http://www-math.umd.edu/research/seminars.html http://www-math.umd.edu/research/seminars.html Thu, 05 Sep 2024 14:00:00 EDT Where: Kirwan Hall 3206
Speaker: Jon DeWitt (UMD) - https://math.umd.edu/~dewitt/
Abstract: We consider exponential mixing for volume preserving random dynamical systems on surfaces. Suppose that (f_1,...,f_m) is a tuple of volume preserving diffeomorphisms of a closed surface M. We now consider the uniform Bernoulli random dynamical system that this tuple generates on M. We assume that this tuple satisfies a condition called being "expanding on average," which means that there exist C,N>0 such that E[\ln \|Df^Nv\|]C for all unit tangent vectors v. From this assumption we show quenched exponential equidistribution as well as quenched exponential mixing. (This is joint work with Dmitry Dolgopyat)
]]> http://www-math.umd.edu/research/seminars.html Thu, 19 Sep 2024 14:00:00 EDT Where: Kirwan Hall 3206
Speaker: Corentin Fierobe (IST Austria) -

]]> http://www-math.umd.edu/research/seminars.html Thu, 03 Oct 2024 14:00:00 EDT Where: Kirwan Hall 3206
Speaker: Giovanni Canestrari (University of Rome Tor Vergata) - https://scholar.google.com/citations?user=jJDzPRYAAAAJ&hl=it

]]> http://www-math.umd.edu/research/seminars.html Thu, 10 Oct 2024 14:00:00 EDT Where: Kirwan Hall 3206
Speaker: Uri Bader (UMD) - https://www.weizmann.ac.il/math/uribader/home

Gromov’s waist inequality for the n-dimensional sphere S^n is a fundamental result in geometry. It says that the maximal volume of a fiber of a (generic) map from S^n to the d-dimensional Euclidean space is at least the (n-d)-dimensional volume of an equator sphere S^{n−d}, which is a constant times the volume of S^n. This constant is the "waist constant".
A question arises: is there an infinite family of n-dimensional compact manifolds satisfying a uniform waist inequality, that is a similar inequality with a uniform waist constant, for a given dimension d?
It is natural to consider the family of all finite covers of a given compact manifold M.
A positive answer to this question in the case d=1 is provided by the Cheeger-Buser inequality, relating the waist constant      with the spectrum of the Laplacian of M.   
In my talk I will survey gently all of the above and explain a recent solution to the case d=2, using a fixed point property for groups acting on L^1-spaces.
Based on a joint work with Roman Sauer.
]]> http://www-math.umd.edu/research/seminars.html Thu, 17 Oct 2024 14:00:00 EDT Where: Kirwan Hall 3206
Speaker: Mark Demers (Fairfield University) - https://faculty.fairfield.edu/mdemers/

We describe the recent construction of Birkhoff cones which are contracted by the action of transfer operators corresponding to dispersing billiard maps.  The explicit contraction provided by this construction permits the study of statistical properties of a variety of sequential and open billiards.  We will discuss some applications of this technique to chaotic scattering and the random Lorentz gas.  This is joint work with C. Liverani.
]]> http://www-math.umd.edu/research/seminars.html Thu, 31 Oct 2024 14:00:00 EDT Where: Kirwan Hall 3206
Speaker: Yi Pan (IST, Austria) -

In classical KAM theory, a certain nondegeneracy of the Hessian of the unperturbed  Hamiltonian is crucial to show the existence of invariant tori. And these invariant tori are graphs of Lipschitz functions. However, when such type of nondegeneracy is violating, very little is known. Inspired by some numerical examples of Simó, we will show the existence of meandering invariant tori which are not graphs and their genericity. This is a joint work in progress with Vadim Kaloshin and Illya Koval.
]]> http://www-math.umd.edu/research/seminars.html Fri, 01 Nov 2024 13:00:00 EDT Where: Kirwan Hall 3206
Speaker: Amie Wilkinson (University of Chicago) - https://math.uchicago.edu/~wilkinso/
Abstract: The centralizer Z(f) of a diffeomorphism f is its group of symmetries, the set of all diffeomorphisms that commute with it. Conjecturally, the generic diffeomorphism f commutes only with its iterates, and thus has the smallest possible centralizer (i.e., Z(f) = ); there significant evidence to support this conjecture in certain settings.

