Author Archives: peterhintz

Mihajlo Cekic (Max Planck Institute for Mathematics, Bonn)

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.

Alexis Drouot (Columbia University)

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.

András Vasy (Stanford)

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.

Semyon Dyatlov (UC Berkeley)

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.

Jeffrey Galkowski (Stanford)

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.

Kiril Datchev (Purdue) and Charles Hadfield (UC Berkeley)

The Bay Area Microlocal Analysis Seminar will take place on Monday, September 25, in room 740, Evans Hall, with two talks, given by Kiril Datchev at 2:40 pm and Charles Hadfield at 4:10 pm.

Speaker: Kiril Datchev (2:40 pm)

Title: Semiclassical resolvent estimates away from trapping

Abstract: Semiclassical resolvent estimates relate dynamics of a particle scattering problem to regularity and decay of waves in a corresponding wave scattering problem. Roughly speaking, more trapping of particles corresponds to a larger resolvent near the trapping. If the trapping is mild, then propagation estimates imply that the larger norm occurs only there. However, in this talk I will show how the effects of heavy trapping can tunnel over long distances, implying that the resolvent can be very large far away as well. This is joint work with Long Jin.


Speaker: Charles Hadfield (4:10 pm)

Title: Resonances on asymptotically hyperbolic manifolds; the ambient metric approach

Abstract: On an asymptotically hyperbolic manifold, the Laplacian has essential spectrum. Since work of Mazzeo and Melrose, this essential spectrum has been studied via the theory of resonances; poles of the meromorphic continuation of the resolvent of the Laplacian (with modified spectral parameter). A recent technique of Vasy provides an alternative construction of this meromorphic continuation which dovetails the ambient metric approach to conformal geometry initiated by Fefferman and Graham. I will discuss the ambient geometry present in this construction, use it to define quantum resonances for the Laplacian acting on natural tensor bundles (forms, symmetric tensors), and mention an application showing a correspondence between Ruelle resonances and quantum resonances on convex cocompact hyperbolic manifolds.

Kiril Datchev (Purdue)

The Analysis and PDE Seminar will take place on Monday, September 18, in room 740, Evans Hall, from 4:10-5:00 pm.

Title: Resolvent estimates and wave asymptotics for manifolds with cylindrical ends

Abstract: Wave oscillation and decay rates on a manifold M are are well known to be related to the geometry of M and to dynamical properties of its geodesic flow. When M is a closed system, such as a bounded Euclidean domain or a compact manifold, there is no decay and the connection is made via the eigenvalues of the Laplacian. When M is an open system, such as the complement of a bounded Euclidean domain, or a suitable more general manifold with large infinite ends, then the spectrum of the Laplacian is continuous and we look instead at resonances, which give rates of both oscillation and decay.

An interesting intermediate situation is a manifold with infinite cylindrical ends, which we call a mixed system. In this case the continuous spectrum has increasing multiplicity as energy grows, and in general it can have embedded resonances and eigenvalues accumulating at infinity, making wave asymptotics more mysterious. However, we prove that if geodesic trapping is sufficiently mild, then such an accumulation is ruled out, and moreover lossless high-energy resolvent bounds hold. We deduce from this the existence of resonance free regions and compute asymptotic expansions for solutions of the wave equation in terms of eigenvalues, resonances, and spectral thresholds. This is joint work with Tanya Christiansen.

Laurent Michel, Stanford and Université Nice

The Analysis and PDE Seminar will take place on Monday, February 13, in room 740, Evans Hall, from 4:10-5:00 pm.

Speaker: Laurent Michel

Title: On the small eigenvalues of the Witten Laplacian

Abstract: The Witten Laplacian, introduced (by E. Witten) in the early 80’s to give an analytic proof of the Morse inequalities, also models the dynamics of the over-damped Langevin equation. The understanding of the so-called metastable states goes through the description of its small eigenvalues. The first result in this direction was obtained in 2004 (Bovier-Gayrard-Klein, Helffer-Klein-Nier) under some generic assumptions on the landscape potential. In this talk, we present the approach of Helffer-Klein-Nier and show some recent progress to get rid of the generic assumption.