Focused Research Group on Hodge Theory, Moduli and Representation Theory: Workshop 4

Speaker List

(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:

  1. For the case of SO(p,1), p >3, the next-to-top homology group Hp-1(M) for the standard arithmetic real hyperbolic p-manifolds M is spanned by totally-geodesic hypersurfaces.
  2. For the case of SU(p,1), the Hodge and Tate conjectures hold away from the middle third of cohomological degrees.
  3. For the case of SO(2,19), the Noether-Lefschetz conjecture of Maulik and Pandharipande holds (Noether-Lefschetz divisors span the Picard group).
2p-4p, Physics 1204. Discussion.

Monday, August 24

9-10, Math1310. Coffee and Donuts.
10-11, Math 1311. Ruddat (Mutations and lowest weight periods 2).