Triangle Seminars

Abstract:
We establish a correspondence between generalized quiver gauge theories in four-dimensions and congruence subgroups of the modular group, hinging upon the trivalent graphs which arise in both. The gauge theories and the graphs are enumerated and their numbers are compared. The correspondence is particularly striking for genus zero torsion-free congruence subgroups which are crucial to Moonshine. We analyze in detail the case of index 24, where modular elliptic K3 surfaces emerge: here, the elliptic j-invariants can be recast as dessins d'enfant which dictate the Seiberg-Witten curves.

Abstract:
We generalise a recent combinatorial description of the Verlinde or WZW fusion algebra of type A by defining cylindric Macdonald functions. The latter arise as weighted sums over non-intersecting paths on a square lattice with periodic boundary conditions. Expanding the cylindric Macdonald functions into Schur functions one obtains generalised Kostka-Foulkes polynomials. The latter contain ordinary Kostka-Foulkes polynomials, which appear in algebraic geometry, representation theory and combinatorics, as special case. We further motivate the cylindric Schur functions by showing that they are connected with a commutative Frobenius algebra which can be interpreted as a deformation of the Verlinde algebra: its structure constants are polynomials whose constant terms are the WZW fusion coefficients.

Abstract:
In the first part of the talk I will discuss the meaning of "geometry" in contemporary mathematics and some general developments. In the second part I will focus on a few particular problems which I know more about, such as questions involving topology and geometric structures on manifolds. The talk will in part be built around an imaginary conversation with Hilbert.