MIT PDE/Analysis Seminar

Abstract: The emergence of complex spatial structures in physical systems often occurs after a simpler background state becomes unstable. Localized fluctuations then grow and spread into the unstable state, forming an invasion front which propagates with a fixed speed and selects a new stable state in its wake. The mathematical study of these invasion processes has historically been limited to systems with restrictive monotonicity properties (in PDE terms, a comparison principle). Such systems, however, inherently cannot describe the formation of complex spatiotemporal patterns, which is of particular interest both in nature and in manufacturing applications. On the other hand, formal calculations in the physics literature have long outlined a universal approach for predicting invasion speeds and associated selected states, valid for systems which do not obey comparison principles and instead exhibit complex spatiotemporal dynamics. This prediction scheme is often referred to as the marginal stability conjecture. In this talk, I will discuss the first proof of the marginal stability conjecture and explore applications to structure formation in physical systems.

Abstract: At low frequencies the acoustic operator with random coefficients essentially behaves like a Laplacian (the so-called homogenized operator). We might thus expect the associated wave operator to display some dispersion. By blending standard dispersive estimates for homogenized operators and quantitative homogenization of the wave equation, we derive some "weak" (say, large-scale) dispersive estimates for waves in disordered media.

Applied to the spreading of low-energy eigenstates, they allow us to relate quantitatively homogenization to Anderson localization for acoustic operators in disordered media. This gives a short and direct proof that the lower spectrum of the acoustic operator is purely absolutely continuous in case of periodic media, and it further provides new lower bounds on the localization length of possible eigenstates in case of quasiperiodic or random media.

Abstract: The study of "small" eigenvalues of the Laplacian on hyperbolic surfaces has a long history and has recently seen many developments. In this talk I will focus on the recent work (joint with Yunhui Wu and Haohao Zhang) on the higher spectral gaps, where we study the differences of consecutive eigenvalues up to $\lambda_{2g-2}$ for genus $g$ hyperbolic surfaces. We show that the supremum of such spectral gaps over the moduli space has infimum limit at least 1/4 as genus goes to infinity. The analysis relies on previous joint works with Richard Melrose on degenerating hyperbolic surfaces.

Abstract: The Phi^4, and generally Phi^p measures, which are extensively studied in quantum field theory, also occur naturally as invariant Gibbs measures for certain (dispersive) Hamiltonian PDEs and parabolic SPDEs. A fundamental question is to rigorously justify the invariance of such measures under said dynamics, which leads to deep questions in the solution theory of random data and stochastic PDEs. In this talk we review some recent progress in the dispersive setting, including the proof of invariance of Phi_2^p under Schrodinger dynamics and of Phi_3^4 under wave dynamics. In the Schrodinger case, we also obtain local well-posedness results in the full probabilistically subcritical regime. These are joint works with Bjoern Bringmann, Andrea R. Nahmod and Haitian Yue.

Abstract: We start by introducing a statistical model for the initial data of an N-body Schrodinger equation, meant to represent a scaled version of an N-particle quantum system with unit-order velocities and interparticle separations. The statistical model yields the expected functional form and scale of the corresponding BBGKY densities. This motivates a general a priori assumption on the Sobolev space norms of the BBGKY densities, which includes quasi-free states. Under this assumption, we prove that the Wigner transformed densities converge to the Boltzmann hierarchy with quadratic collision kernel and quantum scattering cross section. The proof of convergence uses a framework previously applied to the derivation of Bose Einstein condensate from an N-body model, and involves exploiting uniform bounds to obtain compactness and weak convergence. The remaining step is to prove the uniqueness of limits, which is performed using the Hewitt-Savage theorem and an extension of the Klainerman-Machedon board game. Our derivation is optimal with respect to regularity considerations. This is joint work with Xuwen Chen, University of Rochester.

Abstract: Gibbs measures of nonlinear Schrödinger equations are a fundamental object used to study low-regularity solutions with random initial data. In the dispersive PDE community, this point of view was pioneered by Bourgain in the 1990s. On the other hand, the nonlinear Schrödinger equation can be viewed a classical limit of many-body quantum theory. We are interested in the problem of the derivation of Gibbs measures as mean-field limits of Gibbs states in many-body quantum mechanics.

The particular case we consider is when the dimension d=2 and when the interaction potential is the delta function, which corresponds to the Euclidean $\Phi^4_2$ theory. The limit that we consider corresponds to taking the density to be large and the range of the interaction to be small in a controlled way. Our proof is based on two main ingredients.

1. An infinite-dimensional stationary phase argument, based on a functional integral representation.
2. A Nelson-type estimate for a nonlocal field theory in two dimensions.

