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All seminars (unless otherwise stated) will take place on Tuesdays at 5.00pm in Room 707 in the Mathematics Department (25 Gordon Street). There will be tea afterwards in Room 606 in the Mathematics Department (25 Gordon Street) - see how to find us for further details. If you require any more information on the Discrete Geometry and Combinatorics seminars please contact Dr John Talbot e-mail: j.talbot AT ucl.ac.uk or tel: 020-7679-4102.
23 February 2016
Charles Johnson (William and Mary College, USA)
Title: Eigenvalues, Multiplicities and Graphs
Abstract:
Let G be a simple undirected graph on n vertices, and denote by S(G) the set of all real symmetric (or complex Hermitian) matrices, the graph of whose off-diagonal entries is G. G places no restriction on the diagonal entries of A in S(G), other than the necessary reality. We survey recent results about the possible multipliciities of the eigenvalues occurring among matrices in S(G). Though we consider all graphs, an especially interesting case is that in which G is a tree. There are especially strong results here that demonstrably do not occur in general. We discuss maximum multiplicity, the minimum number of distinct eigenvalues, and vertices, the removal of which increases the multiplicity of an eigenvalue. A great deal is known about the lists for trees, though a complete answer is not yet known. Several features of trees (such as path cover number, linear trees, etc) arise and are important, though they have not been studied previously graph-theoretically.
8 March 2016
Mihalis Kolountzakis (University of Crete)
Title: Periodicity for tilings and spectra
Abstract:
We will talk about periodicity (and structure, more generally) in the study of tilings by translation, where the tile is a set or a function in an Abelian group, and also in the study of spectra of sets (sets of characters which form an orthogonal basis for the functions defined on the set). There are connections to harmonic analysis, number theory, combinatorics and computation, and these make this subject so fascinating. Starting from the Fuglede conjecture, now disproved in dimension at least 3, which would connect tilings with spectra, we will go over cases where periodicity always holds, cases where it is optional and cases where it's never true, in one dimension (most positive results) and higher dimension (most interesting questions).