
THE FIELDS
INSTITUTE FOR RESEARCH IN MATHEMATICAL SCIENCES 
Thematic
Program on CalabiYau Varieties: Arithmetic, Geometry
and Physics
September
9–13, 2013
Concentrated Graduate Course
preceeding
the
Workshop 1 on Modular Forms around String Theory
Fields Institute, Room 230


SCHEDULE FOR GRADUATE COURSE
Time

Monday
September 9

Tuesday
September 10

Wednesday September 11

Thursday September 12

Friday
September 13

10:00–11:00






11:15–12:15






Lunch Break

Afternoon Discussion

Speaker 
Title and Abstract 
Clingher, Adrian 
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 ordertwo in a Jacobin elliptic fibration.

Doran, Chuck 
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.

Harder, Andrew 
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.

Perunicic, Andre 
(I) Arithmetic Techniques in Mirror Symmetry
Mirror pairs of certain CalabiYau manifolds defined over
finite fields have their numbers of rational points closely related.
In this talk I will explain padic 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 pointcounting techniques from the
previous lecture, I will present some relations
of zeta functions for mirror pairs of CalabiYau manifolds defined over
finite fields.

Rose, Simon 
(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 GromovWitten theory
We will outline the motivation and definition of GromovWitten
invariants, with a particular focus on the GromovWitten 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 CalabiYau threefolds.

Thompson, Alan 
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.

Zhou, Jie 
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.

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