This is joint work with J.Fröhlich, A. Knowles, and B. Schlein.

Abstract: We are concerned about problems in plasma and fluids with flow velocity being outward at the physical boundary. We consider the Vlasov-Poisson equation on a half line with outflow (completely absorbing) boundary conditions, and present a result on the nonlinear stability of a family of stationary solutions. This is a modeling of a plasma boundary layer (sheath). If time permits, I will also present a result about an outflow problem on the compressible Navier-Stokes equation. This is joint work with M. Suzuki and M. Takayama.

Abstract: I will prove that a discrete harmonic function which is bounded on a large density of the plane is constant. Based on joint work with William Cooperman and Shirshendu Ganguly.

Abstract: The functions sin(kx), cos(kx) are positive on half of the circle. This talk will concern a similar phenomenon of quasi-​symmetry for the sign of Laplace eigenfunctions on Riemannian manifolds. We will talk about the distribution of sign and the question of Nazarov, Polterovich and Sodin at which scale quasi-​symmetry holds and at which scale quasi-​symmetry breaks down. Based on a joint work in progress with Fedya Nazarov.

Abstract: Following Margulis's proof of the Oppenheim conjecture we know that integer values of an irrational indefinite quadratic form, $Q$, in $n\geq 3$ variables are dense. The same is true for an inhomogenous form $Q_\alpha(v)=Q(v+\alpha)$ if either $Q$ or $\alpha$ is irrational. In this talk I will describe effective results that hold for a fixed rational form $Q$ and almost all $\alpha$. This turns out to be related to questions regarding the spectral gap of certain representations of the orthogonal group $\mathrm{SO}_Q(\mathbb{R})$ that I will also discuss. This is based on joint work with Anish Ghosh and Shucheng Yu.

Abstract: Bounds for Dirichlet polynomials help to bound the number of zeroes of the Riemann zeta function in vertical strips, which is relevant to the distribution of primes in short intervals. A Dirichlet polynomial is a trigonometric polynomial of the form

$D(t) = \sum_{n = N}^{2N} b_n n^{it}$.

The main question is about the size of the superlevel sets of $D(t)$. We normalize so that the coefficients have norm at most 1, and then we study the superlevel set $|D(t)| > N^\sigma$ for some exponent sigma between 1/2 and 1.

For large values of sigma, Montgomery proved very strong bounds for the superlevel sets. But for sigma $\le 3/4$, the best known bounds follow from a very simple orthogonality argument (and they don't appear to be sharp). We improve the known bounds for sigma slightly less than 3/4. Work in progress. Joint with James Maynard.

Abstract: In the 70’s, Anderson studied the motion of electrons in materials. If the atomic structure is periodic, electrons can travel freely: the material conducts electricity. On the other hand, if the material has impurities or if the atomic structure is more random, electrons can get trapped: the material is now an insulator. Anderson received the Nobel Prize in Physics for this discovery in ’77. Understanding this question mathematically amounts to understanding the nature of the spectrum for a periodic or random Schrödinger operator. In this talk, we will first illustrate, using results from Kuchment (’12) and Bourgain-Kenig (’05), how this problem is related to the following (deterministic) question going back to Landis (late 60’s): given A elliptic, C^1 (or smoother) and V bounded, how rapidly can a non-trivial solution to −div(A∇u) + V u = 0 decay to zero at infinity?

We will discuss the construction of an operator on the cylinder T^2 × R with an eigenfunction div(A∇u) = −μu, which has double exponential decay at both ±∞, where A is uniformly elliptic and uniformly C^1 smooth in the cylinder. Joint work with S. Krymskii and A. Logunov.

Abstract: Let $\mathcal{N}$ be a closed Riemannian manifold, and $\mathcal{T}$ a smooth nowhere-vanishing Killing vector field. The closure of the orbits of $\mathcal{T}$ forms a torus denoted as $G$, which acts on $\mathcal{N}$. In this talk, I will explore the Lefschetz number of a smooth map $f : \mathcal{N}$ → $\mathcal{N}$ that preserves $\mathcal{T}$ and aim to derive a formula inspired by Atiyah-Bott’s style.

To achieve this, we will examine the de Rham complex of smooth forms on $\mathcal{N}$ in the kernel of the interior multiplication by $\mathcal{T}$ while being invariant under $\mathcal{T}$ ’s action. The cohomology of this complex is finite-dimensional, ensuring the welldefined nature of the Lefschetz number. It is worth noting that these cohomology groups are related to, but not identical to, those of the basic cohomology. In conclusion, I will provide an overview of the proof strategy for the Atiyah-Bott formula.

This is a joint work in progress with Gerardo Mendoza (Tempe University).