Where: Kirwan Hall 3206

Speaker: Dominic Veconi (ICTP) - http://dominic.veconi.com

Abstract: Under mild conditions, hyperbolic maps with singularities in any dimension have at most countably many ergodic Sinai-Ruelle-Bowen (SRB) measures. This class of maps includes the geometric Lorenz attractor, the Lozi map, and the Belykh map. In this talk, I will discuss different results in various settings of singular hyperbolicity in which the number of unique ergodic SRB measures is finite, and examine different examples highlighting the differences used in the techniques for proving these results.

Where: Kirwan Hall 3206

Speaker: Matthew Welsh (UMD) -

Abstract: We consider solutions (roots) x mod m of the quadratic congruence x^2 = D mod m for a fixed, squarefree integer D. Besides these roots being a classical object of study, statistical information on their distribution can be crucial input into methods of analytic number theory, as seen in works by Iwaniec, for example. In joint work with Jens Marklof, we study the fine-scale distribution of these roots by seeing them as return times of the horocycle flow for a specific section in SL(2, Z) \ SL(2, R), analogous to Athreya-Cheung's interpretation of the Boca-Cobeli-Zaharescu map for Farey fractions.

Where: Kirwan Hall 3206

Speaker: Yeor Hafouta (UMD) - https://sites.google.com/mail.huji.ac.il/yeor-hafouta

Abstract:

Transport distances determine exactly how well you can couple two random variables (by means of the L^p norms). Recently S. Bobkov solved a conjecture of E.Rio about optimal central limit theorem (CLT) rates (i.e. a Berry-Esseen theorem) in transport distances for partial sums of independent random variables. One of our main results is an extension of the optimal rates for several classes of weakly dependent processes like chaotic dynamical systems (possibly random or sequential), products of random iid matrices and several classes of Markov chains. A key ingredient in the proof are non-uniform Edgeworth expansions in transport distances. This yields optimal CLT rates in L^p norms (in the sense of coupling), and there are a variety of additional applications.

In the first part of the talk we will introduce the main objects (Edgeworth expansions and transport distances) and review Bobkov's solution, as well as discuss a few immediate (additional) applications of the non-uniform results.

In the second part we will present abstract sufficient conditions for non-uniform optimal CLT rates and Edgeworth expansions, and then we will explain how to verify them for expanding maps (like STF) and for the corresponding random dynamical systems.

Where: Kirwan Hall 3206

Speaker: Jonathan DeWitt (UMD) -

Abstract: Although many rigidity theorems for hyperbolic systems require irreducibility hypotheses, reducible systems may still be rigid. Such hypotheses typically preclude a system from having invariant subsystems such as non-trivial invariant tori. Reducible systems that are rigid may be rigid because they are conformal or, more generally, because they have exactly one Lyapunov exponent. In this talk, we discuss some examples of rigidity for linear cocycles over hyperbolic systems and Anosov automorphisms. We emphasize changes in behavior depending on the reducibility and conformality of the system being studied.

Where: Kirwan Hall 1313

Speaker: Jaime Paradela (Universitat Politècnica de Catalunya) -

Abstract: We study the existence of unstable behavior in the Restricted 3 Body Problem (R3BP), which models the motion of a massless body under gravitational interaction with two massive bodies. In particular, we are interested in the existence of orbits which connect certain arbitrarily far regions of the phase space, in the spirit of what is usually referred to as Arnold Diffusion. The occurence of this kind of unstable behavior has been conjectured by Arnold himself to be "typical" in the complement of integrable systems. Despite presenting strong degeneracies, we construct diffusive orbits in the R3BP: more concretely, we build orbits along which the angular momentum of the massless body (a conserved quantity for the 2 Body Problem) experiments arbitrarily large variations. This is joint work with Marcel Guardia (UB) and Tere M. Seara (UPC).

