# Abstracts

## Charles Doran

Titel: **Calabi-Yau Manifolds, Moduli, and Mirrors**

*Abstract:*

*References:*

Charles F. Doran, Andrew Harder, Alan Thompson, Hodge Numbers from Picard Fuchs equations

Charles F. Doran, Andrew Harder, Andrey Y. Novoseltsev, Alan Thompson, Calabi-Yau threefolds fibred by high rank lattice polarized K3 surfaces

Charles F. Doran, Alan Thompson, Mirror Symmetry for Lattice Polarized del Pezzo surfaces

## Matt Kerr

Titel:**Mixed Hodge theory and mirror symmetry**

*Abstract:*

(1) Asymptotic properties of variations of Hodge structure

I'll
briefly review the A- and B-model origins of VHS, then give an
introduction to limit mixed Hodge structures, extension classes
(including the role of Iritani's Gamma class) and boundary components.

(2) Homogeneity properties of variations of Hodge structure

This
will cover the theory of the Mumford-Tate group/Lie algebra and its
influence on the structure of period maps and boundary components. I'll
include an overview of recent work with Pearlstein and Robles on
several-variable degenerations, with attention to a 2-variable example
arising from a family of CY 3-folds.

(3) Normal functions and LG-models

Motivated
by the 2-variable example, I define admissible normal functions and
discuss where they come from (families of motivic cohomology classes),
and touch on their singularities and limits. I will then discuss several
places where normal functions arise (as "motivic" solutions to
inhomogeneous Picard-Fuchs equations), including local and open mirror
symmetry, Feynman integrals, and Apery constants.

*References:*

Kerr, Matt Algebraic and arithmetic properties of period maps.
Calabi-Yau varieties: arithmetic, geometry and physics, 173–208, Fields
Inst. Monogr., 34, Fields Inst. Res. Math. Sci., Toronto, ON, 2015.

Matt Kerr, Gregory Pearlstein, Colleen Robles, Polarized Relations on Horizontal SL(2)s,

*Prerequisites:*

"some idea of Hodge theory and Lie theory"

## Radu Laza

Titel: **Period maps and moduli***Abstract (provisorial):*

I plan to start with a general lecture on period maps, then I want to discuss essentially what is in my paper with Friedman. Finally, in the third lecture I will talk about my recent preprint with Griffiths et al.

*References:*

Friedman, Robert; Laza, Radu. Semialgebraic horizontal subvarieties of Calabi–Yau type. Duke Math. J. 162 (2013), no. 12, 2077--2148. .

Mark Green, Phillip Griffiths, Radu Laza, Colleen Robles, Completion of Period Mappings and Ampleness of the Hodge Bundle

## Johannes Walcher

Titel: **Arithmetic and modularity of the open topological string***Abstract: *

According to Bershadsky-Cecotti-Ooguri-Vafa, the topological string in the B-model is largely governed by the special geometry on the complex structure moduli space of Calabi-Yau threefolds, whose backbone is the variation of Hodge structure that will be discussed in other lectures at this school. The coupling to an open string sector in the form of (rigid) topological D-branes (of type B) is governed by an extension of both VHS and BCOV holomorphic anomaly. A detailed study of examples reveals that the arithmetic of the situation cannot in general be ignored, and should play an important role in the future. My lectures are built around the motivation, calculations, and open questions.

## Don Zagier

Titel: **Periods of modular forms and differential equations**

*Abstract:*

TBA

*References:*

J. Bruinier, G. Harder and G. van der Geer, D. Zagier The 1-2-3 of Modular Forms: Lectures at a Summer School in Nordfjordeid, Norway (ed. K. Ranestad) Universitext, Springer-Verlag, Berlin-Heidelberg-New York (2008), 280 pages.

W. Kohnen, D. Zagier Modular forms with rational periods in Modular Forms, R.A. Rankin (ed.), Ellis Horwood, Chichechester (1984) 197-249

M. Kontsevich, D. Zagier Periods. In Mathematics Unlimited--2001 and Beyond (B. Engquist and W. Schmid, eds.), Springer, Berlin-Heidelberg-New York (2001), 771-808