Abstract

We will overview some recent developments in enumerative geometry with coefficients in a field of positive characteristic. One goal of this minicourse is to demonstrate how characteristic p techniques can be useful in solving problems in symplectic topology that a priori have no mention of finite fields. The key player in this story is the quantum Steenrod operation, and potential topics include: construction and basic properties of quantum Steenrod operations; relation to the quantum differential equations in characteristic p;  applications of quantum Steenrod operations to Hamiltonian dynamics (e.g. pseudorotations) and/or Gromov-Witten theory (e.g. exponential type conjecture); if time permits, I will also discuss a generalization of quantum Steenrod operations in the context of non-commutative geometry, and the potential role it plays in (arithmetic) mirror symmetry. Notes I II 

We will overview some results from the theory of (mostly genus 0) K-theoretic Gromov-Witten invariants, such as their adelic characterization in terms of cohomological invariants, fixed point localization techniques, reconstruction results via finite-difference operators in Novikov's variables, and the role of q-hypergeometric functions in the K-theoretic version of mirror symmetry. Notes

In the first lecture I will begin by describing classical work of Atiyah-Bott and Kirwan on the cohomology of Hamiltonian G-manifolds and their symplectic quotients. I will then explain refinements of their results to complex-oriented cohomology theories, including integral cohomology. In the second lecture, I will then turn to discussing the quantum GIT conjecture, which predicts a formula for the quantum cohomology of GIT quotients X//G in terms of the equivariant quantum cohomology of X. The formula is motivated by ideas from 3-dimensional gauge theory ("Coulomb branches") and provides a vast generalization of Batyrev's formula for the quantum cohomology 

of a toric Fano variety. I will explain our work with C. Teleman proving this conjecture for certain "anti-canonical" quotients. In the final lecture, I will discuss possible extensions of these ideas to generalized cohomology or non-Fano reductions. Based on joint work with C. Teleman. Notes I II 

I will review the role of the Toda integrable system, and of the Fourier transform between equivariant and Seidel coordinates in equivariant quantum cohomology, with an eye to clarifying established results, ongoing projects and some open conjectures in the subject.