Where: Kirwan Hall 1313

Speaker: Darren Creutz (US Naval Academy) - https://www.dcreutz.com/mathematics

Abstract:

We present recent results on the relationship between word complexity and measure-theoretic mixing properties for subshifts. For strong mixing, we establish that any subshift admitting a strongly mixing probability measure has word complexity p(q) such that p(q)/q \to \infty. We also exhibit subshifts for each f(q) \to \infty such that p(q)/(q f(q)) \to 0, meaning that superlinear complexity p(q)/q \to \infty is the exact cutoff for strong mixing.

For weak mixing, we establish that for rank-one subshifts, weak mixing requires \limsup p(q)/q >= 1.5 and exhibit subshifts admitting weakly mixing measures with \limsup p(q)/q < 1.5 + \epsilon for arbitrary \epsilon > 0. Moreover these subshifts have \liminf p(q)/q = 1.

The talk will primarily focus on the strong mixing result. Part of the talk is based on joint work with S. Rodock and R. Pavlov.

Where: Kirwan Hall 1313

Speaker: Christopher Lutsko (Rutgers) - https://chrislutsko.com

Abstract: Since the early 20th century, spectral methods have been used to obtain effective counting theorems for various objects of interest in number theory, geometry and group theory. In this talk I’ll start by introducing two classical problems: the Gauss circle problem, and the Apollonian counting problem. By surveying results on these problems (and some generalizations), I’ll demonstrate how to use spectral methods to obtain effective asymptotics for some very classical problems. Then I will try and explain how to generalize this method to apply to certain horospherical equidistribution theorems.

Where: Kirwan Hall 3206

Speaker: Francisco Arana-Herrera (University of Maryland) - https://terpconnect.umd.edu/~farana/

Abstract: Cutting a hyperbolic surface along a simple closed multi-geodesic yields a hyperbolic structure on the complementary subsurface. We study the distribution of the shapes of these subsurfaces in moduli space as boundary lengths go to infinity, showing that they equidistribute to the Kontsevich measure on a corresponding moduli space of metric ribbon graphs. In particular, random subsurfaces look like random ribbon graphs, a law which does not depend on the initial choice of hyperbolic surface. This result strengthens Mirzakhani’s famous simple close multi-geodesic counting theorems for hyperbolic surfaces. This is joint work with Aaron Calderon.

Where: Kirwan Hall 3206

Speaker: Stefano Luzzatto (ICTP) - https://www.stefanoluzzatto.net/

Abstract:

Uniformly hyperbolic attractors have been well studied since the 1970s and, due to their structural stability and robustness, it is relatively easy to construct explicit examples. Uniform hyperbolicity conditions are however very strong and and so uniform hyperbolic attractors form a relatively small set in the space of Dynamical Systems. In the 1970s Pesin developed a much more general notion of (non-uniform) hyperbolicity which has been extensively studied and shown to imply a range of ergodic properties similar to the uniform case. It is also conjectured to be much more prevalent and possibly even “generic" in some sense in the space of Dynamical Systems. Paradoxically, however, in most cases it is structurally unstable and it is therefore very difficult to construct explicit examples. The only really non-trivial known examples are for Hénon-like families of maps which are known to exhibit (non-uniformly) hyperbolic attractors for nowhere dense sets of parameters of positive Lebesgue measure!

In this talk I will try to describe, in very broad and general terms, how such examples can be constructed and how we can hope to generalise the construction to other interesting families of systems. In particular I will introduce the notion of “Hyperbolic Coordinates” and explain how they can provide a very useful tool for the construction of (non-uniformly) hyperbolic examples and perhaps be useful in other applications as well.

Where: Kirwan Hall 1313

Speaker: Françoise PÈNE (Université de Brest) - http://lmba.math.univ-brest.fr/perso/francoise.pene/

Where: Kirwan Hall 3206

Speaker: Jairo Bochi (Penn State) - http://www.personal.psu.edu/jzd5895/

Abstract: Mañé Lemma is roughly a Livsic Lemma with the cohomological equation replaced by an inequality. It's the basic tool in Ergodic Optimization. In this talk I'll present several results à la Mañé Lemma, not only for Birkhoff sums, but also for products of matrices and compositions of isometries. I'll mention applications (including a few recent results) and open questions.