This talk will be structured around the broad question: in what settings can one classify the diffeomorphisms with “large” centralizer (where “large” could be construed in many ways, for example: the index of in Z(f) is infinite; the action of Z(f) on M is transitive; or Z(f) is isomorphic to a Lie group of sufficiently high dimension.
]]> http://www-math.umd.edu/research/seminars.html Thu, 07 Nov 2024 14:00:00 EST Where: Kirwan Hall 3206
Speaker: Mathieu Helfter (Jussieu Institute of Mathematics) 

Title:  "Every diffeomorphism is a total renormalization of a close to
identity map"

Abstract:
In a joint work with Pierre Berger and Nicolas Gourmelon, we demonstrate
the existence of a large class of manifolds of dimension at least two
for which every compactly supported smooth diffeomorphism can be
expressed as an exact renormalization of a map arbitrarily close to the
identity. Furthermore, these renormalizations are total, meaning that
the orbit of the renormalization domain covers the entire manifold. This
allows us to establish the existence of global properties for
diffeomorphisms near the identity.
]]> http://www-math.umd.edu/research/seminars.html Thu, 30 Jan 2025 14:00:00 EST Where: Kirwan Hall 3206
Speaker: Andrey Gogolev (Ohio State University) - https://people.math.osu.edu/gogolyev.1/
Abstract: We will discuss some surprising rigidity phenomena for Anosov flows in
dimension 3. For example, in the context of generic transitive 3-dimensional Anosov flows, any
continuous conjugacy is either smooth or reverses the positive and negative SRB measures. This
is joint work with Martin Leguil and Federico Rodriguez Hertz.
]]> http://www-math.umd.edu/research/seminars.html Thu, 06 Feb 2025 14:00:00 EST Where: Kirwan Hall 3206
Speaker: James Yorke (UMD) - https://yorke.umd.edu/
Abstract: Yoshi Saiki and I have been investigating examples of dynamical systems, i.e., maps, that are "heterogeneous": they are ergodic and the system has different numbers of expanding directions in different regions of the space. In this talk I investigate a variant of this problem where we have a set of scalar affine maps of the form
tau_j(x) = a_j x + c_j for j = 1,...,N,
especially when N = 2, and especially when one map is expanding and another is contracting, i.e., a_1 > 1 > a_2 > 0.
A trajectory consists of a random (iid) sequence of maps j_1, j_2, j_3, .... applied iteratively to some initial point x in R. The behaviors of such trajectories is surprising.
We have used extensive numerical simulations to guide our research to find what is likely to be true or important. The most valuable simulation is one that surprises. The title of this talk suggests what Yoshi and I can prove.

]]> http://www-math.umd.edu/research/seminars.html Thu, 13 Feb 2025 14:00:00 EST Where: Kirwan Hall 3206
Speaker: Francisco Arana Herrera (UMD) - https://terpconnect.umd.edu/~farana/
Abstract: Appealing to the theory of geodesic currents, we will explicitly compute the image of the so-called X-ray transform of a negatively curved manifold. Using this we will prove old and new results about marked length spectrum rigidity of negatively curved metrics.
]]> http://www-math.umd.edu/research/seminars.html Thu, 20 Feb 2025 14:00:00 EST Where: Kirwan Hall 3206
Speaker: Marcel Guardia (Universitat de Barcelona) - https://www.ub.edu/dynsys/mguardia/
Abstract: One of the oldest problems in dynamical systems is the stability of the Solar System. That is, consider N bodies moving following Newton's law of gravitation, one of them with large mass (the Sun) and the others with small masses (the planets). If one neglects, the gravitational interaction between planets, the classical Kepler's laws assert that the planets move on ellipses. Then, one wants to understand whether the effect of the planet's mutual attraction causes long term changes on the shape and relative position of the Keplerian ellipses.
In this talk I will explain how to construct unstable motions in a planetary 4 body problem, which lead to drastic changes in the semimajor axes, eccentricity and inclination of these ellipses. The results are based on joint work with Andrew Clarke and Jacques Fejoz.
]]> http://www-math.umd.edu/research/seminars.html Thu, 06 Mar 2025 14:00:00 EST Where: Kirwan Hall 3206
Speaker: Abed Bounemoura (CNRS-Paris Dauphine) -
Abstract: For the linearization problems of circle diffeomorphisms and germs, we will explain optimal results in Gevrey classes. Conjecturally, this should be true for any small divisors problems in any non quasi-analytic class.
]]> http://www-math.umd.edu/research/seminars.html Thu, 13 Mar 2025 14:00:00 EDT Where: Kirwan Hall 3206
Speaker: Yunzhe Li (IST Vienna) -
Abstract: We construct examples of standard maps which admits an analytic deformation preserving the eigendata of infinitely many periodic orbits. These periodic orbits constitute a sequence converging to an invariant curve with a Liouville rotation number. After a brief review of the background and related results, I will discuss key components of the proof, including a resonant normal form for maps and an iterated correction mechanism.
]]> http://www-math.umd.edu/research/seminars.html Thu, 10 Apr 2025 14:00:00 EDT Where: Kirwan Hall 3206
Speaker: Masato Tsujii (Kyushu University) - https://tsujii.wordpress.com/
Abstract: We introduce an open class of discrete dynamical systems generated by differentiable self-covering maps on closed manifolds, which we call virtually expanding. We show that, for such systems, the Perron–Frobenius operator is quasi-compact on a Sobolev space of positive order. We also derive a few consequences of this quasi-compactness. We conjecture that most volume-expanding self-maps on closed manifolds are virtually expanding and present a partial result in support of this conjecture.
]]> http://www-math.umd.edu/research/seminars.html Thu, 17 Apr 2025 14:00:00 EDT Where: Kirwan Hall 3206
Speaker: Svetlana Katok (Penn State) - https://en.wikipedia.org/wiki/Svetlana_Katok
Abstract: Based on extensive numerical experiments, Don Zagier conjectured that for any finitely generated Fuchsian group of the first kind

