School of Mathematics

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.

      Fooling polytopes

      Li-Yang Tan
      Stanford University
      April 1, 2019

      We give a pseudorandom generator that fools $m$-facet polytopes over $\{0,1\}^n$ with seed length $\mathrm{polylog}(m) \cdot \mathrm{log}(n)$. The previous best seed length had superlinear dependence on $m$. An immediate consequence is a deterministic quasipolynomial time algorithm for approximating the number of solutions to any $\{0,1\}$-integer program. Joint work with Ryan O'Donnell and Rocco Servedio.

      A recent perspective on invariant theory

      Viswambhara Makam
      Member, School of Mathematics
      April 1, 2019

      Invariant theory is a fundamental subject in mathematics, and is potentially applicable whenever there is symmetry at hand (group actions). In recent years, new problems and conjectures inspired by complexity have come to light. In this talk, I will describe some of these new problems, and discuss some positive and negative results regarding them.

      Multiplicity One Conjecture in Min-max theory (continued)

      Xin Zhou
      University of California, Santa Barbara; Member, School of Mathematics
      March 27, 2019

      I will present a proof with some substantial details of the Multiplicity One Conjecture in Min-max theory, raised by Marques and Neves. It says that in a closed manifold of dimension between 3 and 7 with a bumpy metric, the min-max minimal hypersurfaces associated with the volume spectrum introduced by Gromov, Guth, Marques-Neves are all two-sided and have multiplicity one.

      Coherence and lattices

      Matthew Stover
      Temple University
      March 27, 2019

      Abstract: I will survey (in)coherence of lattices in semisimple Lie groups, with a view toward open problems and connections with the geometry of locally symmetric spaces. Particular focus will be placed on rank one lattices, where I will discuss connections with reflection groups,  "algebraic" fibrations of lattices, and analogies with classical low-dimensional topology.