Where: Kirwan Hall 3206

Speaker: Krzysztof Frączek (Nicolaus Copernicus University) - https://www-users.mat.umk.pl/~fraczek/

Abstract: The talk will consists of a long historical introduction to the topic of deviation

of ergodic averages for locally Hamiltonian flows on compact surfaces as well as

some current results obtained in collaboration with Corinna Ulcigrai and Minsung Kim.

New developments include a better understanding of the asymptotic of so-called error term

(in non-degenerate regime) and the appearance of new exponents in the deviation spectrum

(in degenerate regime).

Where: Kirwan Hall 1313

Speaker: Sunrose Shrestha (Carleton College) - https://sites.google.com/view/sunroseshrestha/home

Abstract: The dynamics of straight-line flows on compact translation surfaces (surfaces formed by gluing Euclidean polygons edge-to-edge via translations) has been well studied due to its connections to polygonal billiards and Teichmüller theory. However, less is known in general regarding straight-line flows on non-compact infinite area translation surfaces. In this talk, we will consider straight line trajectories on the Mucube -- an infinite Z^3 periodic half-translation surface -- first discovered by Coxeter and Petrie and more recently studied by Athreya-Lee. We will give a complete characterization of the periodic directions on the Mucube in terms of an infinitely generated infinite index subgroup of SL(2,Z). Using the characterization, we show that the characterizing group is in fact the group of derivatives of affine diffeomorphisms of the Mucube. This is joint work (in progress) with Andre P. Oliveira, Felipe A. Ramírez and Chandrika Sadanand.

Where: Kirwan Hall 1313

Speaker: Karen Butt (Michigan) - http://www-personal.umich.edu/~kbutt/index.html

Abstract:

The marked length spectrum of a closed Riemannian manifold of negative curvature is a function on the free homotopy classes of closed curves which assigns to each class the length of its unique geodesic representative. It is known in certain cases that the marked length spectrum determines the metric up to isometry, and this is conjectured to be true in general. In this talk, we explore to what extent the marked length spectrum on a sufficiently large finite set approximately determines the metric.

Where: Kirwan Hall 3206

Speaker: Rigoberto Zelada (UMD) -

Abstract: Let $(X,\mathcal A,\mu)$ be a probability space and let $A_0,...,A_L\in\mathcal A$. In this talk we deal with the asymptotic independence of expressions of the form

$$\mu(A_0\cap T_1^{v_1(n)}A_1\cap \cdots\cap T_L^{v_L(n)}A_L),$$

where $T_1,...,T_L$ are commuting $\mu$-preserving transformations and $v_1,...,v_L$ are polynomials in $\mathbb Z[x]$, and some of their generalizations.

In the first part of this talk we introduce $\Sigma_\ell^*$-limits, $\ell\in\mathbb N$, a new notion of convergence that, as we will see, is strong enough to characterize strong mixing while, simultaneously, being weak enough for avoiding the obstructions to (and, hence, establishing) various strong mixing polynomial ergodic theorems. We will also provide examples which demonstrate that $\Sigma_\ell^*$-limits avoid these obstructions in an "optimal" way.

