Author Archives: murphy

Tanya Christiansen (April 13th)

Speaker: Tanya Christiansen (University of Missouri)

Title: Resonances in even-dimensional Euclidean scattering

Abstract: Resonances may serve as a replacement for discrete spectral data for a class of operators with continuous spectrum. In odd-dimensional Euclidean scattering, the resonances lie on the complex plane, while in even dimensions they lie on the logarithmic cover of the complex plane. In even-dimensional Euclidean scattering there are some surprises for those who are more familiar with the odd-dimensional case. For example, qualitative bounds on the number of “pure imaginary” resonances are very different depending on the parity. Moreover, for Dirichlet or Neumann obstacle scattering or for scattering by a fixed-sign potential one can show there are many resonances in even dimensions. In fact, for these cases the $m$th resonance counting function ($m\in Z, m\neq 0$) has maximal order of growth.

Some of this talk is based on joint work with Peter Hislop.


Raphael Ponge (March 9th)

Speaker: Raphael Ponge (UC Berkeley / Seoul National University)

Title: On the singularities of the Green functions of the conformal powers of the Laplacian.

Abstract: Green functions play a major role in PDEs and conformal geometry. In this talk, I will explain how to compute explicitly the logarithmic singularities of the Green functions of the conformal powers of the Laplacian. This includes the Yamabe and Paneitz operators, as well as the conformal fractional powers originating from the work of Graham-Zworski on scattering theory for AH metrics. The results are formulated in terms of explicit conformal invariants arising from the ambient Lorentzian metric of Fefferman-Graham. As applications we obtain a new characterization of locally conformally flat manifolds and a spectral-theoretic characterization of the conformal class of the round sphere.

Yaiza Canzani (March 2nd)

Speaker: Yaiza Canzani (Harvard)

Title: On the geometry and topology of zero sets of Schrödinger eigenfunctions.

Abstract: In this talk I will present some new results on the structure of the zero sets of Schrödinger eigenfunctions on compact Riemannian manifolds.  I will first explain how wiggly the zero sets can be by studying the number of intersections with a fixed curve as the eigenvalue grows to infinity. Then, I will discuss some results on the topology of the zero sets when the eigenfunctions are randomized. This talk is based on joint works with John Toth and Peter Sarnak.

Dmitry Jakobson (January 26)

Speaker: Dmitry Jakobson (McGill)

Title: Probability measures on manifolds of Riemannian metrics

Abstract: This is joint work with Y. Canzani, B. Clarke, N. Kamran, L. Silberman and J. Taylor. We construct Gaussian measures on the manifold of Riemannian metrics with the fixed volume form. We show that diameter, Laplace eigenvalue and volume entropy functionals are all integrable with respect to our measures. We also compute the characteristic function for the L^2(Ebin) distance from a random metric to the reference metric.

Naoki Saito (December 1)

Note: this talk will not take place in the usual room. Location TBA.

Speaker: Naoki Saito (UC Davis)

Title: Laplacian eigenfunctions that do not feel the boundary: Theory, Computation, and Applications

Abstract: I will discuss Laplacian eigenfunctions defined on a Euclidean
domain of general shape, which “do not feel the boundary.”
These Laplacian eigenfunctions satisfy the Helmholtz equation inside the domain,
and can be extended smoothly and harmonically outside of the domain.
Although these eigenfunctions do not satisfy the usual Dirichlet or Neumann
boundary conditions, they can be computed via the eigenanalysis of the
integral operator (with the potential kernel) commuting with the Laplace
operator. Compared to directly solving the Helmholtz equations on such
domains, the eigenanalysis of this integral operator has several advantages
including the numerical stability and amenability to modern fast numerical
algorithms (e.g., the Fast Multipole Method).
In this talk, I will discuss their properties, the relationship with the
Krein-von Neumann self-adjoint extension of unbounded symmetric operators, and
certain applications including image extrapolation and characterization of
biological shapes.

Alexander Volberg (November 17)

Speaker: Alexander Volberg (MSU)

Title: Beyond the scope of doubling: weighted martingale multipliers and outer measure spaces

Abstract: A new approach to characterizing the unconditional basis property of martingale differences in weighted $L^2(w d\nu)$ spaces is given for arbitrary martingales, resulting in a new version with arbitrary and in particular non-doubling reference measure $\nu$. The approach combines embeddings into outer measure spaces with a core concavity argument of Bellman function type. Specifically, we prove that finiteness of the $A_2$ characteristic of the weight (defined through averages relative to arbitrary reference measure $\nu$) is equivalent to the boundedness of martingale multipliers. Even in the case of the usual dyadic martingales based on dyadic cubes in $\mathbb{R}^d$ our result is new because it is dimension free. In the case of general measures, this result is unexpected. For example, a small change in operator breaks the result immediately. This is a joint work with Christoph Thiele and Sergei Treil.

Sung-Jin Oh (November 10)

Speaker: Sung-Jin Oh (UC Berkeley)

Title: On the energy critical Maxwell-Klein-Gordon equations

In this talk I will present a recent joint work with D. Tataru on the global regularity and scattering for the Maxwell-Klein-Gordon equations on the (4+1)-dimensional Minkowski space, which is energy critical.

Lawrence C Evans (November 3)

Title: Convexity, nonlinear PDE and principal/agent problems

Speaker: Lawrence C Evans (UC Berkeley)

Abstract: I will explain a simple convexity argument that provides an easy derivation of Sannikov’s optimality condition for continuous time principal/agent problems in economics.

Boaz Haberman (October 27)

Speaker: Boaz Haberman (UC Berkeley)

Title: Calderón’s problem for rough conductivities

Abstract: Calderon’s problem asks whether the coefficients of an elliptic equation can be recovered from its Dirichlet-to-Neumann map. Sylvester and Uhlmann introduced the method of complex geometrical optics solutions to solve this problem. In this talk we will discuss how to use some methods from dispersive equations to construct these solutions under more general regularity conditions for the coefficients.

Michael Christ (October 20)

Speaker: Michael Christ (UC Berkeley)

Title: An extremal problem concerning Fourier coefficients

Consider a set in Euclidean space, and consider the $L^q$ norm of its Fourier transform. Among sets of specified measure, what is the largest value of this norm? Is it attained? If so, by which sets?

These natural questions seem to have received little attention. I will state several partial results, and indicate some of the ideas in the proofs. One ingredient is a compactness theorem, whose proof relies on an inverse theorem of additive combinatorics.