How many simple closed curves can you draw on the surface of genus g in such a way that no two are isotopic and no two intersect in more than k points? It is known how to draw a collection in which the number of curves grows as a polynomial in g of degree k+1, and conjecturally, this is the best possible. I will describe a proof of an upper bound that matches this function up to a factor of log(g). It is based on an elegant geometric argument due to Przytycki and employs some novel ideas blending covering spaces and probabilistic combinatorics.

This will be an example-guided and self-contained introduction to Hyperkähler spaces. The examples include (discrete quotients of) quaternionic vector spaces, Nakajima quiver varieties, and Hitchin moduli spaces. No prerequisites necessary.

The Hartree equation is the mean field equation which describes the evolution of a system of particles in interaction in quantum mechanics. It can be proved that it converges in some weak sense to the Vlasov equation when the Planck constant \hbar becomes negligible. In this talk, I will present how this convergence can be quantitatively measured in the case of singular nonlinear interactions such as the Coulomb interaction. To reach this goal, I will introduce the Wigner transform, a semiclassical version of the Wasserstein-Monge-Kantorovitch distance introduced by F. Golse and T. Paul, and also a semiclassical analogue of the kinetic Lebesgue norms. One of the key step to reach this result is the propagation in time of semiclassical moments, in the spirit of the proof of existence for the Vlasov-Poisson equation by P.-L. Lions and B. Perthame, and weighted Schatten norms of the solution, which implies the boundedness of the spatial density of particles. This can be proved by using the formal analogies between the density operator formulation of quantum mechanics and kinetic theory.

Free groups living in SO(n+1,n) act on the vector space R^(2n+1). This makes us sad because they fix 0. However, we can add some translations to our group, making it act affinely on the affine space R^(2n+1). Now we no longer fix 0, so there is at least some hope for the action being properly discontinuous. We will discuss a construction of Smilga's giving rise to properly discontinuous actions and their fundamental domains.

Let G be an almost simple algebraic group with Langlands dual G'. Gaitsgory conjectured that affine Category O for G at a noncritical level should be equivalent to Whittaker D-modules on the affine flag variety of G' at the dual level. We will provide motivation and background for this conjecture, which is some form of geometric Satake for quantum groups. We have proven this conjecture when the level is appropriately integral with Justin Campbell, and the general case is work in progress with Sam Raskin.

Convex cocompact subgroups of rank-one semisimple Lie groups (such as PSL(2,R)) are prototypical examples of hyperbolic groups, and form a structurally stable class of quasi-isometrically embedded discrete subgroups. Anosov subgroups of higher rank groups (such as PSL(n,R) with n2) are an analogue of convex cocompact subgroups of rank-one groups which share many of their good geometric and dynamical properties. I will introduce a higher-rank analogue of geometric finiteness, which may be thought of as a controlled weakening of convex cocompactness to allow for isolated failures of hyperbolicity, and describe some of its geometric and dynamical consequences.

In this talk, I will introduce a generalization of the fast Johnson-Lindenstrauss projection for embedding vectors with Kronecker product structure: the Kronecker fast Johnson-Lindenstrauss transform (KFJLT). This work is motivated by a direct application of the KFJLT to the efficient solution of large-scale Kronecker-structured least squares problems. The KFJLT reduces the embedding cost by an exponential factor of the standard fast Johnson-Lindenstrauss transform (FJLT)'s cost when applied to vectors with Kronecker structure, by avoiding explicitly forming the full Kronecker products. We prove that this computational gain comes with only a small price in embedding power.

We consider a market of financial securities, in which traders may be restricted to trade some of the securities. The market is assumed thin: traders may influence the market and all strategically trade against their price impacts. We prove existence and uniqueness of the equilibrium even when traders are heterogeneous with respect to their beliefs, risk tolerances and endowements. An efficient algorithm is provided to numerically obtain the equilibrium prices and allocations given market's inputs. Surprisingly, restrictions could increase the market's welfare if the traders agree to disagree on the variance-covariance matrix. This is a joint work with C. Kardaras (LSE).