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
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.
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