PureMathematicsAutumn2013
Autumn 2013
3th September Ralf Rueckriemen (Royal Holloway) Trace formulae and heat kernel asymptotics on quantum graphs (extra seminar at 2pm Tuesday 3th September in McCrea 229)
26th September Henning Stichtenoth (Sabanci University) What is the maximal number of points on a curve over IF_l? (extra seminar from 2-3pm Thursday 26th September in McCrea 219)
26th September Alev Topuzoglu (Sabanci University) Permutations of finite fields: A new classification (extra seminar from 3.30-4.30pm Thursday 26th September in McCrea 219)
1st October Ben Fairbairn (Birckbeck) Beauville Surface, Structures and Groups
8th October Eugenio Gianelli (Royal Holloway) On the decomposition matrix of the symmetric group
15th October Tim Browning (Bristol) How frequently
does the Hasse principle fail?
22nd October Jens Bolte (Royal Holloway) Quantum graphs
5th November Peter Kropholler (Southampton) Cohomology of finite rank soluble groups with group theoretic and homological applications
12th November Aditi Kar (Oxford) Rank and deficiency gradient of groups
19th November Ben Martin (Auckland) Complete reducibility for reductive algebraic groups.
26th November Andrei Gagarin (Royal Holloway) The bondage number of graphs on topological surfaces and Teschner's conjecture
3rd December no seminar
10th December Sebastian Egger (Royal Holloway) The small-$t$ asymptotics of the trace of the heat-kernel for arbitrary quantum graphs
Abstract:
Ralf Rueckriemen: I will
introduce quantum graphs and talk about the tools used to study them. I
will first show a trace formula and explain where it comes from and then
present a method to study the heat kernel asymptotics of a quantum
graph.
Ben Fairbairn: Beauville
surfaces are a class of complex surfaces that were first defined by Catanese
about ten years ago and have numerous nice geometric properties. What makes
them particularly good to work with is the fact that their definition can be
translated into entirely group theoretic language. This raises all sort of
questions. Which groups can be used? If a given group can be used what sort of
surfaces can it define? Can other nice geometric properties can be translated
across? In this overview we shall discuss these matters and more. En route we
will meet numerous open questions, problems and conjectures.
Eugenio Gianelli: In this talk I
will give an introduction to the main ideas in the representation theory of the
symmetric groups. Then I will describe the long standing open problem of
finding the so called decomposition matrix of S_n. In the last part
of the talk I will present a new result describing a number of columns of the decomposition matrix in odd prime characteristic.
Tim Browning:
The Hasse
principle, when it holds, gives an algorithm for checking whether a Diophantine
equation has a solution in rationals. It does not hold in general, however,
and counter-examples to the Hasse principle have been known since
the 1970s. We give a snapshot of what is known and discuss how often such
failures arise for some special families.
This is joint
work with Regis de la Breteche.
Jens Bolte: Quantum graphs
are differential operators acting on functions defined on the edges of a
metric graph. I will explain the motivation why one is interested in
these objects
(avoiding physics jargon as much as possible), introduce basic concepts and survey the
main results. In the second part I will describe more recent work on many-particle quantum systems on graphs.
Peter Kropholler: Soluble
groups entered mathematics at the time of Galois because of the
connection between group theory and the problem of solving polynomials
in one equation by radicals. Thus, from a historical perspective, one
could say that soluble groups go right back to the entrance of group
theory onto the mathematical stage. They remain a great testing ground
for theoretical questions throughout group theory and in this talk I
propose to look at some questions about finiteness conditions motivated
not by solving polynomial equations but by certain kinds of geometric
construction, especially spaces like the fundamental group which provide
a connection with topology and homotopy theory. Homology and cohomology
are the natural algebraic theories associated with this territory and
in this talk we will look at some of the features of soluble groups in
this context.
Aditi Kar:I
will survey recent developments in the study of rank gradient and
deficiency gradient in group theory. Rank gradient arose in the study of
3-manifold groups and later, was found to have deep connections with
areas of mathematics like group cohomology and topological dynamics.
Deficiency gradient is a more recent notion but is already at the centre
of some frantic research. I will attempt to give an overview of the
current state of the research and list some interesting questions.
Ben Martin: Let G be a reductive algebraic group over a field k of positive characteristic. The notion of a completely reducible subgroup of G generalises the notion of a completely reducible representation (which is the special case when G=GL_n(k)). I will describe a geometric approach to the theory of complete reducibility, based on ideas of R.W. Richardson, and I will discuss some recent work involving non-algebraically closed fields.
Andrei Gagarin: The
domination number of a graph is the smallest number of its vertices
adjacent to all the other vertices. The bondage number of a graph is the
smallest number of its edges whose removal results in a graph having a
larger domination number. In a sense, the bondage number measures
integrity and reliability of the smallest dominating sets with respect
to edge removals, which may correspond, e.g., to link failures in
communication networks. The decision problem for the bondage number is
known to be NP-hard. We
provide constant upper bounds for the bondage number of graphs on
topological surfaces, and improve upper bounds for the bondage number in
terms of the maximum vertex degree and the orientable and
non-orientable graph genera. Also, we present stronger upper bounds for
graphs with the number of vertices larger than a certain threshold in
terms of graph genera. This settles Teschner's Conjecture in affirmative
for almost all graphs.
This is joint work with Vadim Zverovich, University of the West of England, Bristol, UK.
Sebastian Egger: The
small $t$-asymptotics of heat-kernel is a widely studied object in
mathematical physics. Its asymptotic expansion contains information
about the spectrum of the underlying Schrödinger operator and is an
important approach for its spectral analysis. I will generalize the
known results for the small-$t$ heat-kernel asymptotics of compact
quantum graphs with non-Robin boundary conditions to non-compact quantum
graphs with Robin boundary conditions.