorphisms in dimension 4

When: Thu, November 17, 2016 - 2:00pm
Where: Kirwan Hall 1311
Speaker: Adam Kanigowski (Penn State) - http://www.adkanigowski.cba.pl/en.php
Abstract: We give examples of smooth $K$, non-Bernoulli automorphisms on $\mathbb{T}^4$. They arise by taking a skew-product (with a smooth skewing function) of an Anosov map on $\mathbb{T}^2$ with a smooth Kochergin flow (with high degeneracy of the saddle) on $\mathbb{T}^2$. Joint work with Federico Rodriguez-Hertz and Kurt Vinhage.

Invariant geometric structures for measurable contractions of <b>R</b><sup>n</sup> bundles

When: Thu, December 8, 2016 - 2:10pm
Where: MTH1311
Speaker: Karin Melnick (UMD) - http://www.math.umd.edu/~kmelnick/
Abstract: Normal forms theorems for contracting diffeomorphisms of R^n have a long history, starting with Poincaré and including fundamen​tal results by Sternberg. They state that such diffeomorphisms are conjugate to linear maps or certain polynomials of bounded degree. A nonstationary version for C^0 automorphisms of R^n bundles that are uniformly contracting on fibers, and their centralizers, was proved by Guysinsky and A. Katok in 1998. I will present a differential-geometric approach to polynomial normal forms in the nonuniform setting, for measurable automorphisms of R^n bundles and their centralizers. In the case that the system comes from a contracted foliation in a manifold, these normal forms lead to homogeneous structures on the leaves.

Lattice actions and recent progress in the Zimmer program

When: Fri, January 20, 2017 - 2:00pm
Where: 1308 or 1311
Speaker: Aaron Brown-University of Chicago- awbrown@math.uchicago.edu-http://math.uchicago.edu/~awbrown/

Abstract: The Zimmer Program is a collection of conjectures and questions regarding actions of lattices in higher-rank simple Lie groups on compact manifolds. For instance, it is conjectured that all non-trivial volume-preserving actions are built from algebraic examples using standard constructions. In particular, on manifolds whose dimension is below the dimension of all algebraic examples, Zimmer’s conjecture asserts that every action is finite.

I will present some background, motivation, and selected previous results in the Zimmer program. I will then explain two of my own results within the Zimmer program:
(1) a solution to Zimmer’s conjecture for actions of cocompact lattices in SL(n,R), n>=3 (joint with D. Fisher and S. Hurtado);
(2) a classification (up to topological semiconjugacy) of lattice actions on tori whose induced action on homology satisfies certain criteria (joint with F. Rodriguez Hertz and Z. Wang).

Using Symbolic Dynamics to Determine Similarity Between Regular Languages

When: Thu, January 26, 2017 - 2:10pm
Where: Kirwan Hall 1311
Speaker: Kelly Yancey, Matthew Yancey (Institute for Defense Analyses) -
Abstract: A problem that has emerged in computer science is determining the similarity between regular languages. We will represent a regular language by a deterministic finite automaton (a directed graph with some marked data) and then use ideas from symbolic dynamics to define a metric between the languages. We will also discuss other distances based on the classical Jaccard distance and how they are related to the topological entropy of a regular language. There will be no prior knowledge of automata assumed.

Thermodynamics of the Katok map

When: Thu, February 2, 2017 - 2:00pm
Where: Kirwan Hall 1311
Speaker: Yakov Pesin (Penn State) - https://www.math.psu.edu/pesin/
Abstract: I describe the thermodynamic formalism for the smooth non-uniformly hyperbolic map of the two dimensional torus known as the Katok map. It is a slowdown of a linear Anosov automorphism near the origin and it is a local (but not small) perturbation. I demonstrate existence and uniqueness of equilibrium measures associated with the geometric potential and discuss their ergodic properties including the decay of correlations and the Central Limit Theorem. The underlying techniques is a representation of the Katok map as a Young tower.
This is a joint work with S. Senti and K. Zhang.

A result of Dorothy Maharam from 1965, Then and Now

When: Thu, February 9, 2017 - 2:00pm
Where: Kirwan Hall 1311
Speaker: Terry Adams (Intelligence advanced research projects activity) -
Abstract: In 1965, the AMS Transactions published an article authored by Dorothy Maharam entitled "On orbits under ergodic measure preserving transformations". Professor Maharam gave conditions for sequences of points in a Lebesgue space that uniquely determine a measure preserving transformation with prescribed properties (i.e., ergodic, weak mixing, strong mixing). We will discuss this article, along with recent research on reconstructing or estimating transformations. Also, we will show some surprising research that has descended from Professor Maharam's article.

Characters and Dynamical Properties of Thompson's Group F

When: Thu, March 2, 2017 - 2:00pm
Where: Kirwan Hall 1311
Speaker: Kostya Medynets (U.S. Naval Academy) - https://www.usna.edu/Users/math/medynets/index.php
Abstract: In the talk, we will discuss dynamical properties of the action of Thompson's group F on the unit interval, using which we will show that the group F admits no non-trivial characters. Implications for the structure of invariant random subgroups will be also discussed.