In the second part of the talk we will show how a slight modification to the method introduced by V. Bergelson in "Weakly mixing PET" can be employed to show that

$$\Sigma_{10}^*\text{-}\lim_{n\in\mathbb Z}\mu(A_0\cap T^{n^2}A_1\cap T^{2n^2}A_2)=\mu(A_0)\mu(A_1)\mu(A_2)$$

for any $A_0,A_1,A_2\in\mathcal A$ and any strongly mixing $T$. Then we will explain why such a method no longer works when dealing with an expression of the form $\mu(A_0\cap S^{n^2}A_1\cap T^{n^2}A_2)$ and briefly mention the new ideas and techniques required to prove that, under meager mixing assumptions on $S$ and $T$, one has that for some $m\in\mathbb N$,

$$\Sigma_{m}^*\text{-}\lim_{n\in\mathbb Z}\mu(A_0\cap S^{n^2}A_1\cap T^{n^2}A_2)=\mu(A_0)\mu(A_1)\mu(A_2).$$

This talk is based on joint work with V. Bergelson.

Where: Kirwan Hall 3206

Speaker: Harrison Bray (George Mason University) - https://www.harrisonbray.com/

Abstract: Convex real projective manifolds with the Hilbert metric generalize hyperbolic geometry to a family of metrics with coarse hyperbolicity but low regularity; aside from the hyperbolic case, the Hilbert metric on such structures is not Riemannian and not $C^2$.

I will discuss how a version of the Dirichlet theorem for horoball packings can be adapted to this setting of geometrically finite Hilbert geometries which are not convex cocompact. The methods use Gromov hyperbolicity and an intrinsic definition of horoball depth. Together with a generalized global measure formula, these results imply the logarithm law for divergence of geodesics in the cusps. This is based on joint work with Giulio Tiozzo.

Where: Kirwan Hall 3206

Speaker: Boris Solomyak (Bar-Ilan University) - https://u.math.biu.ac.il/~solomyb/personal.html

Abstract: The problem of absolute continuity for self-similar measures in the "overlapping super-critical regime'' has attracted a lot of attention over the years. Perhaps, best known is the family of Bernoulli convolutions, studied by Erdős (1939, 1940) and many other mathematicians since then. These are self-similar measures on the line, generalizing the classical Cantor-Lebesgue measure, when the contraction ratio exceeds 1/2, so the two "cylinder sets'' overlap.

In spite of the dramatic progress over the last decade, largely due to Hochman, Shmerkin, and Varjú, the problem is still not completely solved. Various generalizations (e.g., more than two cylinders, varying contraction ratios, non-linear self-conformal measures, self-similar and self-affine measures in higher dimensions) have been considered as well, with partial success. After a general introduction and survey, I will discuss results on non-homogeneous self-similar measures (that is, having non-constant contraction ratios) on the line, obtained in joint work with Saglietti and Shmerkin (2018), and their generalization to the planar case, joint with Śpiewak (2023 preprint). Such measures do not have an obvious convolution structure, which makes their study more difficult.

Where: Kirwan Hall 3206

Speaker: Nyima Kao (George Washington University) - https://sites.google.com/view/nyimakao

Abstract: In this talk, we will discuss a sufficient condition for pressure gap over some nonpositively curved manifolds. This condition is optimal in the sense that the geometric potential is on the boundary of this condition. In other words, we can conclude that potentials decaying faster than the geometric potential (towards the singular set) have pressure gaps and no phase transitions. This talk is based on joint work with Dong Chen and Kiho Park.

Where: Kirwan Hall 3206

Speaker: Snir Ben Ovadia (Penn State) - https://sites.psu.edu/snir/

Abstract: We introduce a notion of a point-wise entropy of measures (i.e. local entropy) called em neutralized local entropy and compare it with the Brin-Katok local entropy and with the Ledrappier-Young local entropy on unstable leaves. We show that the neutralized local entropy is bounded from above by the unstable entropy, and so all three notions of local entropy must coincide almost everywhere. Neutralized local entropy is computed by measuring open sets with a relatively simple geometric description. Our proof uses a measure density lemma for Bowen balls, and a version of a Besicovitch covering lemma for Bowen balls. In particular this gives an elementary proof of the fact that the unstable entropy carries all entropy.

