报告人：Zhang Ruixiang（University of Wisconsin-Madison）

题  目：The polynomial partitioning approach to the Fourier restriction problem

综合报告时间：2019.07.02（星期二），09:30-10:30
学术报告时间：2019.07.03（星期三），09:30-10:30、14:00-15:00

地  点：数学院南楼N202室

摘要：The Fourier extension operator has been a central object to study in harmonic analysis. Stein conjectured that it is a bounded linear operator between certain $L^p$ spaces. Recently people have found that auxiliary real polynomials can help one study Stein's above Restriction Conjecture. I will talk about this approach known as "polynomial partitioning" and mention some recent progress on Stein's Restriction Conjecture.In the first lecture (the colloquium), I will state Stein's Fourier Restriction (Extension) Conjecture and talk on why and how auxiliary (real) polynomials might help. In the second lecture, I will explain the framework set up by Guth concerning this approach. In the third lecture, I will talk about three interesting facts about the zero sets of real polynomials (Wongkew's theorem, the Polynomial Wolff Axiom, and a lemma of Guth), and how they can help one get better results in the Fourier Restriction Conjecture. These are relevant to some work in progress joint with Jonathan Hickman, Keith Rogers and Hong Wang.

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报告人：Liu Shinan（University. München）

题  目：Local model of Shimura varieties

时  间：2019.07.04（星期四），10:45-11:45

地  点：数学院南楼N202室

摘  要：Let X be a projective smooth algebraic variety over \mathbb{Q}. It is conjectured that the Hasse-Weil zeta function of X has nice properties, such as an analytic continuation to the whole plane, and admitting a functional equation. When X is a Shimura variety, i.e. a moduli space of abelian varieties, the calculation of zeta function is more accessible. Indeed, to calculate the local zeta factor of X at a good prime p, i.e. when X has a smooth model over \mathbb{Z}_p, one only needs to count abelian varieties over finite fields, which can be done by using Honda-Tate theory. However, to calculate the local factor at a bad prime, one also needs to know the nearby cycle sheaf. In this talk we explain how the local model of Shimura varieties allows us to determine the nearby cycle on a Shimura variety, so to calculate its local zeta factor at bad primes. We will use the modular curve Y_0(p) as our guiding example.
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报告人： Liu Shinan（University.München）

题  目：Local model of Hilbert-Siegel varieties, and the equivariant cotangent complex 1 & 2

时  间：2019.07.05（星期五），10:45-11:45、15:15-16:15

地  点：数学院南楼N202室

摘  要：We present our construction on the local model of Hilbert-Siegel varieties in \Gamma_1(p)-level. This generalizes the previous work in the Siegel case by Haines and Stroh. A key tool used by Haines-Stroh is the cotangent complex. To generalize to the Hilbert-Siegel case, one needs to consider the corresponding equivariant cotangent complex. For this, we define a variant over the Zariski site of the ring-equivariant cotangent complex (or more precisely, the Lie complex) constructed by Illusie in his thesis. In the two talks, we will recall Illusie’s theory and present our new definition, then we explain how our definition is used in the construction of the local model.
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报告人：Xiao Jingwei（MIT）

题  目：Automorphic periods and relative trace formula

时  间：2019.07.02（星期二），10:45-11:45

地  点：数学院南楼N202室

摘  要: Automorphic periods are integrals of automorphic forms over a subgroup. Usually, these periods are related to some L values or Langlands functoriality. In this talk, we discuss these ideas in the language of spherical varieties following the conjecture of Sakellaridis and Venkatesh. We then discuss Jacquet’s relative trace formulas approach that is by far the most powerful tool to study these periods.
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报告人：Xiao Jingwei（MIT）

题  目：Germ expansions in the Jacquet-Rallis relative trace formula and applications

时  间：2019.07.03（星期三），10:45-11:45、15:15-16:15

地  点：数学院南楼N202室

摘  要：I will explain my work on the germ expansion identities for the relative trace formula of Jacquet and Rallis. These identities are encoded in the relative trace formula and we make them explicit. I will explain how this leads to a completely different proof of the endoscopic fundamental lemma for unitary groups (work of Laumon and Ngo) and a new relative trace formula for unitary periods (conjectured by Jacquet).
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报告人：Xue Cong（University of Cambridge）

题  目：Cohomology of stacks of shtukas (千爪兽的上同调)

综合报告时间：2019.07.04（星期四），09:30-10:30
学术报告时间：2019.07.05（星期五），09:30-10:30、14:00-15:00

地  点：数学院南楼N202室

摘  要：The l-adic cohomology with compact support of stacks of shtukas is a generalization of the vector space of automorphic forms with compact support over a function field.
In the series of talks, I will recall the definition of the stacks of shtukas and their cohomology groups, and survey the roles they played in the Langlands correspondence for the function field.
I will also construct the constant term morphisms on the cohomology groups. Using these, I will show some finiteness properties of the cohomology of stacks of shtukas and give an application on the Langlands parametrization of some quotient vector spaces of automorphic forms with compact support.