The energy functional on Besse manifolds

Marco Radeschi
University of Notre Dame
April 9, 2019

A Riemannian manifold is called Besse, if all of its geodesics are periodic. The goal of this talk is to study the energy functional on the free loop space of a Besse manifold. In particular, we show that this is a perfect Morse-Bott function for the rational, relative, S1-equivariant cohomology of the free loop space. We will show how this result is crucial in proving a conjecture of Berger for spheres of dimension at least 4, although it might be useful for proving the conjecture in full generality.

Collective Coin-Flipping Protocols and Influences of Coalitions

Hamed Hatami
University of McGill
April 8, 2019
The seminal result of Kahn, Kalai and Linial implies that a coalition of a small number of players can bias the outcome of a Boolean function with respect to the uniform measure. We extend this result to arbitrary product measures, by combining their argument with a completely different argument that handles very biased coordinates.

Disorder increases almost surely.

Laure Saint-Raymond
University Paris VI Pierre et Marie Curie and Ecole Normale Supérieure
April 8, 2019
Consider a system of small hard spheres, which are initially (almost) independent and identically distributed.

    Then, in the low density limit, their empirical measure $\frac1N \sum_{i=1}^N \delta_{x_i(t), v_i(t)}$ converges
    almost surely to a non reversible dynamics.

      Where is the missing information to go backwards?

      Two-dimensional random field Ising model at zero temperature

      Jian Ding
      The Wharton School, The University of Pennsylvania
      April 5, 2019
      I will discuss random field Ising model on $Z^2$ where the external field is given by i.i.d. Gaussian
      variables with mean zero and positive variance. I will present a recent result that at zero temperature the effect of boundary conditions on the magnetization in a finite box decays exponentially in the distance to the boundary. This is based on joint work with Jiaming Xia.

      Singular moduli for real quadratic fields

      Jan Vonk
      Oxford University
      April 4, 2019

      The theory of complex multiplication describes finite abelian extensions of imaginary quadratic number fields using singular moduli, which are special values of modular functions at CM points. I will describe joint work with Henri Darmon in the setting of real quadratic fields, where we construct p-adic analogues of singular moduli through classes of rigid meromorphic cocycles. I will discuss p-adic counterparts for our proposed RM invariants of classical relations between singular moduli and analytic families of Eisenstein series.

      Stable hypersurfaces with prescribed mean curvature

      Costante Bellettini
      Princeton University; Member, School of Mathematics
      April 2, 2019

      In recent works with N. Wickramasekera we develop a regularity and compactness theory for stable hypersurfaces (technically, integral varifolds) whose generalized mean curvature is prescribed by a (smooth enough) function on the ambient Riemannian manifold. I will describe the relevance of the theory to analytic and geometric problems, and describe some GMT and PDE aspects of the proofs.