Past Lectures

Minimal Surfaces, Hyperbolic Surfaces, and Randomness

Abstract:

Minimal surfaces and hyperbolic surfaces are both "optimal 2d geometries" which are ubiquitous in differential geometry. The first kind is defined by an extrinsic condition (the mean curvature vanishes), while the second kind is defined by an intrinsic condition (the Gaussian curvature is equal to -1). I will discuss a surprising connection between the two geometries coming from randomness. The main statement is that there exists a sequence of closed minimal surfaces in Euclidean spheres, constructed from random permutations, which converges to the hyperbolic plane. This result came from my attempt to bridge minimal surfaces and unitary representations. I will introduce this circle of ideas and mention some general questions. 

Generalizations of the Bernstein Problem

Abstract:

Sergei Bernstein proved (in 1914) that an entire solution to the minimal surface equation on R^2 must be affine. This is a nonlinear version of the Liouville theorem for harmonic functions and turns out to have deep links with the regularity of minimal surfaces. I'll explain what happens in higher dimensions, as well as some natural generalizations.

Nonlinear Harmonic Maps and the Energy Identity

Abstract:

We will begin this talk with a broad introduction to nonlinear harmonic maps with a discussion of some basic examples and background.  We will then focus our attention on a type of singularity formulation, resulting in so called defect measures.   Though not typically studied in such generality, these defect measures are a general construction for understanding the loss of energy in limits of H^1 functions.   In the case of nonlinear harmonic maps, there is a precise conjecture as to the form of these defect measures called the Energy Identity.  

This was recently proved together with Daniele Valorta, and a few words on the proof will be discussed.

Stochastic Quantisation of Yang-Mills

Abstract:

We report on recent progress on the problem of building a stochastic process that admits the still hypothetical 3D Yang-Mills measure as its invariant measure. One interesting feature of our construction is that it preserves gauge-covariance in the limit even though it is broken by our UV regularisation. This is based on joint work with Ajay Chandra, Ilya Chevyrev, and Hao Shen. The talk will be a rather gentle introduction to this area which does not require familiarity with any of the above mentioned objects.

Abstract: 

We discuss the typical behavior of two important quantities on compact manifolds with a Riemannian metric \(g\): the number, \(c(T,g)\), of primitive closed geodesics of length smaller than \(T\), and the error, \(E(L,g)\), in the Weyl law for counting the number of Laplace eigenvalues that are smaller than \(L\). For Baire generic metrics, the qualitative behavior of both of these quantities has been understood since the 1970’s and 1980’s. In terms of quantitative behavior, the only available result is due to Contreras and it says that an exponential lower bound on \(c(T,g)\) holds for \(g\) in a Baire-generic set. Until now, no upper bounds on \(c(T,g)\) or quantitative improvements on \(E(L,g)\) were known to hold for most metrics, not even for a dense set of metrics. In this talk, we will introduce the concept of predominance in the space of Riemannian metrics. This is a notion that is analogous to having full Lebesgue measure in finite dimensions, and which, in particular, implies density. We will then give stretched exponential upper bounds for \(c(T,g)\) and logarithmic improvements for \(E(L,g)\) that hold for a predominant set of metrics. This is based on joint work with J. Galkowski.​

Abstract:

Semiclassical measures are a standard object studied in quantum chaos, capturing macroscopic behavior of sequences of eigenfunctions in the high energy limit. In this talk we discuss the supports of semiclassical measures for Laplacian eigenfunctions and for the toy model of quantum cat maps. For Laplacian eigenfunctions on negatively curved surfaces, semiclassical measures have full support as proved in joint work with Jin and Nonnenmacher. This result is not available in higher dimensions because the key new ingredient, the fractal uncertainty principle (proved in joint work with Bourgain), is only known for subsets of the real line. However, in the setting of higher dimensional quantum cat maps one can still use the one-dimensional fractal uncertainty principle to show the full support property under the assumptions that the quantized matrix has a unique largest eigenvalue and its characteristic polynomial is irreducible over the rationals (joint work with Jézéquel).

Abstract:

The functions \(sin(kx), cos(kx)\) are positive on half of the circle and are negative on another half. Armitage and Gardiner conjectured that the sign of spherical harmonics is always positive on a portion of the sphere bounded below by a positive constant, which depends only on the dimension of the sphere. This phenomenon is called quasi-symmetry of sign and it was proved by Donnelly and Fefferman. Nazarov, Polterovich and Sodin suggested that quasi-symmetry of sign happens on small scales in the regime when the eigenvalue grows to infinity. We will talk about the distribution of sign, based on a joint work in progress with Fedya Nazarov.

Abstract:

Complexity of the geometry, randomness of the potential, and many other irregularities of the system can cause powerful, albeit quite different, manifestations of localization, a phenomenon of sudden confinement of waves, or eigenfunctions, to a small portion of the original domain. In the present talk we show that behind a possibly disordered system there exists a structure, referred to as a landscape function, which can predict the location and shape of the localized eigenfunctions, a pattern of their exponential decay, and deliver accurate bounds for the corresponding eigenvalues. In particular, we establish the first non-asymptotic estimates on the integrated density of states of the Schrödinger operator using a counting function for the minima of the localization landscape.

Abstract:

The celebrated Courant nodal domain theorem implies that the number of nodal domains of a Laplace eigenfunction is controlled by the corresponding eigenvalue. There have been many attempts to find an appropriate generalization of this statement in various directions: to linear combinations of eigenfunctions, to their products, to other operators. It turns out that these and other extensions of Courant's theorem can be obtained if one counts the nodal domains in a coarse way, i.e. ignoring small oscillations. The proof uses multiscale polynomial approximation in Sobolev spaces and the theory of persistence barcodes originating in topological data analysis. The talk is based on a joint work with L. Buhovsky, J. Payette, L. Polterovich, E. Shelukhin and V. Stojisavljevic.

Abstract:

We'll discuss the problem of how much eigenfunctions of the Laplace can concentrate in a point on a compact manifold. This is a classical problem and has been studied since the 1950s: the sharp growth rate is known to be attained on the sphere. What makes the problem intriguing is that on 'most' manifolds, eigenfunctions do not seem to be growing very much (the rate appears to be logarithmic as opposed to polynomial). We survey various aspects of the problem and then prove a new way of characterising growth: really the only way that eigenfunctions can grow is if they care a lot about what they do in another far-away region of the manifold, a type of `spooky action at a distance'. This phenomenon can be directly observed even on manifolds as simple as \(S^1\) (where it becomes a fun elementary fact for sines) or the unit square \([0,1]^2\). The result is consistent with the Berry random wave heuristic. On 'generic' manifolds, we expect no such spooky action at a distance which then forces eigenfunctions to not grow very much.

Abstract:

We consider quantitative unique continuation for a family of second order elliptic operators in divergence form with rapidly oscillating and periodic coefficients, which are used to model various physical phenomena in inhomogeneous or heterogeneous media. We are able to show an approximate three-ball inequality using a representation of the Poisson kernel. We can also obtain quantitative explicit doubling inequalities, which are derived by the combination of convergence rate estimates, three-ball inequalities from large-scale analyticity, and the monotonicity formula of frequency function. Furthermore, the explicit upper bounds of nodal sets are shown using doubling inequalities, approximation of harmonic functions and iteration arguments. Finally, I will report some recent progress in the quantitative unique continuation for Dirichlet eigenfunctions in periodic elliptic homogenization. The talk is based on joint work with Carlos Kenig and Jinping Zhuge.