We introduce a natural symplectic structure on the moduli space of quadratic differentials with simple zeros and describe its Darboux coordinate systems in terms of so-called homological coordinates. We then show that this structure coincides with the canonical Poisson structure on the cotangent bundle of the moduli space of Riemann surfaces, and therefore the homological coordinates provide a new system of Darboux coordinates. The space of projective structures over the moduli space can be identified with the cotangent bundle upon selection of a reference projective connection that varies holomorphically and thus can be naturally endowed with a symplectic structure. Different choices of projective connections of this kind (Bergman, Schottky, Wirtinger) give rise to equivalent symplectic structures on the space of projective connections but different symplectic polarizations: the corresponding generating functions are found. We also study the monodromy representation of the Schwarzian equation associated with a projective connection, and we show that the natural symplectic structure on the space of projective connections induces the Goldman Poisson structure on the character variety. Combined with results of Kawai, this result shows the symplectic equivalence between the embeddings of the cotangent bundle into the space of projective structures given by the Bers and Bergman projective connections.

# Symplectic geometry of the moduli space of projective structures in homological coordinates

# Intersection numbers, integrable hierarchies and matrix models

From the seminal papers of Witten and Kontsevich we know that the intersection theory on the moduli space of complex curves is described by a tau-function of the KdV integrable hierarchy. Moreover, this tau-function is given by a matrix integral and satisfies the Virasoro constraints. Recently, an open version of this intersection theory was investigated. My goal is to show that this open version can also be naturally described by a tau-function of the integrable hierarchy (MKP in this case), and the matrix integral and the Virasoro constraints are also simple.

# Extremal Bounds for Bootstrap Percolation in the Hypercube

The r-neighbour bootstrap process on a graph G begins with an initial set of "infected" vertices and, at each step of the process, a previously healthy vertex becomes infected if it has at least r infected neighbours. The set of initially infected vertices is said to "percolate" if the infection spreads to the entire vertex set. In this talk, we will discuss a recent proof of a conjecture of Balogh and Bollobás which says that the minimum size of a percolating set for the r-neighbour bootstrap process in the d-dimensional hypercube is asymptotically (1/r) (d choose r-1). The proof boils down to defining an analogous process on the edges of the hypercube and doing a nice "linear algebra trick." We will also place this result in context by discussing some of the known results in the field. This is joint work with Natasha Morrison.

# On wall-crossing in $\M_{g,n}$

The term “wall-crossing” is referred to the analysis of the crossing between cells of maximal dimensions in the CW-complex structure provided to the moduli space of marked Riemann surfaces by the Strebel approach. Of particular interest is the homotopy around the co-dimension 2 locus. There is a natural $U(1)$ line bundle on this space and the analysis of its Chern class allows to prove relations between fundamental classes.

Based on a joint work (in progress) with Dmitry Korotkin.

# Reinforced Polya Urns

Consider the following toy model for neural processing in the brain: a large number of neurons are interconnected by synapses, and the brain removes connections which are seldom or never used and reinforces those which are stimulated. We introduce a class of reinforced Polya urn models which aim to describe this dynamics. Our models work as follows: at each time step t, we first choose a random subset A_t of colours (independently of the past) from n colours of balls, and then choose a colour i from the subset A_t with probability proportional to the number of balls of colour i in the urn raised to the power alpha>1. We are mostly interested in stability of equilibria for such models studying phase transitions in a number of examples, including when the colours are the edges of a graph. We conjecture that for any graph G and all alpha sufficiently large, the set of stable equilibria is supported on so-called whisker-forests, which are forests whose components have diameter between 1 and 3.

This talk is based on joint work with Remco van der Hofstad, Mark Holmes and Wioletta Ruszel.

# Homologie persistante et simplification

TBA

# The dual view of Markov Processes II

*To appear on the seminar website: http://www.math.mcgill.ca/rags/seminar/*

# Nodal Sets in Conformal Geometry

We study conformal invariants that arise from nodal sets and negative eigenvalues of conformally covariant operators, which include the Yamabe and Paneitz operators. We give several applications to curvature prescription problems. We establish a version in conformal geometry of Courant's Nodal Domain Theorem. We also show that on any manifold of dimension $n geq 3$, there exist many metrics for which our invariants are nontrivial. We prove that the Yamabe operator can have an arbitrarily large number of negative eigenvalues on any manifold of dimension $n geq 3$. We obtain similar results for some higher order GJMS operators on some Einstein and Heisenberg manifolds. This is joint work with Yaiza Canzani, Rod Gover and Raphael Ponge. If time permits, we shall discuss related results for operators on graphs.

# The dual view of Markov Processes III

# Schlesinger Transformations and Difference Painlevé Equations

The main goal of this talk is to explain an isomonodromic approach to the theory of difference Painlevé equations. We first briefly outline the theory of elementary Schlesinger transformations of Fuchsian systems. Next, we study such transformations for systems that have two-dimensional space of accessory parameters and show that these transformations can be expressed via difference Schlesinger equations. We consider in detail one example of such reductions that corresponds to the equation with the E_6 affine Weyl symmetry group. If time permits, we will show how the resulting equation can be analyzed using the geometric framework of H. Sakai.