# Seminars & Events for Analysis Seminar

##### Grothendieck's inequality and the propeller conjecture

##### Long-time strong instability and unbounded orbits for some nonlinear Schrödinger equations

We establish a relation between long-time strong instability and the existence (in a certain generic sense) of unbounded orbits for dynamical systems on a Banach space. We then discuss some consequences of this relation for nonlinear Schrödinger equations. Namely, we prove long-time strong instability of plane wave solutions for the cubic nonlinearity and the existence of unbounded orbits for certain nonlinearities that are close (but not quite equal) to the cubic one.

##### On the uniqueness of solutions to the 3D periodic Gross-Pitaevskii hierarchy

In this talk, we present a uniqueness result for solutions to the Gross-Pitaevskii hierarchy on the three-dimensional torus, under the assumption of an a priori spacetime bound. We show that this a priori bound is satisfied for factorized solutions coming from a solution of the nonlinear Schrodinger equation, thus obtaining a periodic analogue of the uniqueness result on R3 previously proved by Klainerman and Machedon. This is joint work with Gigliola Staffilani.

##### Regularity of twisted Bergman projections - Part 1

The usual Bergman projection is not globally regular on a general (smoothly bounded) pseudoconvex domain, by a result of Christ. We will discuss a family of "twisted" projections, similar to the Bergman projection, and show they are regular on this class of domains. Because the twist factor defining these projections is related to the boundary defining function of the domains, regularity of the twisted projections implies biholomorphic mappings extend smoothly to the boundary, following the lines of the Bell-Ligocka program.

##### Regularity of twisted Bergman projections - Part 2

The usual Bergman projection is not globally regular on a general (smoothly bounded) pseudoconvex domain, by a result of Christ. We will discuss a family of "twisted" projections, similar to the Bergman projection, and show they are regular on this class of domains. Because the twist factor defining these projections is related to the boundary defining function of the domains, regularity of the twisted projections implies biholomorphic mappings extend smoothly to the boundary, following the lines of the Bell-Ligocka program.

##### Regularity of twisted Bergman projections - Part 3

The usual Bergman projection is not globally regular on a general (smoothly bounded) pseudoconvex domain, by a result of Christ. We will discuss a family of "twisted" projections, similar to the Bergman projection, and show they are regular on this class of domains. Because the twist factor defining these projections is related to the boundary defining function of the domains, regularity of the twisted projections implies biholomorphic mappings extend smoothly to the boundary, following the lines of the Bell-Ligocka program.

##### Gravitational Impulsive Waves

We consider spacetimes satisfying the vacuum Einstein equations with gravitational impulsive waves without symmetry assumptions. These are spacetimes such that some components of the Riemann curvature tensor have delta singularities on a null hypersurface. We prove local existence for the characteristic initial value problem with initial data that has a delta singularity in some components of the curvature tensor. We also prove local existence in the case where two gravitational impulsive waves collide. The proof introduces a new type of energy estimates for the vacuum Einstein equations, allowing the $L2$ norm of some components of the curvature tensor to be infinite. The new estimate allows us to prove local existence for a general class of initial data which is rough in one direction.

##### Global existence for solutions of the energy critical nonlinear Schrodinger equation in the Torus

I will present a joint work with A. Ionescu proving that smooth solutions to the energy critical nonlinear Schrodinger equation on T3 remain global in time. One interest of this is that it clarifies the possible obstructions to global existence of the energy critical equation on manifolds due to trapped geodesics and finite volume. This relies on previous work by Bourgain, Colliander-Keel-Staffilani-Takaoka-Tao and Herr-Tataru-Tzvetkov.

##### Local well-posedness of the KdV equation with almost periodic initial data

We prove the local well-posedness for the Cauchy problem of Korteweg-de Vries equation in an almost periodic function space. The function space contains functions satisfying $f=f_1+f_2+...+f_N$ where $f_j$ is in the Sobolev space of order $s>?1/2N$ of $a_j$ periodic functions. Note that f is not periodic when the ratio of periods $a_i/a_j$ is irrational. The main tool of the proof is the Fourier restriction norm method introduced by Bourgain. We also prove an ill-posedness result in the sense that the flow map (if it exists) is not C2, which is related to the Diophantine problem.

##### Modulation invariant bilinear T(1) theorem

We discuss a T(1) theorem for bilinear singular integral operators with a one-dimensional modulation symmetry.

##### Effective dynamics of a non-linear wave equation

We consider the non-linear wave equation on the real line $i u_t-|D|u=|u|^2u$. Its resonant dynamics is given by the Szego equation, which is a completely integrable non-dispersive non-linear equation. We show that the solution of the wave equation can be approximated by that of the resonant dynamics for a long time. The proof uses the renormalization group method introduced by Chen, Goldenfeld, and Oono in the context of theoretical physics. As a consequence, we obtain growth of high Sobolev norms of certain solutions of the non-linear wave equation, since this phenomenon was already exhibited for the Szego equation.

