Abstract: Consider the following three properties of a general group G:
(1) Algebra: G is abelian and torsion-free.
(2) Analysis: G is a metric space that admits a "norm", namely, a translation-invariant metric d(.,.) satisfying: d(1,g^n) = |n| d(1,g) for all g in G and integers n.
(3) Geometry: G admits a length function with "saturated" subadditivity for equal arguments: l(g^2) = 2 l(g) for all g in G.
While these properties may a priori seem different, in fact they turn out to be equivalent. The nontrivial implication amounts to saying that there does not exist a non-abelian group with a "norm".
We will discuss motivations from analysis, probability, and geometry; then the proof of the above equivalences; and finally, the logistics of how the problem was solved, via a PolyMath project that began on a blogpost of Terence Tao.
(Joint work - as D.H.J. PolyMath - with Tobias Fritz, Siddhartha Gadgil, Pace Nielsen, Lior Silberman, and Terence Tao.)
Abstract: We discuss analogues of 3 important theorems about complex tori, C^n/L, L a lattice, for noncommutative complex tori (which we will define). The 3 basic theorems are the Riemann-Roch Theorem, the Hodge Theorem, and the characterization of when a complex torus is an abelian variety. This is joint work with V. Mathai.
Abstract: Suppose a finite group G acts faithfully on an irreducible variety X. We say that the G-variety X is compressible if there is a dominant rational morphism from X to a faithful G-variety Y of strictly smaller dimension. Otherwise we say that X is incompressible. In a recent preprint, Farb, Kisin and Wolfson (FKW) have proved the incompressibility of a large class of covers related to the moduli space of principally polarized abelian varieties with level structure. Their methods, which rely on the existence of integral models for the moduli space Ag, apply to diverse examples such as moduli spaces of curves and many Shimura varieties of Hodge type. My talk will be about joint work with Fakhruddin and Reichstein, where our goal is to recover some of the results of FKW via the fixed point method from the theory of essential dimension. More specifically, we prove incompressibilty for some Shimura varieties by finding fixed points of finite abelian subgroups of G in their toroidal compactifications. Our results are weaker than the results of FKW for Hodge type Shimura varieties, because the methods of FKW apply in cases where there is no boundary, while we need the boundary to find the fixed points. However, our method has the advantage of extending to many Shimura varieties which are not of Hodge type, in particular, those associated to groups of type E7. Moreover, by using Pink's extension of the Ash, Mumford, Rapoport and Tai theory of toroidal compactifications to mixed Shimura varieties, we are able to prove incompressibility for congruence covers corresponding to certain universal famiiles: e.g., the universal families of principally polarized abelian varieties.
Abstract: We define a class of amenable Weyl group elements in the Lie types B, C, and D, which we propose as the analogues of vexillary permutations in these Lie types. Our amenable signed permutations index flagged theta and eta polynomials, which generalize the double theta and eta polynomials of Wilson and the speaker. In geometry, we obtain corresponding formulas for the cohomology classes of symplectic and orthogonal degeneracy loci.
Abstract: I'll explain recent joint work with Lance Gurney. We prove that any family of ordinary abelian varieties parameterised by p-adic formal scheme S lifts to a unique family over W(S) which admits a delta-structure in the sense of Joyal, Buium, and Bousfield. In the case where S is the spectrum of a perfect field of characteristic p, this specialises to the classical result of Serre-Tate and Messing that every ordinary abelian variety over a perfect field k lifts to a unique one over the ring W(k) of Witt vectors together with a lift of Frobenius.
Abstract: Understanding the structure of Gromov-Witten invariants of Calabi-Yau threefolds is an important problem in enumerative geometry which has been studied since the early 90s. In my talk, I will concentrate on quintic Calabi-Yau threefolds, review the algebraic geometry behind the (now standard) computation of the genus zero invariants, and explain why it cannot be easily extended to higher genus. I will then proceed to discuss a construction (joint with Q. Chen and Y. Ruan) of new moduli spaces that can control the failure of the naive approach. In joint work with S. Guo and Y. Ruan, we use them to prove conjectures from physics about higher genus Gromov-Witten invariants of quintic threefolds, such as the "holomorphic anomaly equations".
Abstract: Lagrangian fibrations play a crucial role in the study of hyper-Kaehler geometry and integrable systems. The P=W conjecture by de Cataldo, Hausel, and Migliorini suggests a surprising connection between the topology of Lagrangian fibrations and Hodge theory. In this talk, we will first discuss a compact version of this phenomenon, based on joint work with Andrew Harder, Zhiyuan Li, and Qizheng Yin. Then we will focus on interactions between compact and noncompact hyper-Kaehler geometry. Such connections lead to new progress on the P=W conjecture for Hitchin systems and character varieties. This is joint work with Mark de Cataldo and Davesh Maulik. If time permits, I will further discuss the connection between P=W and Gopakumar-Vafa invariants for local curves.
Abstract: We define and compute ``analytic'' intersection numbers of quadratic CM-cycles on Lubin-Tate (LT) space at infinite level. This is based on the formalism of tropical (p,q)-forms by Gubler-KÃ¼nnemann and the description of the infinitel level LT-space by Scholze-Weinstein. The intersection problem itself plays a role in the linear Arithmetic Fundamental Lemma conjecture of W. Zhang. Our approach is motivated by a recent result of Q. Li who gave a formula for the corresponding intersection numbers on formal models. A posteriori, we see that our analytically defined numbers coincide the ones from formal models.
Abstract: Given a virtually smooth quasi-projective scheme M, and a morphism from M to a nonsingular quasi-projective variety B, we show it is possible to find an affine bundle M' over M that admits a perfect obstruction theory relative to B. We study the resulting virtual cycles on the fibers of M' over B and relate them to the image of the virtual cycle [M]^vir under refined Gysin homomorphisms. Our main application is when M is a moduli space of stable codimension 1 sheaves on a nonsingular projective surface or Fano threefold.
Abstract: In the second lecture, I want to discuss the theory of non-archimedean integration on the Hitchin fibration due to Groechenig, Wyss and Ziegler. Surprisingly, calculating nonarchimedean integrals is not exactly the same as counting points and this approach gives another proof of the fundamental lemma, and this discrepancy sheds yet new lights on the theory of endoscopy. The proof is also more elementary in the sense that it does not use the theory of perverse sheaves.
Abstract: In my third lecture, I want to report on a completely different development on the moduli space of Higgs bundles. In joint work with T.H. Chen we started exploring the structure of the Hitchin map for the moduli space of Higgs bundles over higher-dimensional varieties, which raises interesting questions on the geometry of commuting varieties.
Abstract: The Langlands program predicts a deep relationship between the world of automorphic representations and that of Galois representations. Understanding this relationship in special cases has led to proofs of many deep theorems in number theory. A central role in the program is played by certain algebraic varieties called Shimura varieties; in special cases, they arise as moduli spaces of abelian varieties with extra structure. In this talk, I will explain a recent joint work with Kisin, using Shimura varieties to prove a result on the independence of l for Frobenius conjugacy classes attached to abelian varieties.