(Joint with Ronnie Nagloo.) I investigate algebraic relations between sets of solutions (and their derivatives) of the "generic" Painlev\'e equations I-VI, proving a somewhat weaker version of ``there are NO algebraic relations".
After a short introduction to homogeneous relational structures (structures such that all local symmetries are global), I will discuss some different topics relating homogeneity to homomorphisms: a family of notions of 'homomorphism-homogeneity' that generalise homogeneity; generic endomorphisms of homogeneous structures; and constraint satisfaction problems.
In the talk I plan to overview several constructions for finite dimensional represenations of DAHA: construction via quantization of Hilbert scheme of points in the plane (after Gordon, Stafford), construction via quantum Hamiltonian reduction (after Gan, Ginzburg), monodromic construction (after Calaque, Enriquez, Etingof). I will discuss the relations of the constructions to the conjectures from the first lecture.
In this talk we show how the computation of the group of components of Prym varieties of spectral covers leads to cohomological results on the moduli space of stable bundles originally due to Harder-Narasimhan. This is joint work with Christian Pauly.
In this talk I will describe Toda's results on deformations of the category Coh(X) of coherent sheaves on a complex manifold X. They come from deformations of X as a complex manifold, non-commutative deformations, and gerby deformations (which can all be interpreted as deformations of X as a generalized complex manifold). Toda also described how to deform Fourier-Mukai equivalences, and I will present some examples coming from mirror SYZ fibrations.