##### Nonlocal Evolution Equations

Nonlocal evolution equations have been around for a long time, but in recent years there have been some nice new developments. The presence of nonlocal terms might originate from modeling physical, biological or social phenomena (incompressibility, Ekman pumping, chemotaxis, micro-micro interactions in complex fluids, collective behavior in social aggregation) or simply from inverting local operators in the analysis of systems of PDE. I will brifly present some regularity results for hydrodynamic models with singular constitutive laws. The main part of the talk will present a nonlinear maximum principle for linear nonlocal dissipative operators and applications.

##### Vinogradov estimates applied to maximal theorems related to Waring's problem

I'll discuss the discrete spherical maximal function of Magyar--Stein--Wainger and it's estimates. Then I'll prove recent results for higher degrees which use Vinogradov type estimates for exponential sums to improve results about maximal functions defined on the hypersurfaces arising in Waring's problem.

##### Global Stability Results for Relativistic Fluids in Expanding Spacetimes

In this talk, I will discuss the future-global nonlinear behavior of relativistic fluids evolving in expanding spacetimes. I will focus on how the global behavior of the fluid is affected by both the spacetime expansion rate and the fluid equation of state. These topics are physically relevant for the following reasons: \textbf{i)} In cosmology, the relativistic fluid model is the most often used model for the ``normal'' matter content of our spacetime. \textbf{ii)} Experimental evidence indicates that our spacetime is expanding. \textbf{iii)} The precise expansion rate is not known, and is in fact a current topic of debate. \textbf{iv)} Prior mathematical results show that the fluid behavior is quite sensitive to the expansion rate. For example, in Minkowski spacetime (which is expansion-free), D.

##### Vanishing/Blow up at infinity for the 3-d critical focussing NLW

##### Description of the blow-up for the semi-linear wave equation

We study the blow-up curve of a (blow-up) solution to the semi-linear wave equation in 1D with power nonlinearity: $u_tt - u_xx = |u|^{p-1} u$. The blow-up curve is a priori 1-Lispchitz. On this curve, we distinguish (geometrically) characteristic points and non-characteristic points. We describe the blow-up behavior in each case, following a series of papers by Frank Merle and Hatem Zaag: in particular, the set of characteristic points is discrete, the blow-up curve is corner-shaped at every characteristic point, and is $C1$ around any non-characteristic point. We also construct construct a blow-up solution with prescribed characteristic point.

##### Applications of Multilinear Restriction and Restriction Estimates

The restriction problem is an important open problem in harmonic analysis. The 2-dimensional case was proven in the 1970's. The case of three (or more) dimensions remains open, with interesting partial results. In 2005, Bennett, Carbery, and Tao proved a ``multilinear" restriction estimate. We can think of this work as a near-optimal estimate for a significant chunk of the terms in the original restriction problem. It had major philosophical impact, giving a new perspective on what part of the problem is most difficult. Recently, the multilinear estimate has also had some practical impact. A paper by Bourgain and me uses the multilinear inequality to prove estimates for the original restriction problem, matching and sometimes improving the best estimates previously known.

##### Well-Posedness and Finite-Time Blowup for the Zakharov System on Two-Dimensional Torus

We consider the Zakharov system on two-dimensional torus. First, we show the local well-posedness of the Cauchy problem in the energy space by a standard iteration argument using the $X^{s,b}$ norms. Our result does not depend on the period of torus. Conservation laws and a sharp Gagliardo-Nirenberg inequality imply an a priori bound of solutions, which enables us to extend the local-in-time solution to a global one if its L2 norm is less than that of the ground state solution of the cubic NLS on R2. We then show that the L2 norm of the ground state is actually the threshold for global solvability, namely, that there exists a finite-time blow-up solution to the Zakharov system on 2d torus with the L2 norm greater than but arbitrarily close to that of the ground state. This is joint work with Masaya Maeda (Tohoku University, Japan).

##### Hardy Spaces With Variable Exponents and Generalized Campanato Spaces

Hardy spaces play an important role not only in harmonic analysis but also in partial differential equations because singular integral operators are bounded on Hardy spaces. The Hardy space H1, which substitute for L1, and the Hardy spaces $H^p$ with $0 < p < 1$, are different in that the latter contains non-regular distributions. Although it will turn out to be an equivalent expression of $L^p$, for $1 < p < \infty$, we can define the Hardy space $H^p$. To have a unified understanding of these situations, we consider and de?ne Hardy spaces with variable exponents on $R^n$. We will connect harmonic analysis with function spaces with variable exponents. We then obtain the atomic decomposition and the molecular decomposition. With these decomposition proved, we investigate the Littlewood?Paley characterization.

##### Finite Point Configurations, Incidence Theory and Multi-linear Operators

A classical problem in geometric combinatorics is to determine how often a single distance may repeat among $n$ points in the plane. A related problem that has received much attention is how often a given triangle can repeat among $n$ points in the plane. We shall report on some partial progress on the second problem using a continuous analog, proved via a Sobolev estimate for a bilinear averaging operator and a conversion mechanism that allows us to deduce a discrete result from a sufficiently robust continuous analog.