# Seminars & Events for Analysis Seminar

##### Time-Periodic Einstein--Klein--Gordon Bifurcations Of Kerr

For an open measure set of Klein--Gordon masses mu^2 > 0, we construct one-parameter families of solutions to the Einstein--Klein--Gordon equations bifurcating off the Kerr solution such that the underlying family of spacetimes are each an asymptotically flat, stationary, axisymmetric, black hole spacetime, and such that the corresponding scalar fields are non-zero and time-periodic. An immediate corollary is that for these Klein--Gordon masses, the Kerr family is not asymptotically stable as a solution to the Einstein--Klein--Gordon equations. (Joint work with O, Chodosh.)

##### Harmonic analysis and intrinsic Diophantine approximation

We will describe a recently developed general approach to Diophantine approximation on homogeneous algebraic varieties, and demonstrate it in some familiar natural examples. The approach utilizes harmonic analysis on Lie groups and some arguments in homogeneous dynamics and ergodic theory. It provides the best possible solution to a host of intrinsic Diophantine approximation problems which were not accessible by previous techniques. Based on joint work with Alex Gorodnik and on joint work with Anish Ghosh and Alex Gorodnik.

##### Hölder continuity of solutions to hypoelliptic equations with rough coefficients

The celebrated De Giorgi-Nash theory about Hölder continuity of solutions to elliptic or parabolic equations with rough --i.e. merely measurable-- coefficients in the late 1950s is a cornerstone of modern PDE analysis. We extend this theory to a class of kinetic equation of Vlasov-Fokker-Planck type ("hypoelliptic of type II" in the terminology of Hörmander) where a first-order hyperbolic operator interacts with a partially elliptic operator with rough coefficients. We also extend the theory of Moser about Harnack inequalities for these equations. This is a joint work with F. Golse, C. Imbert and A. Vasseur.

##### $\ell^p(\mathbf Z^d)$ boundedness for discrete operators of Radon types: maximal and variational estimates

**PLEASE NOTE ROOM CHANGE FROM LAST TERM: SEMINAR WILL NOW BE HELD IN FINE 110.** In recent times - particularly the last two decades - discrete analogues in harmonic analysis have gone through a period of considerable changes and developments. This is due in part to Bourgain's pointwise ergodic theorem for the squares on $L^p(X, \mu)$ for any $p>1$. The main aim of this talk is to discuss recent developments in discrete harmonic analysis. We will be mainly concerned with $\ell^p(\mathbf Z^d)$ estimates $(p>1)$ of $r$-variations $(r>2)$ for discrete averaging operators and singular integral operators along polynomial mappings. All the results are subjects of the ongoing projects with Elias M. Stein and Bartosz Trojan.

##### A restriction estimate using polynomial partitioning

The restriction conjecture is an open problem in Fourier analysis first raised by Stein in the late 60’s. It is about the L^p estimates obeyed by an oscillatory integral operator. I will explain a recent approach to this problem, which gives a slightly better estimate in three dimensions. The polynomial partitioning approach is a divide-and-conquer argument, using a polynomial surface to cut space into pieces and an inductive argument to understand the contribution of each piece. This approach is based on recent ideas from combinatorics. In 2007, Dvir used polynomials in a surprising way to give a very short proof of the finite field Kakeya problem - a kind of cousin of the restriction problem. Building on these ideas, Katz and I used polynomial partitioning to prove new estimates in incidence geometry.

##### On the global dynamics of three dimensional imcompressible magnetohydrodynamics

We construct and study global solutions for the 3-dimensional imcompressible MHD systems with arbitrary small viscosity. In particular, we provide a rigorous justification for the following dynamical phenomenon observed in many contexts: the solution at the beginning behave like non-dispersive waves and the shape of the solution persists for a very long time (proportional to the Reynolds number); thereafter, the solution will be damped due to the long-time accumulation of the diffusive effects; eventually, the total energy of the system becomes extremely small compared to the viscosity so that the diffusion takes over and the solution afterwards decays fast in time. We do not assume any symmetry condition. The size of data and the a priori estimates do not depend on viscosity.

##### A New Analytic Approach to Wave Turbulence

In this talk we discuss improvements to a new approach to wave turbulence instigated by Zaher Hani, Pierre Germain and Erwan Faou. This approach will combine techniques from analytic number theory and dispersive PDE theory to study an example of discrete turbulence, for which dynamics is dominated by the exact resonances of the equation. Specifically we will study the large box limit of the Nonlinear Schr\"odinger Equation in the weakly nonlinear regime. This is joint work with Zaher Hani, Pierre Germain and Jalal Shatah.

##### Absolute continuity and rectifiability of harmonic measure

**Please note special location. **The properties of harmonic measure (most importantly, absolute continuity and rectifiability) are key to many problems in Analysis, Probability, Geometric Measure Theory, as well as PDEs. In this talk we will establish precise connections between the structure of the harmonic measure, geometry of the set, and well-posedness of the underlying boundary problems. The central results to be presented are as follows. (1) We prove that for any open connected set $\Omega\subset\mathbb{R}^{n+1}$, and any $E\subset \partial \Omega$ with $0<\mathbb{H}^n(E)<\infty$ absolute continuity of the harmonic measure $\omega$ with respect to the Hausdorff measure on $E$ implies that $\omega|_E$ is rectifiable.

