Sergio García Quiles:15 minutes of Mixed Integer Linear Programming. Broadly speaking, Operational Research consists in formulating and solving optimization problems. And sooner or later we will have to solve a mixed integer linear problem, that is, a problem where the function to optimize and the constraints are linear and some variables are restricted to take integer values. Being everything linear, it would seem that these problems can be solved easily. But this is not usually the case. Why are they difficult then? This talk will try to give a brief insight on this topic showing several examples.
Ben Goddard: Complex, multiscale modelling Using a range of examples from interdisciplinary research, I'll discuss some of the challenges of modelling complex systems with multiple scales, e.g. in time or length. Whilst such problems are routinely tackled in applied sciences, most approaches pay little attention to the underlying mathematical properties or structure. I'll describe some ways in which a more mathematical treatment can significantly improve the quality of these models, often whilst also reducing the computational cost. If time allows, I'll give an example of my approach to such problems, which combines mathematical modelling with rigorous analysis and numerics.
David Jordan: Fundamental groups, quantizations, and topological field theories The character variety of a surface is the space of representations of its fundamental group (into GL_n, say). It was observed by Atiyah thatthe character variety behaves like a classical physical system in twointeresting ways: firstly, it carries a natural "symplectic form" (i.e.it has position and momentum coordinates in a neighborhood of everypoint), and secondly it exhibits "locality": for every decomposition ofthe surface into smaller bits, we get a corresponding decomposition ofthe character variety. The mathematical framework for this type oflocality was also introduced by Atiyah, and Segal, and goes under thename "topological field theory". In physics, the passage from classical to quantum mechanics is called"quantization," and involves replacing the position and momentumcoordinates by non-commuting "quantum observables" obeying Heisenberg'suncertainty principle. I'll describe a mechanism for quantizingcharacter varieties, in such a way that locality is preserved.
Hiro Oh: On nonlinear dispersive Hamiltonian PDEs I will briefly describe central questions on the study of nonlinear dispersive Hamiltonian PDEs, both in the past and in the future, in particular related to analysis and probability.