In this talk we consider compact holomorphic symplectic manifolds (aka hyperkahler manifolds), particular those fibred by Lagrangian submanifolds. The general fibre must be an abelian variety. Focusing on dimension four, we will describe how to construct examples whose fibres are abelian surfaces that are Jacobians of genus two curves or Prym varieties of double covers of curves. We will also describe some classification results for these kinds of fibrations.

I will give a tour of some places in modern mathematical physics where classical combinatorial objects of discrete math arise. More specifically, I will explore how counting points (of various moduli spaces) and surfaces (holomorphic curves) in symplectic topology can be related to the enumeration of graph colorings (chromatic polynomial), triangulations (Catalan numbers) and quiver structures (clusters).

TBA

Spaces of quantum theories are the fundamental objects in modern mathematical physics. Outside of some basic examples, little is known about the geometry and topology of these spaces. In this talk, I will begin with spaces built from quantum mechanics. We will find that twisted equivariant K-theory encodes the homotopy type of the space of (supersymmetric) quantum mechanical systems. Viewing quantum systems as 1-dimensional quantum field theories, a natural generalization suggests a connection between 2-dimensional (supersymmetric) quantum field theories and twisted equivariant elliptic cohomology. This builds on ideas of Segal, Stolz and Teichner, and is intended as an introduction to their program.

The geometry of propagating strings in Conformal Field Theory (i.e. string theory), naturally gives rise to an algebraic structure called a vertex operator algebra describing the interactions of fields. The graded traces and pseudo-traces of families of linear operators corresponding to components of the fields (vertex operators) of interacting particles exhibit certain invariance properties with respect to action of the modular group SL(2,Z). We discuss aspects of modular symmetries that appear in graded trace and pseudo-trace functions for various classes of algebras of fields (vertex algebras) and their modules.