Lyapunov exponents and decay of correlations for random perturbations of some 2D surface maps, including the standard map

When: Thu, March 16, 2017 - 2:00pm
Where: Kirwan Hall 1311
Speaker: Alex Blumenthal (UMD) - http://www.mrl.nyu.edu/~alex/
Abstract: Estimating Lyapunov exponents of a deterministic dynamical system from below is an extremely difficult cancellation problem, especially for systems which are hyperbolic on a "large" but noninvariant subset of phase space. The Chirikov standard map (with coefficient L) has come to symbolize this challenge: for large L, it is an area preserving map of the torus exhibiting expansion of order L on a subset of phase space of area \geq 1 - O(L^{-1}), whereas it remains an open problem to show that the Lyapunov exponent of the standard map is > 0 on any positive-area subset.

We show that for sufficiently large (although extremely small, of amplitude greater than e^{-L}) random perturbations, the standard map has a positive Lyapunov exponent of size approximately \log L. For larger perturbation sizes (of amplitude greater than L^{-1}), we obtain averaged decay of correlations at a rate proportional to L.

Approximate orthogonality of powers for ergodic affine unipotent diffeomorphisms on nilmanifolds

When: Tue, March 28, 2017 - 2:00pm
Where: MTH B0427
Speaker: Livio Flaminio (Universite de Lille 1, France) - http://math.univ-lille1.fr/~flaminio/
Abstract: We prove that any ergodic affine unipotent diffeomorphisms of a compact
nilmanifold enjoys the property of asymptotically orthogonal
powers (AOP). Two consequences follow: (i) Sarnak's conjecture on
M\"obius orthogonality holds in every uniquely ergodic model of an
ergodic affine unipotent diffeomorphism; (ii) For ergodic affine
unipotent diffeomorphisms themselves, the M\"obius orthogonality holds
on so called typical short intervals.

(Joint work with K.\ Fr\k{a}czek, J.\ Ku\l aga-Przymus and
M.\ Lema\'nczyk.)

Sequences modulo one: convergence of local statistics

When: Thu, March 30, 2017 - 2:00pm
Where: Kirwan Hall 1311
Speaker: Ilya Vinogradov (Princeton University) - https://web.math.princeton.edu/~ivinogra/
Abstract: The study of randomness of fixed objects is an area of active
research with many exciting developments in the last few years. We will
discuss recent results about sequences in the unit interval specializing to
directions in affine lattices, \sqrt n modulo 1, and directions in hyperbolic lattices.
Theorems about these sequences address convergence of moments as well as rates of convergence, and their proofs showcase a beautiful interplay between dynamical systems and number theory. I plan to focus on hyperbolic lattices for the more technical part of the talk.

Compactifications of strata of translation surfaces

When: Thu, April 6, 2017 - 2:00pm
Where: Math 1311
Speaker: Matt Bainbridge- Indiana University-Bloomington
Abstract: A Riemann surface X equipped with a holomorphic one-form w has a
natural flat metric with cone points at the zeros of w. Such an
object is otherwise known as a translation surface. The moduli space
of pairs (X,w) where the cone angles (equivalently the orders of the
zeros) have been fixed is a stratum of translation surfaces. These
strata are the natural setting for Teichmuller dynamics: there is an
action of SL(2,R) which has been intensely studied in recent years.

In this talk, we discuss the problem of compactifying these strata.
We propose a smooth compactification of these strata. We speculate on
connections between topological invariants of this compactified moduli
space and dynamical invariants of the Teichmuller geodesic flow.

This is part of joint work with Chen, Gendron, Gruchevsky, and Moeller.

A Few Fairy Math Tales

When: Thu, April 13, 2017 - 2:00pm
Where: Kirwan Hall 1311
Speaker: Dmitri Burago (Penn State) -
Abstract: The format of this talk is rather non-standard. It is actually a combination of
several mini-talks. They would include only motivations, formulations, basic
ideas of proof if feasible, and open problems. No technicalities. Each topic
would be enough for 2+ lectures. However I know the hard way that in diverse
audience, after 1/3 of allocated time 2/3 of people fall asleep or start playing
with their tablets. I hope to switch to new topics at approximately right times.
I include more topics that I plan to cover for I would be happy to discuss others
after the talk or by email/skype. I may make short announcements on these
extra topics. The topics will probably be chosen from the list below. I sure will
not talk on topics I have spoken already at your university.

“A survival guide for feeble fish”. How fish can get from A to B in turbulent
waters which maybe much fasted than the locomotive speed of the fish provided
that there is no large-scale drift of the water flow. This is related to homogenization of G-equation which is believed to govern many combustion processes. Based on a joint work with S. Ivanov and A. Novikov.