Where: MATH 1313

Speaker: Philipp Kunde (Jagiellonian University) - https://sites.google.com/view/pkunde

Abstract: An important question in ergodic theory dating back to the foundational paper of von Neumann is the so-called smooth realization problem: Are there smooth versions to the objects and concepts of abstract ergodic theory? Does every ergodic measure-preserving transformation have a smooth model?

Here, a smooth model of an MPT $(\Omega, \mu, T)$ is a smooth diffeomorphism f of a compact manifold $M$ preserving a measure $\lambda$ equivalent to the volume element such that the MPT $(M, \lambda, f)$ is isomorphic to the MPT $(\Omega, \mu, T)$. One of the most powerful tools of constructing smooth volume-preserving diffeomorphisms with prescribed ergodic or topological properties is the Approximation by Conjugation method introduced by Dmitri Anosov and Anatole Katok. We give an overview of the method and its contributions to the smooth realization problem.

Where: Kirwan Hall 1313

Speaker: Abror Pirnapasov (École normale supérieure de Lyon) - https://sites.google.com/view/abrorpirnapasov/home

Abstract: Hutchings used Embedded Contact Homology to show the following for area-preserving disc diffeomorphisms that are a rotation near the boundary of the disc: if the asymptotic mean action on the boundary is greater than the Calabi invariant, then the infimum of the mean action of the periodic points is less than or equal to the Calabi invariant. In this talk, I explain how to extend this result to all orientation and area-preserving disc diffeomorphisms. I also introduce a more general result for area-preserving disc diffeomorphisms with only one periodic point. This is joint work with David Bechara, Barney Bramham, and Patrice Le Calvez.

Where: Kirwan Hall 3206

Speaker: Scott Schmieding (Penn State) - https://s-schmieding.github.io/

Abstract: I will discuss relationships between cohomology and degrees of factor maps between certain types of Smale systems. This is motivated in part by various Pisot conjectures which ask whether certain irreducibility conditions on a substitution implies the flow on the corresponding tiling space must have pure discrete spectrum.

Where: Kirwan Hall 3206

Speaker: May Mei (Denison) - http://personal.denison.edu/~meim/

Abstract: The Penrose tiling is among the most popular 2-D models for quasicrystals, materials exhibiting aperiodic order. But the article "the" in front is a misnomer. In this talk, we look at the graph Laplacian associated with four variations from the mutual local derivability class of the Penrose tiling. Two tilings are said to be mutually locally derivable (MLD) if each can be obtained from the other using local rules. From many perspectives, two MLD tilings can be thought of as "the same.'' However, local derivability greatly impacts the adjacency relationship, which in turn has ramifications for the graph Laplacian. In each of the four cases, we exhibit locally-supported eigenfunctions, which necessarily cause jump discontinuities in the integrated density of states for these models. Moreover, in several cases we provide concrete lower bounds on this jump. These results suggest a host of questions about spectral properties of the Laplacian on aperiodic tilings, which we will also present.

Where: Kirwan Hall 3206

Speaker: Karl Winsor (Fields Institute) - https://people.math.harvard.edu/~kwinsor/

Abstract: Strata of abelian differentials (or translation surfaces) carry a natural foliation called the absolute period foliation. A leaf of this foliation is navigated by varying an abelian differential without changing its integrals along closed loops. Roughly speaking, this is done by moving the zeros of the differential relative to each other. In certain strata in low genus, the dynamics of this foliation are well-understood due to a close connection with homogeneous dynamics. However, for most strata in higher genus, the situation is more mysterious. In this talk, I will present a method for completely classifying the closures of leaves of the absolute period foliation of most strata. Our results suggest a version of Ratner's orbit closure theorem in this setting, and a possible analogy between SL(2,R) dynamics and isoperiodic dynamics.

