NTS ABSTRACTFall2020
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Sep 3
Yifeng Liu 
BeilinsonBloch conjecture and arithmetic inner product formula 
In this talk, we study the Chow group of the motive associated to a tempered global Lpacket $\pi$ of unitary groups of even rank with respect to a CM extension, whose global root number is 1. We show that, under some restrictions on the ramification of $\pi$, if the central derivative $L'(1/2,\pi)$ is nonvanishing, then the $\pi$nearly isotypic localization of the Chow group of a certain unitary Shimura variety over its reflex field does not vanish. This proves part of the BeilinsonBloch conjecture for Chow groups and Lfunctions. Moreover, assuming the modularity of Kudla's generating functions of special cycles, we explicitly construct elements in a certain $\pi$nearly isotypic subspace of the Chow group by arithmetic theta lifting, and compute their heights in terms of the central derivative $L'(1/2,\pi)$ and local doubling zeta integrals. This is a joint work with Chao Li. 
Sep 10
Yufei Zhao 
The joints problem for varieties 
We generalize the GuthKatz joints theorem from lines to varieties. A special case of our result says that N planes (2flats) in 6 dimensions (over any field) have $O(N^{3/2})$ joints, where a joint is a point contained in a triple of these planes not all lying in some hyperplane. Our most general result gives upper bounds, tight up to constant factors, for joints with multiplicities for several sets of varieties of arbitrary dimensions (known as Carbery's conjecture). Our main innovation is a new way to extend the polynomial method to higher dimensional objects. A simple, yet key step in many applications of the polynomial method is the "vanishing lemma": a singlevariable degreed polynomial has at most d zeros. In this talk, I will explain how we generalize the vanishing lemma to multivariable polynomials, for our application to the joints problem. Joint work with Jonathan Tidor and HungHsun Hans Yu (https://arxiv.org/abs/2008.01610) 
Sep 17
Ziquan Yang 
A Crystalline Torelli Theorem for Supersingular K3^[n]type Varieties 
In 1983, Ogus proved that a supersingular K3 surface in characteristic at least 5 is determined up to isomorphism by the Frobenius action and the Poincaré pairing on its second crystalline cohomology. This is an analogue of the classical Torelli theorem for K3's, due to Shapiro and Shafarevich, which says that a complex algebraic K3 surface is determined up to isomorphism by the Hodge structure and the Poinaré pairing on its second singular cohomology. I will explain how to reinterpret Ogus' theorem from a motivic point of view and generalize the stronger form of the theorem to a class of higher dimensional analogues of K3 surfaces, called K3^[n]type varieties. This is also an analogue of Verbitsky's global Torelli theorem for general irreducible symplectic manifolds. A new feature in Verbitsky's theorem, which did not appear in the classical Torelli theorem for K3's, is the notion of "parallel transport operators". I will explain how to work with this notion in an arithmetic setting. As an application, I will also present a similar crystalline Torelli theorem for supersingular cubic fourfolds, the Hodge theoretic counterpart of which is a theorem of Voisin.

Sep 24
Yousheng Shi 
Kudla Rapoport conjecture over the ramified primes 
In the nineties, Kudla formulated a conjecture relating central derivative of certain Eisenstein series to arithmetic intersection numbers of special cycles on Shimura varieties. Later Kudla and Rapoport formulated a local version of the conjecture which compares intersection numbers of special cycles on the unitary Rapoport Zink spaces over an inert prime of an imaginary quadratic field with derivatives of local density of hermitian forms. In this talk, I will review KudlaRapoport conjecture and its global motivation. Then I will talk about an attempt to formulate a KudlaRapoport type of conjecture over the ramified primes. In case of unitary Shimura curves, this new conjecture can be proved. This is a joint work with Qiao He and Tonghai Yang. 