The Analysis and PDE seminar will take place Monday Sept 10 in 740 Evans from 4-5pm.

Title: Quantitative additive energy estimates for regular sets and connections to discretized sum-product theorems

Abstract: We prove new quantitative additive energy estimates for a large class of porous measures which include, for example, all Hausdorff measures of Ahlfors-David subsets of the real line of dimension strictly between 0 and 1. We are able to obtain improved quantitative results over existing additive energy bounds for Ahlfors-David sets by avoiding the use of inverse theorems in additive combinatorics and instead opting for a more direct approach which involves the use of concentration of measure inequalities. We discuss some connections with Bourgain’s sum-product theorem.

The Analysis and PDE seminar will take place Monday, Aug 27, in 740 Evans from 4-5pm.

Title: Balanced geometric Weyl quantization with applications to QFT on curved spacetimes

Abstract: First I will describe a new pseudodifferential calculus for (pseudo-)Riemannian spaces, which in our opinion (my, D.Siemssen’s and A.Latosiński’s) is the most appropriate way to study operators on such a manifold. I will briefly describe its applications to computations of the asymptotics the heat kernel and Green’s operator on Riemannian manifolds. Then I will discuss analogous applications to Lorentzian manifolds, relevant for QFT on curved spaces. I will mention an intriguing question of the self-adjointness of the Klein-Gordon operator. I will describe the construction of the (distinguished) Feynman propagator on asymptotically static spacetimes. I will show how our pseudodifferential calculus can be used to compute the full asymptotics around the diagonal of various inverses and bisolutions of the Klein-Gordon operator.

The Analysis and PDE seminar will take place Monday, September 10, in 740 Evans from 4:10 to 5pm.

Title: TBA

Abstract: TBA

The Analysis and PDE seminar will take place Monday, March 19, in 740 Evans from 4:10 to 5pm.

Title: Calderon problem for Yang-Mills connections

Abstract: In the classical Calderon conjecture we want to recover a metric on a compact manifold, up to diffeomorphism fixing the boundary, from the Dirichlet-to-Neumann (DN) map of the metric Laplacian. This problem is routinely motivated by applications and is open in dimension 3 and higher. In this talk, we fix the metric and consider the DN map of the connection Laplacian for Yang-Mills connections. We sketch the proof of uniqueness up to gauges fixing the boundary for smooth line bundles. The proof uses new techniques, involving unique continuation principles for degenerate elliptic equations and an analysis of the zero set of solutions to an elliptic PDE. Time permitting, we will also discuss some (counter) examples concerning zero sets of determinants of matrix solutions to elliptic PDE.

The Analysis and PDE seminar will take place Monday, March 5, in 740 Evans from 4:10 to 5pm.

Title: Several questions on Laplace eigenfunctions

Abstract: Let $(M,g)$ be a compact Riemannian manifold without boundary. We are interested in asymptotic properties of Laplace eigenfunctions on $M$ as the eigenvalue $\lambda$ tends to infinity. The advances of the last few years will be discussed and a survey of interesting open questions will be given.

The Analysis and PDE seminar will take place on Monday, March 12, in 740 Evans from 4:10 to 5 pm.

Title: Honeycomb Structures, Edge States, and the Strong Binding Regime

Abstract: We review recent progress on the propagation of waves for the 2D Schrödinger and Maxwell equations for media with the symmetry of a hexagonal tiling of the plane.

The Analysis and PDE seminar will take place Monday, February 12, in 740 Evans from 4:10 to 5 pm.

Title: Edge (resonant) states for 1D bi-periodic systems

Abstract: We study the bifurcation of Dirac points under the introduction of edges in certain periodic systems. For honeycomb Schrodinger operators, Fefferman, Lee-Thorp and Weinstein showed that if introducing the edge opens an essential spectral gap near the Dirac energy, then the perturbed operator has an edge-localized eigenstate. This state is associated to the topologically protected zero-mode of a Dirac operator, which emerges from a formal multiscale approach (one of the two scales being the size of the edge).

We consider 1D models where the introduction of a large edge does not necessarily open an essential spectral gap near the Dirac energy. We approach such systems with Fredholm analytic tools and show that Dirac points bifurcate to resonant states. When the edge perturbation happens to open an essential spectral gap, this improves a previous result of Fefferman–Lee-Thorp–Weinstein by (a) proving the validity of the multiscale approach; (b) relating each eigenvalue in the gap to an eigenvalue of the above Dirac operator; (c) deriving full expansions of the associated states.

Joint work with Michael Weinstein and Charles Fefferman.

The Analysis and PDE seminar will take place Monday, February 5, in 740 Evans from 4:10 to 5 pm.

Title: Fredholm theory and the resolvent of the Laplacian near zero energy on asymptotically conic spaces

Abstract: We consider geometric generalizations of Euclidean low energy resolvent estimates, such as estimates for the resolvent of the Euclidean Laplacian plus a decaying potential, in a Fredholm framework. More precisely, the setting is that of perturbations $P(\sigma)$ of the spectral family of the Laplacian $\Delta_g-\sigma^2$ on asymptotically conic spaces $(X,g)$ of dimension at least $3$, and the main result is uniform estimates for $P(\sigma)^{-1}$ as $\sigma\to 0$ on microlocal variable order spaces under an assumption on the nullspace of $P(0)$ on the appropriate function space (which in the Euclidean case translates to $0$ not being an $L^2$-eigenvalue or having a half-bound state). These spaces capture the limiting absorption principle for $\sigma\neq 0$ in a lossless, in terms of decay, manner.

The Analysis and PDE seminar will take place Monday, January 29, in 740 Evans from 4:10 to 5 pm.

Title: Fourier dimension for limit sets

Abstract: For a finite measure $\mu$ on the real line, its Fourier dimension is defined using the rate of polynomial decay of the Fourier transform $\hat \mu$. The Fourier dimension of $\mu$ may be much smaller than the Hausdorff dimension of the support of $\mu$: a classical example is the Cantor measure on the mid-third Cantor set which has Fourier dimension equal to 0.

I will present a joint result with J. Bourgain showing that the Patterson-Sullivan measure on the limit set of a convex co-compact group of fractional linear transformations has positive Fourier dimension. The proof uses advanced tools from additive combinatorics (the discretized sum-product theorem) and exploits the fact that fractional linear transformations are (generally) not linear. An application is a new spectral gap result for convex co-compact hyperbolic surfaces.

The Analysis and PDE seminar will take place Monday, January 15, in 740 Evans from 4:10 to 5pm.

Title: Concentration of eigenfunctions: Averages and Sup-norms

Abstract: In this talk, we relate microlocal concentration of eigenfunctions to sup-norms and sub-manifold averages. In particular, we characterize the microlocal concentration of eigenfunctions with maximal sup-norm and average growth. We then exploit this characterization to derive geometric conditions under which maximal growth cannot occur. This talk is based on joint works with Yaiza Canzani and John Toth.