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Abstracts

Charles Doran

Matt Kerr

Radu Laza

Johannes Walcher

Don Zagier

Charles Doran

Titel: Calabi-Yau Manifolds, Moduli, and Mirrors

Abstract:

Lecture 1:  We will introduce the main themes of construction and classification of Calabi-Yau manifolds, building up our understanding in dimensions one (elliptic curves), two (K3 surfaces), and beyond (especially CY3-folds).  The role of "mirror symmetry as gadfly" is established.
 
Lecture 2:  We will study families of and fibrations on Calabi-Yau manifolds, and use a mix of Hodge theory and geometry to construct and classify them.  Moduli spaces of Calabi-Yau manifolds emerge naturally "from the inside out".
 
Lecture 3:  We will motivate, state, and describe evidence for the DHT mirror conjecture mirroring internal (fibration) and external (degeneration) structures on Calabi-Yau manifolds.  This unifies the a priori unrelated Fano/Landau-Ginzburg and Calabi-Yau mirror proposals.
 

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


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