Abstracts

Week of March 7, 2021

March 8, 2021
Geometry, Symmetry and Physics Higher Siegel-Weil formulas over function fields 4:30pm -
https://yale.zoom.us/j/92811265790 (Password is the same as last semester)

The Siegel-Weil formula relates the integral of a theta function along a classical group H to a special value of a Siegel-Eisenstein series on another group G. Kudla proposed an "arithmetic analogue" of the Siegel-Weil formula, relating intersection numbers of special cycles on Shimura varieties for H to the first derivative at a special value of a Siegel-Eisenstein series on G. We study a function field analogue of this problem in joint work with Zhiwei Yun and Wei Zhang. We define special cycles on moduli stacks of unitary shtukas, construct associated virtual fundamental classes, and relate their degrees to the derivatives to *all* orders of Siegel-Eisenstein series. The results can be seen as “higher derivative” analogues of the Kudla-Rapoport Conjecture. Themes of geometric representation theory play an important role; in particular, a key to the proof is a categorification of local density formulas for Fourier coefficients of Eisenstein series, and a parallel categorification of the degrees of virtual fundamental classes of special cycles, in terms of a global variant of Springer theory.
March 9, 2021
Algebra and Number Theory Seminar Endoscopy for affine Hecke category 8:00pm -
Zoom

Affine Hecke categories are categorifications of Iwahori-Hecke algebras, which are essential in the classification of irreducible representations of loop group LG with Iwahori-fixed vectors. The affine Hecke category has a monodromic counterpart, which contains sheaves with prescribed monodromy under the left and right actions of the maximal torus. We show that the neutral block of this monoidal category is equivalent to the neutral block of the affine Hecke category (with trivial torus monodromy) for the endoscopic group H. It is consistent with the Langlands functoriality conjecture.
March 10, 2021
Applied Mathematics Learning in Graph Neural Networks 1:00pm -
Zoom Meeting ID: 97670014308

Abstract: Graph Neural Networks (GNNs) have become a popular tool for learning representations of graph-structured inputs, with applications in computational chemistry, recommendation, pharmacy, reasoning, and many other areas. In this talk, I will show some recent results on learning with message-passing GNNs. In particular, GNNs possess important invariances and inductive biases that affect learning and generalization. We relate these properties and the choice of the “aggregation function” to predictions within and outside the training distribution.

This talk is based on joint work with Keyulu Xu, Jingling Li, Mozhi Zhang, Simon S. Du, Ken-ichi Kawarabayashi, Vikas Garg and Tommi Jaakkola.

email tatianna.curtis@yale.edu for info.
March 11, 2021
Geometry & Topology Knot concordance and exotica 4:00pm -
https://yale.zoom.us/j/96501374645

One well-known strategy for distinguishing smooth structures on closed 4-manifolds is to produce a knot $K$ in $S^3$ which is (smoothly) slice in one smooth filling $W$ of $S^3$ but not slice in some homeomorphic smooth filling $W’$. There are many techniques for distinguishing smooth structures on complicated closed 4-manifolds, but this strategy stands out for its potential to work for 4-manifolds $W$ with very little algebraic topology. However, this strategy had never actually been used in practice, even for complicated $W$. I’ll discuss joint work with Manolescu and Marengon which gives the first application of this strategy. I’ll also discuss joint work with Manolescu which gives a systematic approach towards using this strategy to produce exotic definite closed 4-manifolds.