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**Discrete Geometry and Combinatorics Seminar**- Previous Discrete Geometry and Combinatorics Seminars
- Discrete Geometry and Combinatorics Seminar Autumn 2015
- Discrete Geometry and Combinatorics Seminar Summer 2015
- Discrete Geometry and Combinatorics Seminar Spring 2015
- Discrete Geometry and Combinatorics Seminar Autumn 2014
- Discrete Geometry and Combinatorics Seminar Summer 2014
- Discrete Geometry and Combinatorics Seminar Spring 2014
- Discrete Geometry and Combinatorics Seminar Autumn 2013
- Discrete Geometry and Combinatorics Seminar Spring 2013
- Discrete Geometry and Combinatorics Seminar Autumn 2012
- Discrete Geometry and Combinatorics Seminar Spring 2012
- Discrete Geometry and Combinatorics Seminar Autumn 2011
- Discrete Geometry and Combinatorics Seminar Spring 2011
- Discrete Geometry and Combinatorics Seminar Autumn 2010
- Discrete Geometry and Combinatorics Seminar Spring 2010
- Discrete Geometry and Combinatorics Seminar Autumn 2009
- Discrete Geometry and Combinatorics Seminar Spring 2009

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## Discrete Geometry and Combinatorics Seminar

### Spring 2016

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).

Page last modified on 08 mar 16 16:38