### Speaker List

- Matthew Ballard, University of South Carolina
- Charles Doran, University of Alberta
- Helge Ruddat, Johannes Gutenberg-Universität Mainz
- Ryan Keast, Washington University in St. Louis
- John Millson, University of Maryland

### (Tentative) Schedule

** On Friday and Monday ** talks and discussions will be held
in
Math
1311, and there will be
additional space for discussion in Math 1310.

** On Saturday and Sunday ** talks and discussions will be in
Physics ** 1204 **. There will be additional discussion
space in
Physics ** 1303, 1304 and 1219.**

#### Friday, August 21, 2015

##### 9-10a, Math 1310. Coffee and Donuts.

##### 10-11, Math 1311. Ballard (Windows, compactifications, and kernels).

Abstract: During the past few years, an interesting new perspective on establishing equivalences (and more generally semi-orthogonal decompositions) has emerged. A common name for this idea is windows since it allows one to pass through a wall separating one chamber from another in a given moduli problem. More precisely, it allows one to compare derived categories in the adjacent chambers. In this talk, I will introduce windows and give the audience the current lay of the land. Then, I will focus on joint work with Diemer, Favero, Katzarkov, and Kontsevich on a new approach to windows coming from partial compactifications of the diagonal of a groupoid.

##### 11:30am-12:30pm, Math 1311. Doran (Calabi-Yau Manifolds Realizing Symplectically Rigid Monodromy Tuples).

Abstract: We define an iterative construction that produces families of elliptically fibered Calabi-Yau n-folds with section from families of elliptic Calabi-Yau varieties of one dimension lower by a combination of a quadratic twist and a rational base transformation encoded in a generalized functional invariant. Simultaneously, we relate the respective Picard-Fuchs operators and their holomorphic solutions through a generalization of the classical Euler transform for hypergeometric functions. This is achieved by constructing the transcendental cycle that, upon integration with the holomorphic n-form, produces the holomorphic period as the warped product of the corresponding transcendental cycle in dimension n-1 with a Pochhammer contour. In particular, we construct one-parameter families of elliptically fibered Calabi-Yau manifolds with section such that their Picard-Fuchs operators realize all symplectically rigid Calabi-Yau differential operators that were classified by Bogner and Reiter.##### 2p-5p, Math 1311. Discussion.

##### 6-8p, The Common, Marriott Inn and Conference Center. Dinner.

#### Saturday, August 22

##### 9-10, Physics 1303. Coffee and Pastries.

##### 10:00-11, Physics 1204. Ruddat (Mutations and lowest weight periods).

Abstract: I am going to explain how period integrals can be patched together under mutations. A mutation is a certain type of transformation that is used to glue charts in cluster varieties as well as for the reconstruction of Calabi-Yau manifolds from degeneration data. This is part of a joint work with Bernd Siebert where we compute canonical coordinates for Calabi-Yau families by explicitly evaluating period integrals.

##### 11:30-12:30, Physics 1204. Keast (Normal functions over locally symmetric spaces).

Abstract: We classify the irreducible Hermitian real variations of Hodge structure admitting an infinitesimal normal function, and draw conclusions for cycle-class maps on families of abelian varieties with a given Mumford-Tate group.

##### 2-5, Unstructured Collaboration Time.

#### Sunday, August 23

##### 9-10, Physics 1303. Coffee and Pastries.

##### 10-11, Physics 1204. Doran.

##### 11:30-12:30, Physics 1204. Millson (Hodge type theorems for arithmetic manifolds associated to orthogonal and unitary groups).

Abstract: I will speak about two papers with Nicolas Bergeron and Colette Moeglin (and one paper with Bergeron, Zhiyuan Li and Moeglin). For each n between 1 and p, the (standard) arithmetic manifolds M associated to the orthogonal groups SO(p,q) resp. the unitary groups SU(p,q) contain many totally geodesic submanifolds N of codimension nq associated to embedded subgroups SO(p-n,q) resp SU(p-n,q). In the 1980's Steve Kudla and I proved that the cohomology classes dual to such N could be represented by differential forms constructed using the Weil or oscillator representation. More precisely, for the case of SO(p,q), the theory of the oscillator representation provides an integral transform ( "geometric theta lifting") from the space of holomorphic Siegel modular forms for Sp(2n,R) and weight (p+q)/2 to harmonic differential nq-forms on the above manifolds M. Recently using Arthur's work on the Selberg trace formula we proved that the geometric theta lifting was onto. In my talk I will explain the basic differential-geometric principles behind the geometric theta lifting.

The theory of the previous paragraph has the following applications:

- For the case of SO(p,1), p >3, the next-to-top homology group
H
_{p-1}(M) for the standard arithmetic real hyperbolic p-manifolds M is spanned by totally-geodesic hypersurfaces. - For the case of SU(p,1), the Hodge and Tate conjectures hold away from the middle third of cohomological degrees.
- For the case of SO(2,19), the Noether-Lefschetz conjecture of Maulik and Pandharipande holds (Noether-Lefschetz divisors span the Picard group).