INSTITUTE FOR RESEARCH IN MATHEMATICAL SCIENCES
Program on Calabi-Yau Varieties: Arithmetic, Geometry
Concentrated Graduate Course
Workshop 1 on Modular Forms around String Theory
Fields Institute, Room 230
SCHEDULE FOR GRADUATE COURSE
||Title and Abstract
Nikulin involutions in the context of lattice polarized K3 surfaces
I will review the notion of Nikulin involution on a K3 surface
X. Then, I will discuss a special type of such involution, obtained
from translations by a section of order-two in a Jacobin elliptic fibration.
Introduction to K3 surfaces
I will describe geometric constructions, periods, and moduli
for K3 surfaces, by way of introduction to the more technical lectures
by Thompson, Harder, and Clingher.
Lattice theory and K3 surfaces
Following the publication of the proof of the global Torelli
theorem for K3 surfaces, it became evident that large portions of the
theory of K3 surfaces and their moduli reduce to the theory of a specific
even unimodular lattice of signature (3,19) and its associated orthogonal
group. In this talk, I will discuss some basic lattice theory and outline
how it can be used to prove geometric statements about K3 surfaces.
(I) Arithmetic Techniques in Mirror Symmetry
Mirror pairs of certain Calabi-Yau manifolds defined over
finite fields have their numbers of rational points closely related.
In this talk I will explain p-adic techniques which can be used to count
rational points on such mirror pairs. We will compare the the number
of rational points on a manifold and its mirror modulo p.
(II) Mirror Symmetry for Zeta Functions
As an application of the point-counting techniques from the
previous lecture, I will present some relations
of zeta functions for mirror pairs of Calabi-Yau manifolds defined over
(I) Introduction to modular forms (and their enumerative significance)
We will introduce the notion of a modular form, with a focus
on those forms which arise in an enumerative setting.
(II) Introduction to Gromov-Witten theory
We will outline the motivation and definition of Gromov-Witten
invariants, with a particular focus on the Gromov-Witten theory of P2
and its role in counting plane curves. We will also try to talk about
many of the interesting structures that come naturally from these constructions,
and highlight the role of Calabi-Yau three-folds.
Moduli of K3 surfaces
I will discuss the construction of the moduli space of K3
surfaces and some of its properties, before moving on to talk about
degenerations of K3 surfaces and the compactification problem.
Special Kähler geometry and BCOV holomorphic anomaly equations,
I and II
In this talk, first we will introduce the basics of mirror
symmetry, special K¨ahler geometry and BCOV holomorphic anomaly
equations. We will then construct the special polynomial ring and sketch
how to solve the BCOV anomaly equations using the polynomial recursion
technique, by showing some examples.
Back to top