# Seminars & Events for 2017-2018

##### Nonlinear stability of Minkowski spacetime for self-gravitating massive fields

I will discuss the global evolution problem for self-gravitating massive matter in the context of Einstein's theory and, more generally, of the f(R)-theory of gravity. In collaboration with Yue Ma (Xian), by analyzing the Einstein equations in wave gauge coupled to Klein-Gordon equations, I have established that Minkowski spacetime is globally nonlinearly stable in presence of massive fields.

##### Symmetry methods for quantum variational principles and expectation value dynamics

Inspired by previous works by Kramer & Saraceno and Shi & Rabitz, this talk exploits symmetry methods for the variational formulation of different problems in physics and chemistry.

##### Morse-Bott cohomology from homological perturbation

In this talk, I will give a new construction of the Morse-Bott cochain complex, where the underlying vector space is generated by the cohomology of the critical manifolds. This new construction has two nice features: (1) It requires the minimum amount of transversality. (2) The choices made in the construction do not depend on the moduli spaces.

##### Princeton-Tokyo Fluid Mechanics Workshop

A 3-day workshop bringing together researchers from Princeton and Tokyo working in the field of mathematical fluid dynamics. New recent results and possible future directions will be discussed.

Workshop Organizers: Peter Constantin, Yoshikazu Giga, Vlad Vicol.

##### On some problems in random discrete matrices

In this talk we survey some interesting problems related to the singularity problem of random discrete matrices. In particular, we discuss the following problems:

1. an extension of an old problem of Odlyzko about the probability for randomly chosen $\pm 1$ vectors to intersect the hypercube \{1,-1\}^n in a non-trivial way, and

##### Hydrodynamics of integrable classical and quantum systems

Discussed is the Euler-type hydrodynamics for one-dimensional integrable quantum systems, as the Lieb-Liniger delta Bose gas and the XXZ chain. Of particular interest are domain wall initial states. We will use classical hard rods as an illustration of the underlying structure.

##### Dominating varieties by liftable ones.

Given a smooth projective variety over an algebraically closed field of positive characteristic, can we dominate it by another smooth projective variety that lifts to characteristic 0? We give a negative answer to this question.

##### Princeton-Tokyo Fluid Mechanics Workshop

A 3-day workshop bringing together researchers from Princeton and Tokyo working in the field of mathematical fluid dynamics. New recent results and possible future directions will be discussed.

Workshop Organizers: Peter Constantin, Yoshikazu Giga, Vlad Vicol.

##### Provably good convex methods for mapping problems

Computing mappings or correspondences between surfaces is an important tool for many applications in computer graphics, computer vision, medical imaging, morphology and related fields. Mappings of least angle distortion (conformal) and distance distortion (isometric) are of particular interest.

##### Fluids and Boundaries

I will present recent work on two kinds of boundary interactions for incompressible fluids. I will first describe results concerning the evolution of the free boundary surrounding an elastic body immersed in a viscous incompressible fluid. At the interface elasticity meets the Navier-Stokes equations. I will then describe results concerning a nonlocal equation (SQG) in a bounded domain.

##### Princeton-Tokyo Fluid Mechanics Workshop

A 3-day workshop bringing together researchers from Princeton and Tokyo working in the field of mathematical fluid dynamics. New recent results and possible future directions will be discussed.

Workshop Organizers: Peter Constantin, Yoshikazu Giga, Vlad Vicol.

##### Restriction problem with polynomial partitioning

In harmonic analysis, people are interested in the following problem: up to a constant, for any function, can we control the L_q norm of its Fourier transform restricted to the unit sphere, by the L_p norm of the function itself? The restriction conjecture is about all possible pairs (p, q) such that this statement holds.

##### Quasi-periodic Solutions for Nonlinear Klein-Gordon Equations

We construct time quasi-periodic solutions to the nonlinear Klein-Gordon equations on the torus in arbitrary dimensions. This generalizes the method developed in the limit-elliptic setting to the hyperelliptic one, by using in addition, a Diophantine property of algebraic numbers. It presents a general direct approach to second-order in time equations.

##### Hook formulas for counting skew Standard Young Tableaux with applications

Enumeration of linear extensions (total orderings) of partially ordered sets (posets) is a classical topic in discrete mathematics.

##### Basic String Topology

Let M be a compact, oriented manifold and LM the space of maps of the circle into M, the *free loop space* of M. I will give simplified, chain level definitions for the Chas-Sullivan "loop" product and coproduct on the homology of LM, and a lift of the coproduct from relative to absolute homology. Interactions between the product and coproduct will be discussed.

##### Peculiar modules for 4-ended tangles

A peculiar module is a certain algebraic invariant of 4-ended tangles that I developed in my PhD thesis as a tool for studying the local behaviour of Heegaard Floer homology for knots and links. I will briefly explain its construction and describe its classification in terms of immersed curves on a 4-punctured sphere as well as a glueing formula.

##### The ACC for log canonical threshold polytopes

The LCT (log canonical thresholds) is an invariant which measures the complexity of singularities. A conjecture due to Shokurov predicts that LCT for a single divisor satisfy the ACC (ascending chain condition), and it was proven by Hacon-McKernan-Xu.

##### Almost global existence of solutions for space periodic capillarity-gravity water waves equations

We prove that any solution of the Cauchy problem for the capillarity-gravity water waves equations, in one space dimension, with periodic, even in space, initial data of small size \ep , is almost globally defined in time on Sobolev spaces, i.e.

##### 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.

##### Recent developments in dimensional free estimates in harmonic analysis

**Please note the different time.**

We will discuss some recent developments in dimensional-free bounds for the Hardy--Littlewood averaging operators defined over convex symmetric bodies in $\mathbb R^d$. Specifically we will