# Seminars & Events for 2015-2016

##### Quasi-periodic standing wave solutions of gravity-capillary water waves

We prove the existence and the linear stability of Cantor families of small amplitude time quasi-periodic standing wave solutions (i.e. periodic and even in the space variable x) of a 2-dimensional ocean with infinite depth under the action of gravity and surface tension, joint work with R.Montalto.

##### Active subspaces: Emerging ideas for dimension reduction in parameter studies

Scientists and engineers use computer simulations to study relationships between a physical model's input parameters and its outputs. However, thorough parameter studies---e.g., constructing response surfaces, optimizing, or averaging---are challenging, if not impossible, when the simulation is expensive and the model has several inputs. To enable studies in these instances, the engineer may

##### Hilbert scheme of points on singular surfaces

The Hilbert scheme of points on a quasi-projective variety parameterizes its zero-dimensional subschemes. These Hilbert schemes are smooth and irreducible for smooth surfaces but will eventually become reducible for sufficiently singular surfaces.

##### New directions in statistical mechanics and dynamical systems A conference dedicated to 80th birthday of D. Ruelle and Y. Sinai.

The meeting is supported by Princeton University and IAMP.

**Wednesday, December 16, 2015 (talks in McDonnell A01):**

**10:00 am: **Hillel Furstenberg (Hebrew University of Jerusalem) - Affine Representations and Harmonic Functions

##### Polynomials vanishing on Cartesian products

Let F(x,y,z) be a real trivariate polynomial of constant degree, and let A,B,C be three sets of real numbers, each of size n.

##### Modularity and potential modularity theorems in the function field setting

**Please note slight change in time (4:15). ** Let G be a reductive group over a global field of positive characteristic. In a major breakthrough, Vincent Lafforgue has recently shown how to assign a Langlands parameter to a cuspidal automorphic representation of G. The parameter is a homomorphism of the global Galois group into the Langlands L-group $^LG$ of G.

##### The Seiberg-Witten equations and knots in 3-manifolds

Given a null-homologous knot inside a closed oriented 3-manifold, I will describe a filtered chain complex using the Seiberg-Witten equations, and discuss its relationship with homological invariants of knots defined by Ozsvath-Szabo and Kronheimer-Mrowka. This is work in preparation.

##### Decoupling in harmonic analysis and the Vinogradov mean value theorem

**This is the second Number Theory seminar on this date. Please note the special time.** Based on a new decoupling inequality for curves in R^d, we obtain the essentially optimal form of Vinogradov's mean value theorem in all dimensions (the case d=3 is due to T. Wooley).

##### Absolute vs. Relative Gromov-Witten invariants

We compare absolute and relative Gromov-Witten invariants with the basic contact vector for very positive divisors. For such divisors, one might expect that these invariants are the same up to a natural multiple. We show that this is indeed the case outside of a narrow range of the dimension of the target and the genus of the domain.

##### From symplectic geometry to combinatorics and back

A much studied combinatorial object associated to a polytope is its "Ehrhart function". This is typically studied for integral or rational polytopes.

##### Analysis, PDEs, and Geometry: A Conference in Honor Sergiu Klainerman

##### Analysis, PDEs, and Geometry: A Conference in Honor of Sergiu Klainerman

##### Analysis, PDEs, and Geometry: A Conference in Honor of Sergiu Klainerman

##### Analysis, PDEs, and Geometry: A Conference in Honor of Sergiu Klainerman

##### Degenerate evolutionary PDE and applications to stability of geometric singularities

We are going to study a family of degenerate parabolic and hyperbolic differential equations which model stability problems of singularities encountered in geometric evolutionary PDE such as the Ricci flow and the Einstein equations.

##### Completing the Square

L-functions (e.g., Riemann zeta function) constitute a special class of functions in one complex variable. It is “natural" to look at the Taylor expansion of L-functions (suitably normalized) at their “centers”.

##### Floer theory revisited

I will describe a formalism for (Lagrangian) Floer theory wherein the output is not a deformation of the cohomology ring, but of the Pontryagin algebra of based loops, or of the analogous algebra of based discs (with boundary on the Lagrangian). I will explain the consequences of quantum cohomology, and the expected applications of this theory.

##### An invitation to conformal geometry – from Gaussian curvature to Q-curvature

In the study of surface geometry, the uniformization theorem and the Gauss-Bonnet formula are of central importance. The former provides standard geometric models while the latter connects geometric and topological properties of a surface. In this talk, I will discuss higher-dimensional generalizations of these two results in a conformal-geometry context.

##### Modern developments in probabilistic combinatorics

The use of the probabilistic method, pioneered by P. Erd\H{o}s, has led to many remarkable developments in modern mathematics, including such recent breakthroughs as the existence of combinatorial designs and solutions to old problems in Ramsey Theory.

##### Extremal problems for cycles in graphs

**Please note different time this week.** In this talk, I will give a selective survey of extremal problems for cycles in graphs. In particular, proofs of a number of conjectures of Erdos on this topic will be presented, employing a variety of methods from different branches of mathematics.