How can one discretize elliptic PDEs without using finite elements, triangulations
and such? On manifolds and even reasonably "nice" mm–spaces. A notion of
\rho-Laplacian and its stability. Joint with S. Ivanov and Kurylev.

"What is inside?" Imagine a body with some intrinsic structure, which, as usual, can be thought of as a metric. One knows distances between boundary points (say, by sending waves and measuring how long it takes them to reach specific points on the boundary). One may think of medical imaging or geophysics. This topic is related to another topic on minimal fillings. Joint work with S. Ivanov.

How well can we approximate an (unbounded) space by a metric graph whose
parameters (degree of vertices, length of edges, density of vertices etc) are uniformly bounded? We want to control the ADDITIVE error. Some answers (the most difficult one is for $\R^2$) are given using dynamics and Fourier series.

Ellipticity of surface area in normed spaces. An array of problems which go back to Busemann. They include minimality of linear subspaces in normed spaces and constructing surfaces with prescribed weighted image under the Gauss map. I would try to give a report of recent developments, in a nutshell. Our interest to these problems came from our attempts to attack questions discussed in “what is inside?” mini-talk. Joint with S. Ivanov

More stories are left in my left pocket. Possibly: On making decisions under uncertain information, both because we do not know the result of our decisions and we have only probabilistic data; On the intrinsic geometry of surfaces in normed spaces; Unbounded groups with bi-invariant metrics; Area spaces.

On rank and isomorphism of von Neumann special flows

When: Tue, April 25, 2017 - 2:00pm
Where: Kirwan Hall B0427
Speaker: Anton Solomko (University of Bristol, UK) - https://people.maths.bris.ac.uk/~as14789/index.html
Abstract: A von Neumann flow is a special flow over an irrational rotation of the circle and under a piecewise smooth roof function with a non-zero sum of jumps. Such flows appear naturally as special representations of Hamiltonian flows on the torus with critical points. We consider the class of von Neumann flows with one discontinuity. I will show that any such flow has infinite rank and that the absolute value of the jump of the roof function is a measure theoretic invariant. The main ingredient in the proofs is a Ranter type property of parabolic divergence of orbits of two nearby points in the flow direction.
Joint work with Adam Kanigowski.

An afternoon in honor of Sergei Novikov (notice special schedule)

When: Wed, April 26, 2017 - 2:00pm
Where: MATH 3206
Speakers: Gennadi Kasparov (Vanderbilt University), Igor Krichever (Columbia University), and Anton Zorich (Paris Jussieu)

Abstract: see https://www-math.umd.edu/research/conferences/afternoon-of-geometric-analysis.html

Lower bounds for Lyapunov exponents of flat bundles on curves

When: Thu, April 27, 2017 - 2:00pm
Where: Kirwan Hall 1311
Speaker: Anton Zorich (Institut Mathematiques de Jussieu, Paris, France) - https://webusers.imj-prg.fr/~anton.zorich/
Abstract: (Joint work with A. Eskin, M. Kontsevich, M. Moeller)

Consider a flat bundle over a complex curve. Associated Lyapunov
exponents describe mean monodromy of the flat bundle along a random
hyperbolic geodesic on the base; they play a role of dynamical
analogs of characteristic numbers of the bundle.

We prove a conjecture of Fei Yu that the sum of the top k Lyapunov
exponents of the flat bundle is always greater or equal to the
degree of any rank k holomorphic subbundle. We generalize the
original context from Teichmuller curves to any local system over a
curve with non-expanding cusp monodromies. As an application we
obtain the large genus limits of individual Lyapunov exponents in
hyperelliptic strata of Abelian differentials.

Understanding the case of equality with the degrees of subbundle
coming from the Hodge filtration seems challenging, e.g. for
Calabi-Yau type families. We conjecture that equality of the sum of
Lyapunov exponents and the degree is related to the monodromy group
being a thin subgroup of its Zariski closure.

Large Time Behavior of Randomly and Deterministically Perturbed Dynamical Systems

When: Thu, May 4, 2017 - 2:00pm
Where: Kirwan Hall 1311
Speaker: Leonid Koralov (UMD) - https://www.math.umd.edu/~koralov/
Abstract: We study several asymptotic problems: averaging of incompressible flows with ergodic components, regularization of deterministically perturbed incompressible flows, transition from homogenization to averaging regimes in periodic flows, randomly perturbed flows with regions where a strong flow creates a trapping mechanism. These problems are related by a common set of probabilistic techniques that are used to solve them.

Hunting for the Fourth class of dynamical systems

When: Thu, May 11, 2017 - 2:00pm
Where: Kirwan Hall 1311
Speaker: James Yorke (UMD) - http://www.chaos.umd.edu/~yorke/
Abstract: It has recently been proved that there are 3 open sets of dynamical systems that together are C^1 dense. My collaborators and I believe they are not C^3 dense and that there is a fourth open set of dynamical systems which is disjoint from the other 3. I will describe the supposed 4th kind.