Spring-13. Math 277 (ccn 54472): Topics in Differential Geometry CANCELLED

Instructor: Alexander Givental
ThTh 11-12:30 p.m. in 5 Evans
Office hours: TuTh 4-6 p.m. in 701 Evans

Riemann-Roch Formulas in Gromov-Witten Theory

Outline: The ultimate goal of the course is to give an exposition of the recent joint work of the instructor with Valentin Tonita "The quantum Riemann-Roch-Hirzebruch theorem in true genus-0 Gromov-Witten theory". The theorem completely characterises genus-0 Gromov-Witten invariants of K-theoretic origin in terms of the usual, cohomological ones. This recent development opens many new, largely unexplored opportunities in enumerative algebraic geometry. On the other hand, the proof (and even formulation) of the theorem is "technology-dependent", that is, relies on a series of previous advances by many authors (e.g. - among many others - Chen-Ruan's Gromov-Witten invariants of orbifolds and Jarvis-Kimura's computation of such for the orbifolds Point/Finite Group).

We plan to begin with an overview of classical Riemann-Roch formulas in homotopic topology and algebraic geometry. Then we will give an introduction to Gromov-Witten invariants and study in detal general properties of such invariants in genus 0. At this point we will have to develop the so-called "symplectic loop space formalism" - a combinatorial framework (replacing the notion of Frobenius structures) suitable for comparing Gromov-Witten invariants of various sorts. Our intermediate goal will be to establish a number of general "twisting" formulas (they typically show that various attempts to invent new GW-invariants are unsuccessful, by expressing those new invariants in terms of the old ones), including the Quantum Hirzebruch-Riemann-Roch formula of Tom Coates, expressing certain GW-invariants with values in complex cobordisms in terms of cohomologica ones. We'll conclude with proving the title theorem, providing some applications (among them - the intrinsic role of finite-difference equations in quantum K-theory), and outlining a number of open problems and directions promising for further research. Prerequisites: For courses of this level, I used to write: A dense subset in Griffitth-Harris' "Principles of Algebraic Geometry", but then somebody added in handwriting: ... in discrete topology. Well, the truth is, that by now I myself have so firmly forgotten those Principles, that saying "a nowhere dense set" (in that topology) would be more appropriate.

Examples and Exercises (PDF)