Reinforced random walks and statistical physics

Pierre Tarres
Université Paris-Dauphine
January 24, 2017
We explain how the Edge-reinforced random walk, introduced by Coppersmith and Diaconis in 1986, is related to several models in statistical physics, namely the supersymmetric hyperbolic sigma model studied by Disertori, Spencer and Zirnbauer (2010), the random Schrödinger operator and Dynkin's isomorphism. These correspondences enable us to show recurrence/transience results on the Edge-reinforced random walk, and they also allow us to provide insight into these models.

Active learning with "simple" membership queries

Shachar Lovett
University of California, San Diego
January 23, 2017
An active learning algorithm for a classification problem has access to many unlabelled samples. The algorithm asks for the labels of a small number of samples, carefully chosen, such that that it can leverage this information to correctly label most of the unlabelled samples. It is motivated by many real world applications, where it is much easier and cheaper to obtain access to unlabelled data compared to labelled data. The main question is: can active learning algorithms out-perform classic passive learning algorithms?

Constructible sheaves in mirror symmetry

David Treumann
Boston College; von Neumann Fellow, School of Mathematics
January 20, 2017
I will survey the coherent-constructible correspondence of Bondal, which embeds the derived category of coherent sheaves on a toric variety into the derived category of constructible sheaves on a compact torus. The tools of the first lecture turn this into a homological mirror theorem -- a pretty strong one, after recent results of Ike and Kuwagaki.

Constructible sheaves in symplectic topology

David Treumann
Boston College; von Neumann Fellow, School of Mathematics
January 18, 2017

I will give an introduction to the microlocal theory of sheaves after Kashiwara and Schapira, and some of its recent applications in symplectic topology. I'll start with the basics, but target applications for the 75 minutes are Tamarkin's proof of nondisplaceability, and Shende's proof that a Legendrian isotopy between conormals of knots implies a smooth isotopy between the knots.

Numerical invariants from bounding chains

Jake Solomon
Hebrew University of Jerusalem; Visitor, School of Mathematics
December 16, 2016
I'll begin with a leisurely introduction to Fukaya A-infinity algebras and their bounding chains. Then I'll explain how to use bounding chains to define open Gromov-Witten invariants. The bounding chain invariants can be computed using an open analog of the WDVV equations. This leads to an explicit understanding of the homotopy type of certain Fukaya A-infinity algebras. Also, the bounding chain invariants generalize Welschinger's real enumerative invariants. A nice example is real projective space considered as a Lagrangian submanifold of complex projective space.

Numerical invariants from bounding chains

Jake Solomon
Hebrew University of Jerusalem; Visitor, School of Mathematics
December 14, 2016
I'll begin with a leisurely introduction to Fukaya A-infinity algebras and their bounding chains. Then I'll explain how to use bounding chains to define open Gromov-Witten invariants. The bounding chain invariants can be computed using an open analog of the WDVV equations. This leads to an explicit understanding of the homotopy type of certain Fukaya A-infinity algebras. Also, the bounding chain invariants generalize Welschinger's real enumerative invariants. A nice example is real projective space considered as a Lagrangian submanifold of complex projective space.