Abstract: Consider a smooth compact connected oriented surface with boundary of Anosov type, i.e., it has a strictly convex boundary, no conjugate points, and hyperbolic trapped set. We prove that if two metrics of Anosov type have same marked boundary distance, then they are isometric. One of the main ingredients is a new transfer principle showing that the marked length spectrum rigidity conjecture implies the marked boundary distance rigidity conjecture under the existence of a suitable isometric embedding into a closed Anosov manifold. This is joint work with Thibault Lefeuvre.
Abstract: In the talk we will develop a natural definition (in the categorical sense) of an algebra structure on a reproducing kernel Hilbert space. This definition is also equivalent to subconvolutivity of weight functions in examples from harmonic analysis. We will then show the category of reproducing kernel Hilbert algebras (RKHA) is closed under orthogonal sums, tensor products, pushouts, and pullbacks such that the spectrum is a functor compatible with these constructions. Furthermore, the image of the spectrum as a functor from RKHA to Top contains all compact subspaces of R^n, n>0.
Abstract: We will discuss the relationship between spectra of elements and states in unital Banach and C*-algebras, and how these relate to the numerical radii of elements. For example, we will discuss elementary proofs that every member of the spectrum of any element in a unital Banach algebra is realized by some state (i.e., unital linear functional of norm one) on the algebra. Then, for various algebras, we will show how to explicitly construct the states that realize these spectral elements. We will then strengthen this result using the holomorphic functional calculus.
Abstract: On a closed Riemannian manifold, the marked length spectrum rigidity problem consists in recovering the metric from the knowledge of the lengths of its closed geodesics (marked by the free homotopy classes of the manifold). I will present a solution to this problem for Anosov surfaces namely, surfaces with uniformly hyperbolic geodesic flow (such as negatively-curved surfaces). The proof blends hyperbolic dynamics, complex geometry and microlocal analysis. Joint work with Gabriel P. Paternain and Colin Guillarmou.
October 12, 2023: Erik van Erp "Introduction to Noncommutative Geometry, Part I: The roots of NCG in Quantum Theory"
Abstract: Book I of Euclid's Elements opens with "A point is that which has no part." Ever since Euclid, our theories of space have been based on the idea of a dimensionless point. For all their revolutionary impact, non-Euclidean geometries do not challenge this foundation. Likewise, the space-time continuum of general relativity is a "point set". Philosophically, the concept of space as a continuum of points is intimately related to the principle of locality in physics: causality travels "from point to point". With the violation of locality in quantum theory, the idea that physical space should be modeled as a point set becomes questionable. Space, at subatomic scales, is stranger than curved spacetime. Noncommutative Geometry is a theory of space that is compatible with quantum theory. If there are no points, how do we define coordinates? geometry? vector bundles? curvature? differential forms? integrals? field theories? This is Part I of a series of introductory talks about NCG. I will discuss the roots of NCG in quantum theory.
October 19, 2023: Nisha Chandramoorthy "Rigorous computations for linear response and Bayesian inference in chaotic systems"
Abstract: Linear response refers to the smooth change in the statistics of an observable in a dynamical system in response to a smooth parameter change in the dynamics. The computation of linear response has been a challenge, despite work pioneered by Ruelle giving a rigorous formula in Anosov systems. This is because typical linear perturbation-based methods are not applicable due to their instability in chaotic systems. Here, we give a new differentiable splitting of the parameter perturbation vector field, which leaves the resulting split Ruelle’s formula amenable to efficient computation. A key ingredient of the overall algorithm, called space-split sensitivity, is a new recursive method to differentiate quantities along the unstable manifold. Of particular importance is the score – gradient of log density – of the conditional density of the SRB measure along the unstable manifold. This fast algorithm for the conditional scores motivates our attack of another longstanding computational challenge in high-dimensional dynamics – sampling from Bayesian filtering distributions, which we discuss in the second half of the talk. Bayesian filtering is repeated Bayesian inference, and we develop a new transport-based formulation of Bayesian inference, in the setting where the score of the target distribution is known, such as in the above scenario. Transport maps are transformations between the sample space of a source (which is generally easy to sample) and a target (typically non-Gaussian) probability distribution. The new transport map is a constructive solution of an infinite-dimensional generalization of a Newton method to find the zero of a “score operator”. We define such a score operator that gives the difference of the score of a transported distribution from the target score. The Newton iteration enjoys fast convergence under smoothness assumptions and does not make a parametric ansatz on the transport map.
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.
Abstract: We shall survey some classical and more recent results on quantum ergodicity for manifolds and billiards.
November 16, 2023: JinCheng Wang "Unique measure of maximal entropy for magnetic geodesic flows of surfaces with negative curvature"
Abstract: The existence and uniqueness of the measure of maximal entropy (MME) for a given system are questions dynamicists are interested in. In this talk, I will introduce magnetic geodesic flows of negatively curved surfaces and investigate their MMEs. Some geometric properties will be presented, which contrast with that of the geodesic flow.