there is a partition of the boundary of the hyperbolic plane (circle at infinity) and a Bowen-Series-like boundary map acting in a piecewise manner by generators of the group such that its two-dimensional natural extension has an attractor with finite rectangular structure which every point enters in finite time. The finite rectangular structure property along with other properties (conjecturally equivalent to it) form, in Zagier’s terminology, a reduction theory for the group. He conjectured that the rectangular structure persists even when the partition points used in defining the boundary map are perturbed in a continuous manner.



I will talk about several results, joint with Adam Abrams and Ilie Ugarcovici, in various combinations, where the finite rectangular structure property was proved.

For the modular group, for all (a,b)-continued fractions algorithms, where a and b are real numbers satisfying b-a ≤ 1, ab ≤ -1, for surface groups, for an open set of partitions, and recently, for all finitely generated Fuchsian groups with at least one cusp, a large class of Fuchsian group which contains all subgroups of the modular group, congruence or not. If time permits, I will talk about applications of the reduction theory to symbolic coding of geodesics.
]]> http://www-math.umd.edu/research/seminars.html Thu, 24 Apr 2025 14:00:00 EDT Where: Kirwan Hall 3206
Speaker: Henry Talbott (University of Michigan) - https://htalbott.math.lsa.umich.edu/
Abstract: The spine graph construction is an explicit geometric construction that can be used to show moduli spaces of hyperbolic surfaces with boundary are homeomorphic to certain moduli spaces of ‘metric ribbon graphs’. In this talk, I will explain why these homeomorphisms roughly preserve the geometry of these moduli spaces, in particular with respect to the Weil-Petersson and Kontsevich volume forms. Furthermore, I will explain how this result can be used to obtain a convergence-in-mean result on critical exponent random variables.
]]> http://www-math.umd.edu/research/seminars.html Tue, 29 Apr 2025 14:00:00 EDT Where: Kirwan Hall 3206
Speaker: Zhiyuan Zhang (Imperial College) - https://sites.google.com/site/homepageofzhiyuanzhang/home
Abstract: We give a geometric characterization of the quantitative joint non-integrability, introduced by Asaf Katz, of strong stable and unstable bundles of partially hyperbolic measures and sets in dimension 3. This is done via the use of higher order templates for the invariant bundles. Using the recent work of Katz, we derive some consequences, including the measure rigidity of uu-states and the existence of physical measures. This is a joint work with Alex Eskin and Rafael Potrie. We also discuss a work-in-progress with Artur Avila, Sylvain Crovisier, Alex Eskin, Rafael Potrie and Amie Wilkinson on the unstable foliation and u-states of Anosov diffeomorphism.
]]> http://www-math.umd.edu/research/seminars.html Thu, 01 May 2025 14:00:00 EDT Where: Kirwan Hall 3206
Speaker: Kostiantyn Drach (University of Barcelona) - https://sites.google.com/view/kdrach
Abstract: For a smooth expanding map of the circle, its (unmarked) length spectrum is defined as the set of logarithms of multipliers along all periodic orbits. This set is analogous to the set of lengths of all closed geodesics on negatively curved surfaces -- the classical length spectrum. In the talk, I will present a length spectral rigidity result for expanding circle maps. Namely, I will show that a smooth expanding circle map, under certain assumptions on the sparsity of its length spectrum, cannot be perturbed with an arbitrarily small smooth perturbation (depending on the map) so that the length spectrum stays the same. The proof uses the Whitney extension theorem, a quantitative Livcis-type theorem, and a novel iterative scheme. This is joint work with Vadim Kaloshin.
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