Mathematics 416-B1, Mirror Symmetry

Instructor: Sheldon Katz
Office and Hours: 231 IH, Wed 10am, Thurs 9am. Phone/email: 265-6258, katz@math.uiuc.edu
Time and location:9am MWF, 441 Altgeld Hall
Text:Mirror Symmetry and Algebraic Geometry, by D.A. Cox and S. Katz.
Prerequisites: Algebraic Geometry (Math 422) or permission of the instructor.
Course Description: This is an introductory course on the algebro-geometric aspects of mirror symmetry, with an emphasis on Gromov-Witten theory.

This subject was inspired by string theory. To certain algebraic varieties X together with attached geometric data, certain string theories can be associated. Mirror symmetry is the assertion that there are other varieties Y with the property that the physical string theories are the same. While this assertion is mathematically imprecise, there are precise mathematical assertions that can be distilled, and many of these can be proven. The most famous result is the computation of the genus 0 Gromov-Witten invariants of the quintic threefold (the ``number'' of rational curves of arbitrary degree). In this sense, much of the subject is now on firm mathematical footing. The course will focus on rigorous mathematical aspects.

Topics include: Toric geometry, complex moduli, Kaehler moduli, Gromov-Witten theory, localization, proof of mirror theorem. If time allows, some speculative explorations on D-branes and open string Gromov-Witten invariants will be included as well.

Some Related Reading Material

Seminars of interest