##### Almost optimal local existence for radially symmetric time like minimal surface equation in 1+3 dimensional Minkowski space

**Please note special time and location.**

##### Needle decomposition and Ricci curvature

**Please note special day and time. **Needle decomposition is a technique in convex geometry, which enables one to prove isoperimetric and spectral gap inequalities, by reducing an n-dimensional problem to a 1-dimensional one. This technique was promoted by Payne-Weinberger, Gromov-Milman and Kannan-Lovasz-Simonovits. In this lecture we will explain what needles are, what they are good for, and why the technique works under lower bounds on the Ricci curvature."

##### Global existence for quasilinear wave equations close to Schwarzschild

We study the quasilinear wave equation $[g] u = 0$, where the metric $g$ depends on $u$ and equals the Schwarzschild metric when u is identically 0. Under a couple of extra assumptions on the metric $g$ near the trapped set and the light cone, we prove global existence of solutions. This is joint work with Hans Lindblad.

##### Geometric control: from damped waves to microfluid mixing

The damped wave equation is a prototype of a non-selfadjoint PDE, which means that the stationary problem has complex spectrum. The asymptotic distribution of the spectrum is heavily influenced by properties of the geodesic flow, and in part determines the rate of decay to equilibrium. In this talk, I will survey recent results on such decay estimates for the damped and over-damped wave equation, followed by a description of work-in-progress applying damped wave methods to systems arriving in the analysis of certain microfluidic mixing devices.

##### 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.

##### 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. We will discuss the issue of local existence for these types of equations and time permitting understand the main features of each geometric problem separately.

##### On the subcritical transition of the 3D Couette flow

We discuss the dynamics of small perturbations of the plane, periodic Couette flow in the 3D incompressible Navier-Stokes equations at high Reynolds number. For sufficiently regular initial data, we determine the stability threshold for small perturbations and characterize the long time dynamics of solutions near this threshold. For rougher data, we obtain an estimate of the stability threshold which agrees closely with numerical experiments. The primary linear stability mechanism is an anisotropic enhanced dissipation resulting from the mixing caused by the large mean shear; the main linear instability is a non-normal instability known as the lift-up effect. Understanding the variety of nonlinear resonances and devising the correct norms to estimate them form the core of the analysis we undertake.

##### variance of sums of arithmetic functions over primes in short intervals

**This is a special Analysis/Princeton-IAS Number Theory seminar. **Goldston & Montgomery and Montgomery & Soundararajan have established formulae for the variance of sums of the von Magoldt function over short intervals (i.e. for the variance of the number of primes in these intervals) assuming, respectively, the pair-correlation conjecture and the Hardy-Littlewood conjecture. I will discuss the generalisation of these formulae to other arithmetic functions associated with the Selberg class of L-functions, in the context of both zero statistics and arithmetic correlations. I also hope to discuss the function-field analogues of these generalisations.

##### On the Boltzmann Equation for Non-spherical Particles

The classical Boltzmann equation is a well-studied mathematical model for the evolution of rarified gases whose constituent particles have perfect spherical symmetry. As most matter in the universe that is in the gaseous phase is not made up of molecules or atoms which are perfect spheres, it is a natural question to ask how the structure of the classical Boltzmann equation changes when one modifies the geometry of the underlying gas particles. In this talk, we present some new results on the characterisation of collision invariants for compact, strictly-convex, non-spherical particles. These results allow one to establish local conservation laws, to characterise Maxwellia, and to perform hydrodynamic limits for the analogue of the classical Boltzmann equation for non-spherical particles.

##### Schottky Implies Poincare in any genus at least 4

The Schottky problem is the problem of finding holomorphic equations on the Siegel upper half plane of degree g ( at least 4) which cut out the space of Jacobi varieties in the Siegel upper half plane. The Poincare problem is the problem of cutting out the same space but in a neighborhood of the diagonal matrices. Poincare solved his problem for all g at least 4. The Schottky problem was completely solved by Schottky only in the case g=4. In this talk we shall show for all genus at least 4 how to write a set of (g-3)(g-2)/2 equations ( of Schottky type ) which vanish on the space of Jacobi varieties and reduce to the Poincare equations near the diagonal matrices. The main ideas of the proof will involve the Riemann theta formula, The Schottky Jung proportionalities, and the transformation theory of theta constants.

##### Existence of Pollicott--Ruelle resonances

Pollicott--Ruelle resonances are the modes appearing in correlations for Anosov flows and as singularities of dynamical zeta functions. A resonance free strip implies exponential decay of correlations and, thanks to the work of Dolgopyat, Liverani and Tsujii, it is known to exist for contact Anosov flows. Using methods of microlocal analysis and scattering theory we show that the size of that strip is finite for any Anosov flow, that is, there exist strips with infinite number of resonances. (Joint work with Long Jin and Frederic Naud).

##### Graviton propagator on Schwarzschild spacetime

**Please note the special day, time and location. **The study of graviton Hawking radiation, renormalized interactions of dynamical gravitons with other matter fields, and quantum back reaction of Hawking radiation on the black hole horizon all require the knowledge of the graviton propagator on a black hole spacetime. I will discuss an explicit mode-decomposed construction of the de Donder gauge graviton propagator for the Unruh state on the eternal Schwarzschild spacetime. De Donder gauge is chosen for its covariance and locality properties. This is joint work with F. Bussola (Pavia).