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Every compact connected 2-manifold N can be expressed as the orbit space determined by a group of isometries G acting freely and properly discontinuously on a Riemannian 2-manifold M which is either the 2-sphere, the Euclidean plane, or the Poincaré disk. In each case, M is simply connected, and so the fundamental group of N is isomorphic to G. In this talk, we express the isometries of the Poincaré disk as complex-valued functions, and determine the groups of isometries which yield as orbit spaces the orientable surfaces of genus at least 2 and the nonorientable surfaces of genus at least 3. We then use these functions to find relations which must hold in the fundamental groups of these surfaces.
Complex Systems interview talk.
Plasmonic structures are made of dielectrics and metals, and at optical frequencies metals exhibit unusual electromagnetic properties like a negative dielectric permittivity whereas dielectrics have a positive one. This change of sign allows the propagation of electromagnetic surface waves at the metal-dielectric interface, where strange phenomena appear if it presents corners. Recently theoretical studies have been carried out, combining results of the T-coercivity approach and the analysis of corner singularities : it has been shown the existence of two states, depending on the problem's parameters. In this presentation we present the theory of T-coercivity and a stable numerical method adapted to each state, with a specific treatment at the corners. In the first state (the solutions are of "classical energy”, H^1) we develop meshing rules adapted to the geometry to guarantee the convergence’s optimality of the approximation with the finite element method. For the second state (the solutions are no longer in H^1), we propose an original numerical method using Perfectly Matched Layers at corners to capture the singularities. These techniques will be applied to a 2D plasmonic scattering problem.
With Calta and Kraaikamp, we define and study $\alpha$-type maps $T_{n,\alpha}$ for each of a countable number of (Fuchsian triangle) groups (thus playing the role of $\text{PSL}_2(\mathbb Z)$); these groups are defined over number fields (whose degree is unbounded). We show that the key property of orbit synchronization of the endpoints of the interval of definition holds for each $n$ on a set of $\alpha$ of full measure, and identify cross-sections for these $T_{n,\alpha}$.
In this second talk, we aim to sketch proofs of some of the main results.
We are happy to announce the second talk in the SIAM PNW Section 2016-17 Seminar Series. See https://sites.google.com/site/siampnwsection/home for more informaiton about the seciton and how to contact the officers.
Who: Prof. Nilima Nigam, Mathematics, Simon Fraser University
When: Thursday, January 26, 3pm (PST)
Where: KEAR 212 (for local attendees)
Title: A MODIFICATION OF SCHIFFER’S CONJECTURE, AND A PROOF VIA FINITE ELEMENTS.
Abstract:
Approximations via conforming and non-conforming finite elements can be used to construct validated and computable bounds on eigenvalues
for the Dirichlet Laplacian in certain domains. If these are to be used as part of a proof, care must be taken to ensure each step of the
computation is validated and verifiable. In this talk we present a long-standing conjecture in spectral geometry, and its resolution
using validated finite element computations. Schiffer’s conjecture states that if a bounded domain Ω in R^n has any nontrivial Neumann
eigenfunction which is a constant on the boundary, then Ω must be a ball. This conjecture is open. A modification of Schiffer’s conjecture
is: for regular polygons of at least 5 sides, we can demonstrate the existence of a Neumann eigenfunction which is
does not change sign on the boundary. In this talk, we provide a recent proof using finite element calculations for the regular
pentagon. The strategy involves iteratively bounding eigenvalues for a sequence of polygonal subdomains of the triangle with mixed Dirichlet
and Neumann boundary conditions. We use a learning algorithm to find and optimize this sequence of subdomains, and use non-
conforming linear FEM to compute validated lower bounds for the lowest eigenvalue in each of these domains. The linear algebra is performed
within interval arithmetic. This is joint work with Bartlomiej Siudeja and Ben Green, at U. Oregon.
Notes: The seminar can be viewed via the online link provided below. However, we strongly encourage local attendees to watch it from
Kear 212. (The first talk in the series in October was quite heavily subscribed and some groups from various universities were unable to watch
due to the limitations on the number of available connections).