12 MON

◆ Philip Candelas (Oxford)

◆ Sugwoo Shin (UC Berkeley/KIAS)

- Title: ell-adic Galois representations valued in spin groups

- Abstract: The global Langlands conjecture (made precise by Clozel, Fontaine-Mazur, and Buzzard-Gee) predicts that an automorphic representation of a connected reductive group G (if contributing to cohomology) gives rise to an ell-adic Galois representation valued in the L-group of G with compatible local information. The ell-adic cohomology of Shimura varieties plays a central role in most available constructions. After an overview I will report on some recent progress, in collaboration with Arno Kret, when G is (a form of) GSp(2n) or GSO(2n), in which case its Langlands dual group is the general spin group (a.k.a. special Clifford group) GSpin(2n+1) or GSpin(2n). 

13 TUE

◆ Sarah Harrison (McGill)

- Title: Quantum Modularity and log CFTs from 3 dimensions

◆ John Voight (Dartmouth)

- Title: The Faltings-Serre method, with applications

- Abstract: From the point of view of general algorithmic theory and with a practical point of view, we describe the Faltings-Serre method to show that two ell-adic representations are equivalent.  

14 WED

◆ Tudor Dimofte (UC Davis)

- Title: Higher categorical structures in 3d and 4d gauge theories

- Abstarct: I will discuss aspects of recent work on (categories of) line operators and (two-categories of) boundary conditions in 3d N=4 gauge theories. They sit at the center of a rich network of physics and mathematics, connecting to older topics such as Seiberg-Witten and Rozansky-Witten invariants; and more modern topics such as HOMFLY knot homology, symplectic duality, mathematical constructions of Coulomb branches, and geometric Langlands. My aim is to give an overview of the basic structures involved, and to mention a few of these connections.

◆ Kiran Kedlaya (UCSD)

- Title: Frobenius structures on hypergeometric equations: a computational approach

- Abstract: Hypergeometric equations form an important class of examples of Picard-Fuchs equations; on the arithmetic side, this means that they correspond to families of L-functions. It also means that for each hypergeometric equation, for each prime p (with an explicit finite list of exceptions), there exists a p-adic analytic Frobenius structure on the hypergeometric equation that interpolates the Euler factors at p for all L-functions in the corresponding family. Computing this structure gives an efficient algorithm for computing hypergeometric L-functions particularly as the order of the equation increasess; this requires as input an explicit formula for the specialization at one fiber. For this we use one of the singular fibers, at which the Frobenius action is given by a formula of Dwork generalizing the Gross-Koblitz formula for Gauss sums.

15 THU

◆ Atish Dabholkar (ICTP)

- Title: APS η-invariant, path integrals, and mock modularity

- Abstract: In this talk, I'll describe how the Atiyah-Patodi-Singer η-invariant can be related to the temperature dependent Witten index of a noncompact theory and will give a new proof of the APS theorem using scattering theory. I'll relate the η-invariant to a Callias index and compute it using localization of a supersymmetric path integral and show that the η-invariant for the elliptic genus of a finite cigar is related to quantum modular forms obtained from the completion of a mock Jacobi form which can be  computed from the noncompact path integral. 

◆ Adriana Salerno (Bates)

Title: Hasse-Witt matrices and mirror toric pencils

Abstract: Mirror symmetry predicts unexpected relationships between arithmetic properties of distinct families of algebraic varieties. For example, Wan and others have shown that for some mirror pairs, the number of rational points over a finite field matches modulo the order of the field. In this talk, we obtain a similar result for certain mirror pairs of toric varieties. We use recent results by Huang, Lian, Yau and Yu describing the relationship between the Picard-Fuchs equations and the Hasse-Witt matrix of these varieties, which encapsulates information about the number of points. The result allows us to compute the number of points modulo the order of the field explicitly, and we illustrate this by computing K3 surface examples related to hypergeometric functions. This is joint work with Ursula Whitcher (AMS). 

16 FRI

◆ Albrecht Klemm (Bonn)

-Title: Topological string on elliptic Calabi--Yau 3-folds with N -sections and modular forms

-Abstract: The all genus topological string amplitudes on Calabi--Yau 3-folds are generating functions of the symplectic invariants of holomorphic curves. We argue that the Fourier--Mukai transform on the A-model category as well as the holomorphic anomaly of the B-model restrict these amplitudes to be meromorphic Jacobi forms of À0(N) subgroups of SL(2;Z) . Vanishing conditions and Nakajima's blow up equations allow then to fix these amplitudes explicitly.

◆ Duco van Straten (Mainz)

-Title: Dimensional interpolation: exploring the algebraic geometry of fractional dimension

-Abstract: The subject of interpolation has a long and rich history that includes famously the discovery of the Gamma function. In the talk I will report on current joint work with Golyshev and Zagier on dimensional interpolation and the first step into the algebraic geometry of projective spaces and grassmanians of fractional dimension. In order to formulate an Hirzebruch-Riemann-Roch formalism one is directed to the Gamma-class and quantum cohomology.