Triangle Seminars

Abstract:
Many if not all models of disease transmission on networks can be
linked to the exact state-based Markovian formulation. However the large
number of equations for any system of realistic size limits their
applicability to small populations. As a result, most modelling work relies
on simulation and pairwise models. In this talk, for a simple
SIS dynamics on an arbitrary network, we formalise the link between a well
known pairwise model and the exact Markovian formulation and we formalise
lumping and its direct link to graph automorphism. Lumping is a powerful
technique that exploits graph symmetry and allows to keep the model exact
while considerably reducing the number of equations. Finally, for pairwise
model two different closures are presented, one well established and one
that has been recently proposed. The closed dynamical systems are solved
numerically and the results are compared to output from individual-based
stochastic simulations. This is done for a range of networks
with the same average degree and clustering coefficient but generated using
different algorithms. It is shown that the ability of the pairwise system
to accurately model an epidemic is fundamentally dependent on the underlying
large-scale network structure. We show that the existing pairwise models
work well for certain types of network but have to be used with caution as
higher-order network structures may compromise their effectiveness.

Keywords: network, epidemic, Markov chain, moment closure.

Abstract:
Though defects in a general sense are ubiquitous and much-studied
within statistical mechanics models it is only recently that they have been
considered within integrable field theory. At first sight, defects could be
considered disastrous since the property of integrability might be lost.
However, it turns out that not only is it possible to have 'integrable
defects' but they have a range of interesting properties and cast some new
light on traditional features. Several examples will be described, together
with their properties in classical and quantum versions of the models.

Abstract:
We review a method of the construction of cubic off-shell interaction vertices for Higher Spin fields. This method is based on the BRST approach and is valid both for flat and AdS backgrounds. As a particular illustration of this method we construct an off-shell extension of the vertices which are related to perturbative bosonic string theory. We discus a generalization of this method for higher order vertices and examine whether BCFW recursion relations are applicable for interacting Higher Spin gage theories.

Abstract:
Mumford and Thurston observed that the theory of Jenkins-Strebel differentials can be used to triangulate a compactification of the moduli space of n-pointed Riemann surfaces of given genus. This was later exploited by Kontsevich to prove the Witten conjecture. We review this construction, make a comparison with the Deligne-Mumford compacfication and discuss some of the conjectures in algebraic geometry it led to.

Abstract:
Quantized complex curves play a central role in both topological string
theory and Chern-Simons theory with complexified gauge group. In both cases these
quantum curves yield operators that annihilate partition functions. However, in
both cases, the actual quantization of these curves has only been understood
indirectly (via matrix models on one hand, via recursion relations for the Jones
polynomial on the other). I will discuss an intrinsic, geometric quantization scheme
that should produce such quantum Riemann surfaces directly.

N.B. Such quantum curves also show up in conformal field theory, as the operators
that annihilate correlators with degenerate insertions.