Special Day on Eigenfunctions of the Laplacian on Manifolds
Mathematics Department
Northwestern University
Saturday October 25, 2014
Harris Hall, L28
Local Map
All are invited to attend, there is no registration.
Schedule
10.00am  11.00am

Daniele Valtorta (EPFL Lausanne)
Minkowski estimates on critical and nodal sets of harmonic functions
 Abstract
Given a nonconstant harmonic function, we obtain Minkowski
bounds on its critical and almost critical set. The proof relies on a
refined blowup analysis for harmonic functions based on the properties
of Almgren's frequency.
With minor modifications, these estimates are valid also for solutions
to a very general class of elliptic PDEs. Given the link between
harmonic functions and eigenfunctions of the Laplacians, with the
necessary modifications these results apply also to nodal and singular
sets of eigenfunctions. This is joint work with Aaron Naber.

11.30am  12.30pm

Chris Sogge (Johns Hopkins University)
Focal points and supnorms of eigenfunctions  Abstract
If (M,g) is a compact real analytic Riemannian manifold,
we give a necessary and sufficient condition for there to be a sequence of
quasimodes saturating supnorm estimates. The condition is that there
exists a selffocal point x_{0} in M for the geodesic flow at which
the associated PerronFrobenius operator
U: L^{2}(S^{*}_{x0}M) →L^{2}(S^{*}_{x0}M) has a nontrivial invariant
function. The proof is based on von Neumann's ergodic theorem and
stationary phase. In two dimensions, the condition simplifies
and is equivalent to the condition that there be a point through which the
geodesic flow is periodic.
This is joint work with Steve Zelditch.

2.30pm  3.30pm

John Toth (McGill University)
L^{2}restriction lower bounds for Schrodinger eigenfunctions in classically forbidden regions  Abstract
Let (M,g) be a compact, closed, realanalytic Riemannian manifold and P(h) = h^{2} Δ_{g} + V be a Schrodinger operator with
(g,V) realanalytic. Given a regular energy value E with consider
L^{2}normalized eigenfunctions φ_{h} satisfying
P(h) φ_{h} = ( E + o(1)) φ_{h} . We first prove that for any hypersurface H in
the forbidden region Ω(E) = { V > E } there exists a constant
c_{H} > 0 such that
(a) ∫_{H}  φ_{h}^{2} ≥ e^{cH/h}.
We then use the estimate in (a) to prove that when dim M=2 and H is a
C^{ω} hypersurface in Ω(E), the nodal set Z_{ φh} = { φ_{h} = 0 }
satifies the bound
(b) # { Z_{ φh} ∩ H } = O_{H} (h^{1}).
This is joint work with Yaiza Canzani.

4.00pm  5.00pm

Iosif Polterovich (Université de Montréal)
Spectral geometry of the Steklov problem  Abstract
The Steklov problem is an elliptic eigenvalue problem with the spectral
parameter in the boundary conditions.
While this problem shares some common properties with its more well known Dirichlet
and Neumann cousins, the Steklov eigenvalues and eigenfunctions have a number of
distinctive geometric features. We will discuss some recent advances in the
subject, particularly in the study of spectral asymptotics, spectral invariants,
eigenvalue estimates, and nodal geometry. The talk is based on a joint survey
article with A. Girouard.

Organizers:
Questions? email to: emphasisGA@math.northwestern.edu