Where: Kirwan Hall 3206

Speaker: Ethan Ackelsberg (IAS) - https://sites.google.com/view/ethanackelsberg

Abstract: A central result in additive combinatorics is Szemerédi’s theorem, which says that any set of integers of positive density contains arbitrarily long arithmetic progressions. An ergodic-theoretic proof due to Furstenberg has led to many interesting refinements of Szemerédi’s theorem. One such refinement, obtained by Bergelson–Host–Kra (2005), states that any set of positive density in the integers contains many (at least as many as in a random set of the same density, up to an arbitrarily small error) 3-term arithmetic progressions $\{a, a+d, a+2d\}$ with the same common difference $a$, sometimes referred to as a popular difference. They also showed that one can always find popular differences for 4-term arithmetic progressions, but this stops being true for 5-term and longer progressions.

I will discuss several recent extensions and generalizations of the results of Bergelson–Host–Kra in the context of linear configurations in abelian groups. After translating the problem into a dynamical setting, these results rely on a description of the structure of general ergodic measure-preserving actions of abelian groups.

This talk draws on joint works with V. Bergelson, A. Best, and O. Shalom.

Where: Kirwan Hall 3206

Speaker: Daniel Mitsutani (University of Chicago) - https://math.uchicago.edu/~mitsutani/

Abstract: The study of the Lyapunov exponents of random matrix products has a long history dating back to Furstenberg. In this talk we will discuss extensions of these classical results for some linear cocycles over hyperbolic systems. We consider linear cocycles over hyperbolic systems with canonical holonomies. These holonomies always exist for locally constant or fiber bunched cocycles. Under an irreducibility condition we show the following: If there exists a reasonable measure such that for that measure all Lyapunov exponentshave multiplicity 1, then for every reasonable measure all Lyapunov exponents have multiplicity 1. This provides a counterpart to a theorem of Guivarch and Raugi for random matrix products.

Where: Kirwan Hall 1313

Speaker: Jing Zhou (Penn State) - https://sites.google.com/view/mathjingzhou233/

Abstract: In the early 60's J. B. Keller and D. Levy discovered a fundamental property: the instability tongues in Mathieu-type equations lose sharpness with the addition of higher-frequency harmonics in the Mathieu potentials. 20 years later V. Arnold rediscovered a similar phenomenon on sharpness of Arnold tongues in circle maps (and rediscovered the result of Keller and Levy). In this paper we find a third class of objects where a similarly flavored behavior takes place: area-preserving maps of the cylinder. Speaking loosely, we show that periodic orbits of standard maps are extra fragile with respect to added drift (i.e. non-exactness) if the potential of the map is a trigonometric polynomial. That is, higher-frequency harmonics make periodic orbits more robust with respect to "drift". The observation was motivated by the study of traveling waves in the discretized sine-Gordon equation which in turn models a wide variety of physical systems. This is a joint work with Mark Levi.

Where: Kirwan Hall 1308

Speaker: Mariusz Lemanczyk (Nicolaus Copernicus University in Toruń) -

Abstract: In 2010, P. Sarnak formulated the conjecture on the orthogonality of deterministic sequences with the (arithmetic) Moebius function. In 2015, W. Veech formulated a conjecture on the equivalence of Sarnak's conjecture with a special ergodic property of so called Furstenberg systems of the Moebius function. The talk will be devoted to the proof of Veech's conjecture achieved in my joint work with A. Kanigowski, J. Kułaga-Przymus and T. de la Rue.

Where: Kirwan Hall 3206

Speaker: Rachel Greenfeld (IAS) - https://www.math.ias.edu/~rgreenfeld/

Abstract: Translational tiling is a covering of a space (e.g., Euclidean space) using translated copies of a building block, called a "tile'', without any positive measure overlaps. What are the possible ways that a space can be tiled? One of the most well known conjectures in this area is the periodic tiling conjecture. It asserts that any tile of Euclidean space can tile the space periodically. In a joint work with Terence Tao, we disprove the periodic tiling conjecture in high dimensions. In the talk, I will discuss and motivate the study of this conjecture, our recent result, as well as new developments.