April 6: Daniel Thompson (Ohio State)
Title: Unique Equilibrium states for the robustly transitive diffeomorphisms of Mañé and Bonatti-Viana
Abstract: We establish results on uniqueness of equilibrium states for the well-known Mañé and Bonatti-Viana familes of robustly transitive diffeomorphisms. 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 Mañé examples are partially hyperbolic, whereas the Bonatti-Viana examples are not partially hyperbolic but admit a dominated splitting. I'll explain why these maps satisfy our hypotheses. This is joint work with Vaughn Climenhaga (Houston) and Todd Fisher (Brigham Young).
April 20: Stephen Kleene (Brown University)
Title: Constructing complete embedded minimal surface with prescribed geometry and topology
Abstract: Singular perturbation is a powerful technique that has found applications to a wide range of geometric problems, from the study of Einstein metrics, of moduli spaces of finite topology minimal surfaces in $R^3$, to the regularity theory for Mean Curvature Flow and to the study of Yao's conjecture for minimal surfaces in three manifolds. Nonetheless, it remains a somewhat ad hoc and poorly understood collection of techniques. In this talk, I will outline my work relating to this field, discuss the broad outlines of the technique, several of its applications, and recent advances.
April 27: Francesco Cellarosi (University of Illinois, Urbana-Champaign)
Title: Quadratic Weyl sums, Automorphic Functions, and Invariance Principles
Abstract: In 1914, Hardy and Littlewood published their celebrated approximate functional equation for quadratic Weyl sums (theta sums). Their result provides, by iterative application, a powerful tool for the asymptotic analysis of such sums. The classical Jacobi theta function, on the other hand, satisfies an exact functional equation, and extends to an automorphic function on the Jacobi group. We construct a related, almost everywhere non-differentiable automorphic function, which approximates quadratic Weyl sums up to an error of order one, uniformly in the summation range. This not only implies the approximate functional equation, but allows us to replace Hardy and Littlewood's renormalization approach by the dynamics of a certain homogeneous flow. The great advantage of this construction is that the approximation is global, i.e., there is no need to keep track of the error terms accumulating in an iterative procedure. Our main application is a new functional limit theorem, or invariance principle, for theta sums. The interesting observation is that the paths of the limiting process share a number of key features with Brownian motion (scale invariance, invariance under time inversion, non-differentiability), although time increments are not independent, the value distribution at each fixed time is distinctly different from a normal distribution. Joint work with Jens Marklof.
May 4: Federico Rodriguez Hertz (Penn State)
Title: Random Dynamics and a formula for Furstenberg, Kullback-Ledrappier Entropy
Abstract: Joint with Aaron Brown, we found the following property for stationary measures of random composition of surface diffeomorphisms. Either the stable space is non-random, or the stationary measure is atomic or it is and SRB measure. This result generalize the work of Y. Benoist and J-F. Quint as well as the ones by A. Eskin and M. Mirzakhani to non-homogeneous, non affine setting. In the meantime we found a formula for the Furstenberg or Kullback-Ledrappier entropy involving Lyapunov exponents and dimensions. In this talk I will describe the results and some consequences of it.
May 11: Yuri Lima (University of Maryland)
Title: Geodesic flows on surfaces with nonpositive curvature that are Bernoulli
Abstract: We prove that the measure of maximal entropy of the geodesic flow on a nonflat smooth surface with nonpositive curvature is Bernoulli. This is consequence of a more general result for smooth flows without fixed points on three dimensional manifolds: if an equilibrium measure of a Holder potential has positive metric entropy then the flow is either Bernoulli, or it is isomorphic to a Bernoulli flow times a rotational flow. The proof has two parts. The first part is to code the flow by a suspension over a countable Markov shift (joint work with Sarig), and the second part is to understand equilibrium measures on the symbolic space (joint work with Ledrappier and Sarig).
May 14: Alex Blumenthal (Courant Institute, NYU)
Title: Entropy, volume growth and SRB measures for Banach space mappings
Abstract: (Joint with Lai-Sang Young) In this talk, we discuss the extension of the characterization of SRB measures, as those for which volume growth on the unstable foliation coincides with metric entropy, to the setting of C^2 Fréchet differentiable Banach space mappings leaving invariant compactly supported Borel probability measures with finitely-many positive Lyapunov exponents. The results discussed generalize previously known results, due to Ledrappier and others, for diffeomorphisms of finite dimensional Riemannian manifolds, and are applicable to dynamical systems defined by large classes of dissipative parabolic PDEs.
There are two major differences in this setting. The first is that there is no natural notion of d-dimensional volume element, and whatever notion volume we use must not only be compatible with the MET, but should also be regular enough to support distortion estimates. The second is that the map need not be invertible away from its attractor, and contraction in stable directions may be arbitrarily strong.
May 21: Holly Krieger (MIT)
Title: A case of the dynamical André-Oort conjecture
Abstract: I will discuss the dynamical André-Oort conjecture proposed by Baker-DeMarco, focusing on a case suggested by Ingram and recently proved by Ghioca, Nguyen, Ye, and myself: if C is an irreducible plane curve containing infinitely many points (a,b) for which z^d+a and z^d+b are both post-critically finite, then C is a fiber of a projection, or the zero locus of h(x) = Ax for some (d-1)st root of unity A. The proof includes an analysis of the existence of linear maps on angles of external rays of the Mandelbrot set.
June 1: Semyon Dyatlov (MIT) [Joint with Calderon-Zygmund Seminar]
Title: Spectral gaps via additive combinatorics
Abstract: A spectral gap on a noncompact Riemannian manifold is an asymptotic strip free of resonances (poles of the meromorphic continuation of the resolvent of the Laplacian). The existence of such gap implies exponential decay of linear waves, modulo a finite dimensional space; in a related case of Pollicott--Ruelle resonances, a spectral gap gives an exponential remainder in the prime geodesic theorem.
We study spectral gaps in the classical setting of convex co-compact hyperbolic surfaces, where the trapped trajectories form a fractal set of dimension $2\delta + 1$. We obtain a spectral gap when $\delta=1/2$ (as well as for some more general cases). Using a fractal uncertainty principle, we express the size of this gap via an improved bound on the additive energy of the limit set. This improved bound relies on the fractal structure of the limit set, more precisely on its Ahlfors-David regularity, and makes it possible to calculate the size of the gap for a given surface.
June 8: Steven Frankel (Yale University)
Title: Quasigeodesic flows, coarse hyperbolicity, and closed orbits
Abstract: A flow on a manifold is called quasigeodesic if each orbit is coarsely comparable to a geodesic. A flow is called pseudo-Anosov if it has a transverse attracting-repelling structure. If the ambient manifold is hyperbolic, these two conditions are mysteriously related. In fact, Calegari conjectures that every quasigeodesic flow on a closed hyperbolic manifold can be deformed into a pseudo-Anosov flow.
The transverse attracting-repelling structure of a pseudo-Anosov flow lends its orbits a form of rigidity. For example, the Anosov closing lemma says that if a point returns close to itself after a long time, then there's a nearby point that returns exactly to itself.
The goal of this talk is clarify the relationship between quasigeodesic and pseudo-Anosov flows. We will show that a quasigeodesic flow has a *coarse* transverse attracting-repelling structure, and use this to prove a closing lemma. In particular, we will show that every quasigeodesic flow on a closed hyperbolic manifold has periodic orbits, answering a question of Calegari's.