We look at bond percolation on “graph blowups” of a graph G. This is a generalization of the Erdos-Renyi random graph, and it features an analogous phase transition with respect to the appearance of a giant component. We show that the vector multiplicities of the giant component converge (after suitably centering and rescaling) to a Gaussian field on G whose covariance can be computed explicitly as the square of a massive Green’s function on G. The proof strategy is combinatorial and relies on the combinatorics of spanning trees on blowup graphs, whose generating function has a beautiful analytic structure.

I will explain some results and conjectures relating the representation theory of simple affine vertex algebras to the geometry of homogeneous elliptic affine Springer fibres associated with the Langlands dual affine Lie algebra. I will also explain generalizations of this relation to chiral algebras associated with an n-punctured Riemann surface of genus g, constructed by T. Arakawa, and the geometry of the Hitchin moduli space on the same Riemann surface. 

The Fukaya-Seidel category is classically defined in terms of a Kähler manifold X equipped with a holomorphic function W, with the relevant data encoded in the exact one-form dW. However, a natural generalization suggests itself: to replace dW with an arbitrary closed (but not necessarily exact) holomorphic one-form α on X. This more general version arises in many constructions in low-dimensional topology. In this talk, I will explore how the framework of the algebra of the infrared provides a means to understand this extension and illuminates the structure of the resulting category. 

A relaxed-pace seminar on impromptu subjects related to the interests of the audience. Everyone is welcome. The subjects are geometry, probability, combinatorics, dynamics, and more!

I will present some criteria (necessary or sufficient) for the action on the affine space of a group Gamma of affine transformations to be proper. This is joint work with Fanny Kassel.

The main of these criteria links properness of action to the divergence of a parameter called the Margulis invariant. This invariant measures roughly the translation part of an affine transformation, but in a way that is invariant by conjugation.

This link was already known in some special cases (and has often been exploited to construct proper actions). We tried to establish it in as general setting as possible. We proved it in particular if Gamma has some suitable Anosov property (with respect to some natural parabolic subgroup, that depends on the affine group we are working in).

I will possibly also evoke some other invariants similar to the Margulis invariant, that could lead to criteria that work in even more general settings.

We formulate and report on the problem of the Bass-Note 

Spectrum of an invariant operator as one varies over locally uniform 

geometries. In the Euclidean setting this recasts classical problems of

Mahler from the geometry of numbers in a new light. For certain operators 

homogeneous dynamics can be used decisively. In the non-Euclidea setting 

of hyperbolic manifolds we review some recent developments using the

conformal bootstrap method and of random covers to study the Bass- Note

spectra. We highlight the theme and impact of rigidity.

The small quantum connection of a Fano variety is among the most accessible and fundamental objects in enumerative geometry. In this talk, I’ll survey recent results on the structure of the quantum connection, with an emphasis on bounds for the sizes of Jordan blocks of the “regularized monodromy”. These bounds can be viewed as mirror analogues of classical results by Borel, Katz, and Varchenko concerning the Jordan blocks of monodromy for Gauss–Manin connections associated to families of varieties. This is joint work—partially in progress—with P. Seidel.

A relaxed-pace seminar on impromptu subjects related to the interests of the audience. Everyone is welcome. The subjects are geometry, probability, combinatorics, dynamics, and more!

In this talk, we will discuss smooth random dynamical systems and group actions on surfaces. Random dynamical systems, especially understanding stationary measures, can play an important role for understanding a group action. For instance, when a group action on torus is given by toral automorphisms, using random dynamics, Benoist-Quint classified all orbit closures under a mild assumption. We will study group actions on surfaces by diffeomorphisms, using random dynamics. We will discuss absolute continuity of stationary measures, classification of orbit closure, and exact dimensionality of stationary measures. This talk will be mostly about the ongoing joint work with Aaron Brown, Davi Obata, and Yuping Ruan.

The mapping class group and it’s subgroups are often illuminated by actions on graphs built from curves on the surface. These actions allow for a variety of questions about the group to be translated into either combinatorial or geometric information about these graphs. We will examine this approach in the case of the stabilizer of a framing of the surface. These are subgroups that Calderon and Salter have shown are important in the algebraic geometry of Moduli space. This work also suggests that the appropriate graph for these subgroups to act on is the graph of curves with winding number zero. We show the geometry of this graph can be well understood using Masur and Minsky’s subsurface projections. As a consequence, we learn that, unlike the traditional curve graph, this admissible curve graph is not hyperbolic.

After Tea, join us in a walk to Beinecke library where we will peruse some of the collection of mathematical rare manuscripts

A relaxed-pace seminar on impromptu subjects related to the interests of the audience. Everyone is welcome. The subjects are geometry, probability, combinatorics, dynamics, and more!

I will tell (part of) the origin story of geometric Langlands. Motivated by zeta functions, we will discuss algebraic geometry over finite fields and the “sheaf-function dictionary” following Weil, Grothendieck, and Deligne. We will introduce geometric Langlands as part of this philosophy. Finally, we will survey some recent developments extending these older ideas. The goal of the talk is to give a big picture perspective without getting bogged down in technicalities.