Monday, July 2 |
07:00 - 08:45 |
Breakfast ↓ Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building. (Vistas Dining Room) |
08:45 - 09:00 |
Introduction and Welcome by BIRS Staff ↓ A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions. (TCPL 201) |
09:00 - 09:50 |
Michael Levitin: Spectral geometry - from the 19th to 21st century in 50 minutes ↓ I will give a very pedestrian overview of some problems in spectral geometry — a vast topic covering relations between eigenvalues of boundary value problems for the Laplacian (or for other differential operators) in a Euclidean domain or on a Riemannian manifold, and the underlying geometry. They will include some open problems of various degrees of difficulty. (TCPL 201) |
09:50 - 10:20 |
Coffee Break (TCPL Foyer) |
10:20 - 11:00 |
Dorin Bucur: Maximization of Neumann eigenvalues ↓ Abstract: In this talk I will discuss the question of the maximization of the k-th eigenvalue of the Neumann-Laplacian under a volume constraint. After an introduction to the topic I will discuss the question of existence of optimal geometries. For now, there is no a general existence result, but one can prove existence of an optimal {\it (over) relaxed domain}, view as a density function. These results are an on-going work with E. Oudet. In the second part of the talk, I will focus on the low eigenvalues. The first non-trivial one is maximized by the ball, the result being due to Szego and Weinberger in the fifties. Concerning the second non-trivial eigenvalue, Girouard, Nadirashvili and Polterovich proved that the supremum in the family of planar simply connected domains of R2 is attained by the union of two disjoint, equal discs. I will show that a similar statement holds in any dimension and without topological restrictions. This last result is jointly obtained with A. Henrot. (TCPL 201) |
11:10 - 11:30 |
Etienne Vouga: Hearing the Shape of the Bunny ↓ It is well-known that one cannot generally hear the shape of a drum:
the metric of a compact surface is not uniquely determined by its
Laplace-Beltrami spectrum. But one can still seek computational
solutions to the inverse problem: given a sequence of eigenvalues, can
we compute a surface whose Laplace-Beltrami spectrum approximates the
sequence? I will discuss some numerical experiments related to this
problem for the case of surfaces of sphere topology, whose discrete
conformal parameterization leads to an especially simple formulation
of the inverse problem. (TCPL 201) |
11:30 - 13:00 |
Lunch ↓ Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building. (Vistas Dining Room) |
13:00 - 14:00 |
Guided Tour of The Banff Centre ↓ Meet in the Corbett Hall Lounge for a guided tour of The Banff Centre campus. (Corbett Hall Lounge (CH 2110)) |
14:00 - 14:20 |
Group Photo ↓ Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo! (TCPL 201) |
14:20 - 15:00 |
Coffee Break (TCPL Foyer) |
15:00 - 15:40 |
Ron Kimmel: Invariant Representations of Shapes and Forms: Self Functional Maps ↓ A classical approach for surface classification is to find a compact algebraic representation for each surface that would be similar for objects within the same class and preserve dissimilarities between classes. We introduce self functional maps as a novel surface representation that satisfies these properties, translating the geometric problem of surface classification into an algebraic form of classifying matrices. The proposed map transforms a given surface into a universal isometry invariant form defined by a unique matrix. The suggested representation is realized by applying the functional maps framework to map the surface into itself. The key idea is to use two different metric spaces of the same surface for which the functional map serves as a signature. Specifically, in this lecture, we suggest the regular and the scale invariant surface laplacian operators to construct two families of eigenfunctions. The result is a matrix that encodes the interaction between the eigenfunctions resulted from two different Riemannian manifolds of the same surface. Using this representation, geometric shape similarity is converted into algebraic distances between matrices.
In contrast to geometry understanding there is the emerging field of deep learning. Learning systems are rapidly dominating the areas of audio, textual, and visual analysis. Recent efforts to convert these successes over to geometry processing indicate that encoding geometric intuition into modeling, training, and testing is a non-trivial task. It appears as if approaches based on geometric understanding are orthogonal to those of data-heavy computational learning. We propose to unify these two methodologies by computationally learning geometric representations and invariants and thereby take a small step towards a new perspective on geometry processing. If time permits I will present examples of shape matching, facial surface reconstruction from a single image, reading facial expressions, shape representation, and finally definition and computation of invariant operators and signatures. (TCPL 201) |
15:50 - 16:30 |
David Colton: Spectral Theory for the Transmission Eigenvalue Problem ↓ The transmission eigenvalue problem plays a central role in inverse scattering theory. This is a non-selfadjoint problem for a coupled pair of partial differential equations in a bounded domain corresponding to the support of the scattering object. Unfortunately, relatively little is known about the spectrum of this problem. In this talk I will consider the simplest case of the transmission eigenvalue problem for which the domain and eigenfunctions are spherically symmetric. In this case the transmission eigenvalue problem reduces to an eigenvalue problem for ordinary differential equations. Through the use of the theory of entire functions of a complex variable, I will show that there is a remarkable diversity in the behavior of the spectrum of this problem depending on the behavior of the refractive index near the boundary. Included in my talk will be results on the existence of complex eigenvalues, the inverse spectral problem and a remarkable connection (due to Fioralba Cakoni and Sagun Chanillo) between the location of transmission eigenvalues for automorphic solutions of the wave equation in the hyperbolic plane and the Riemann hypothesis. (TCPL 201) |
16:50 - 17:30 |
Fioralba Cakoni: Discussion and open problem session on numerical aspects of spectral geometry (TCPL 201) |
17:30 - 19:30 |
Dinner ↓ A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building. (Vistas Dining Room) |