# Seminars & Events for 2017-2018

##### Odd sphere bundles and symplectic manifolds

I will motivate the consideration of a special class of odd dimensional sphere bundles over symplectic manifolds. These bundles give a novel topological perspective for symplectic geometry. In particular, the symplectic A-infinity algebra recently found by Tsai-Tseng-Yau turns out to be equivalent to the standard de Rham differential graded algebra of forms on the sphere bundles.

##### Lecture 1: The games of Steiner and Poncelet and algebraic group schemes

We shall briefly present in very elementary terms the `games' of Steiner and Poncelet, amusing mathematical solitaires of the XIX Century, also related to elliptic billiards. We shall recall that the finiteness of the game is related to torsion in tori or elliptic curves.

##### Lecture 1: The games of Steiner and Poncelet and algebraic group schemes

We shall briefly present in very elementary terms the `games' of Steiner and Poncelet, amusing mathematical solitaires of the XIX Century, also related to elliptic billiards. We shall recall that the finiteness of the game is related to torsion in tori or elliptic curves.

##### Lecture 2: Torsion values for sections in abelian schemes and the Betti map

We shall consider variations in the games, related to the so-called `Betti-map', which we shall describe. We shall also illustrate some links of the Betti map with several other contexts and state some theorems on torsion values, both of existence type and finiteness type (obtained mainly in joint work with David Masser).

##### Lecture 2: Torsion values for sections in abelian schemes and the Betti map

We shall consider variations in the games, related to the so-called `Betti-map', which we shall describe. We shall also illustrate some links of the Betti map with several other contexts and state some theorems on torsion values, both of existence type and finiteness type (obtained mainly in joint work with David Masser).

##### IDeAS Seminar: Computational Algebraic Geometry and Applications to Computer Vision

Many models in science and engineering are described by polynomials. Computational algebraic geometry gives tools to analyze and exploit algebraic structure. In this talk, we offer a user-friendly introduction to some of these notions, including dimension (formalizing degrees of freedom), degree (formalizing the number of solutions to a polynomial system) and 0-1 laws in algebraic geometry (s

##### a polyhedron comparison theorem for 3-manifolds with positive scalar curvature

We establish a comparison theorem for polyhedra in 3-manifolds with nonnegative scalar curvature, answering affirmatively a dihedral rigidity conjecture by Gromov.

##### The KPZ fixed point

The KPZ universality class contains one dimensional growth models,

##### Index and spectrum of minimal hypersurfaces arising from the Allen-Cahn construction

The Allen-Cahn construction is a method for constructing minimal surfaces of codimension 1 in closed manifolds. In this approach, minimal hypersurfaces arise as the weak limits of level sets of critical points of the Allen-Cahn energy functional. This talk will relate the variational

##### Equivalence of Liouville quantum gravity and the Brownian map

Over the past several decades, two natural random surface models have emerged within physics and mathematics. The first is Liouville quantum gravity, which has roots in string theory and conformal field theory from the 1980s and 1990s. The second is the Brownian map, which has roots in planar map combinatorics from the 1960s.

##### Chabauty's method and effectivity in Diophantine geometry

I will talk about the method of Chabauty, an approach towards Siegel's finiteness theorem and Mordell's conjecture, whose idea is along the lines of those of Weil and Lang. Extended by Coleman, the method of Chabauty can produce good estimates on the number of rational points on a higher genus curve in certain cases.

##### Hadamard well-posedness of the gravity water waves equations

The gravity water waves equations consist of the incompressible Euler equations and an evolution equation for the free boundary of the fluid domain. Assuming the flow is irrotational, Alazard-Burq-Zuily (Invent.

##### Graph searches on structured families of graphs

Graph searching, a mechanism to traverse a graph visiting one vertex at a time in a specific manner, is a powerful tool used to extract structure from various families of graphs. Some families of graphs have a vertex ordering characterization, and we review how graph searching is used to produce such vertex orderings .

##### Smooth structures, stable homotopy groups of spheres and motivic homotopy theory

Following Kervaire-Milnor, Browder and Hill-Hopkins-Ravenel, Guozhen Wang and I showed that the 61-sphere has a unique smooth structure and is the last odd dimensional case: S^1, S^3, S^5 and S^{61} are the only odd dimensional spheres with a unique smooth structure. The proof is a computation of stable homotopy groups of spheres.

##### Well-posedness problems for the magneto-hydrodynamics models

We will talk about some recent results on the well-posedness problems in Sobolev spaces for the magneto-hydrodynamics with and without Hall effect, i.e., the Hall MHD and classical MHD models. One of the purposes of the work is to search the optimal Sobolev space of well-posedness for the two models.

##### Lecture 3: Ambients for the Betti map and the question of its rank

In this last lecture we shall consider in more detail some of the mentioned contexts involving the Betti map. We shall also discuss in short some recent work with Yves André and Pietro Corvaja, extending what comes from Manin's theorem of the kernel.

##### Lecture 3: Ambients for the Betti map and the question of its rank

In this last lecture we shall consider in more detail some of the mentioned contexts involving the Betti map. We shall also discuss in short some recent work with Yves André and Pietro Corvaja, extending what comes from Manin's theorem of the kernel.

##### CR invariant area functionals and the singular CR Yamabe solutions on the Heisenberg of H^{1}.

Joint seminar with Rutgers University.

##### Isoperimetric inequalities on surfaces

I will describe several recent results concerning extremal metrics and values of isoperimetric constants on different surfaces like the 2-sphere, the real projective plane or the Klein bottle. The idea is to find a metric realizing the supremum of a given eigenvalue over the whole set of Riemannian metrics on the surface.

##### Regularity and structure of scalar conservation laws

Scalar conservation law equations develop jump discontinuities even when the initial data is smooth. Ideally, we would expect these discontinuities to be confined to a collection of codimension-one surfaces, and the solution to be relatively smoother away from these jumps. The picture is less clear for rough initial data which is merely bounded.