I'll give a brief survey of what is known about the density of rational points on definable sets in o-minimal expansions of the real field, then discuss improving these results in certain cases.
The main aim is to incorporate the nonlinear term into non-Markovian Master equations for a continuous time random walk (CTRW) with non-exponential waiting time distributions. We derive new nonlinear evolution equations for the mesoscopic density of reacting particles corresponding to CTRW with arbitrary jump and waiting time distributions. We apply these equations to the problem of front propagation in the reaction-transport systems of KPP-type.
We find an explicit expression for the speed of a propagating front in the case of subdiffusive transport.
I show that the expansion of the real field by a total Pfaffian chain is model complete in a language with symbols for the functions in the chain, the exponential and all real constants. In particular, the expansion of the reals by all total Pfaffian functions is model complete.
Cherednik algebras (always of type A in this talk) are an intriguing class of algebras that have been used to answer questions in a range of different areas, including integrable systems, combinatorics and the (non)existence of crepant resolutions. A couple of years ago Iain Gordon and I proved that they form a non-commutative deformation of the Hilbert scheme of points in the plane. This can be used to obtain detailed information about the representation theory of these algebras.
In the first part of the talk I will survey some of these results. In the second part of the talk I will discuss recent work with Gordon and Victor Ginzburg. This shows that the approach of Gordon and myself is closely related to Gan and Ginzburg's quantum Hamiltonian reduction. This again has applications to representation theory; for example it can be used to prove the equidimensionality of characteristic varieties.