Talk Schedule
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September 2: Chun-Hsien Hsu, University of Chicago
- Title: Braverman-Kazhdan-Ngô program over non-Archimedean local fields
- Abstract:
The Poisson summation conjecture, initiated by Braverman and Kazhdan, later refined by Lafforgue, Ngô, and many others, and broadened by Sakellaridis to the setting of affine spherical varieties, predicts Poisson summation formulae on affine spherical varieties (and beyond) that generalize classical harmonic analysis on vector spaces. It provides a uniform framework for establishing analytic properties of L-functions and, ultimately, for establishing Langlands functoriality in great generality. In the setting of L-monoids, it is often referred to as the Braverman–Kazhdan–Ngô program.
Recent work of Getz et al. explains how Fourier transforms for L-monoids can be constructed using the Harish-Chandra Plancherel formulae, assuming enough of tempered local Langlands correspondence. In joint work with HaoYun Yao, we construct the Schwartz spaces over non-Archimedean local fields under the same assumption. With an additional assumption on \(\gamma\)-factors, we also establish the Tate–Godement–Jacquet theory in this setting. Our approach is spectral in nature. In this talk, I will explain how Schwartz spaces and Fourier transforms can be constructed spectrally for general linear groups. If time permits, I will sketch a proof of the Tate–Godement–Jacquet theory for \(\mathrm{GL}_2\).
(Warmup talk for graduate students at 11:00pm-11:45pm in Krieger 204.)
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September 9: Pranjal Warade, Purdue University
- Title: Beyond Endoscopy for \(\mathrm{GL}_2\) : The \(k\)-th Symmetric Power Trace Formula
- Abstract:
The Beyond Endoscopy program of Langlands proposes to use trace formulas to detect functorial transfers through the analytic behavior of automorphic \(L\)-functions. In this talk, we study the Beyond Endoscopy trace formula for \(\mathrm{GL}_2/\mathbb{Q}\) associated with the \(k\)-th symmetric power representation \(\sigma_k\) of the dual group \(\mathrm{GL}_2(\mathbb{C})\), for arbitrary \(k\). Following a suggestion of Sarnak, the corresponding \(L\)-functions \(L(s_B, \pi, \sigma_k)\) are incorporated directly into the trace formula through suitable basic functions at the finite places. We use the Jacquet−Zagier approach of integrating the cuspidal kernel against an Eisenstein series and recovering the trace formula as a residue at \(s=1\), replacing the usual truncation by a continuous deformation. We focus on the elliptic contribution. After Poisson summation in the trace variable, we study its dominant term, corresponding to the zero dual variable \(\xi=0\). Our main result is that this term admits a meromorphic continuation to \(\mathrm{Re}(s_B) \geq 0\), with a pole of order \(k\) at \(s_B=1\). Following Frenkel−Langlands−Ngô and Altûg, the dominant term is expected to isolate the contribution of the trivial representation.
A major analytical difficulty in carrying out Poisson summation comes from singularities of the archimedean orbital integrals at central elements. We address this using theta series associated with the Zagier \(L\)-functions and their functional equations. For \(k=1\), our result recovers Altûg's analysis for the standard representation; more generally, the pole is expected to cancel against the other geometric contributions.
(Warmup talk for graduate students at 11:00pm-11:45pm in Krieger 204.)
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September 16: Luis Garcia, University College London
- Title: Vertex algebras and the Noether-Lefschetz locus
- Abstract:
Consider a family of complex smooth projective surfaces parametrized by a quasi-projective variety. How many fibers contain curves that do not deform to the whole family? Remarkably, in certain instances these counts turn out to be the coefficients of a modular form. Despite much recent progress, this fascinating phenomenon remains poorly understood. However, a hint comes from the fact that in several ways these generating series are reminiscent of the partition functions of vertex algebras, a mathematical structure that formalizes the notion of a chiral conformal field theory and which naturally gives rise to modular forms. I will explain that these modularity phenomena are indeed related, by constructing a vertex algebra associated with an appropriate variation of Hodge structure. These terms will be defined during the talk and no previous knowledge of vertex algebras of variations of Hodge structure should be necessary to follow it.
(Warmup talk for graduate students at 11:00pm-11:45pm in Krieger 204.)
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September 30: Salim Tayou, Dartmouth College
- Title: Modularity of special cycles in orthogonal and unitary Shimura varieties
- Abstract:
Since the work of Jacobi and Siegel, it is well known that Theta series of quadratic lattices produce modular forms. In a vast generalization, Kudla and Millson have proved that the generating series of special cycles in orthogonal and unitary Shimura varieties are modular forms. In this talk, I will explain an extension of these results to toroidal compactifications where we prove that, when these cycles are corrected by certain boundary cycles, the resulting generating series is still a modular form in the case of divisors in orthogonal Shimura varieties and cycles of codimension up to the middle degree in the cohomology of unitary Shimura varieties, thereby partially answering a conjecture of Kudla.
The results of this talk are joint work with Philip Engel and François Greer, and joint work with François Greer.
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October 7: Yan Pan, University of Arizona
- Title: The Gan-Gross-Prasad conjecture for GSpin groups
- Abstract:
The Gan-Gross-Prasad (GGP) conjecture relates the non-vanishing of certain automorphic period integrals to the non-vanishing of central values of L-functions. We will discuss recent progress in certain cases of the global GGP conjecture for GSpin groups. Our approach is based on two new constructions: a family of Rankin-Selberg integrals for the period-to-L-value direction and a twisted automorphic descent construction for the \(L\)-value-to-period direction.
Warmup talk for graduate students at 11:00pm-11:45pm in Krieger 204.
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October 14: Guy Shtotland , Ben-Gurion University
- Title: Distinction of the Steinberg representation with respect to symmetric subgroups
- Abstract:
We study the distinction of the Steinberg representation of a split \(p\)-adic group \(G\) with respect to a split symmetric subgroup \(H\). We relate this problem to the problem of determining the existence of a non zero harmonic function on a certain hyper graph related to \(X=G/H\). We show that the Steinberg representation is distinguished only if \(X\) is quasi-split and that \(\mathrm{GL}_{2n+1}/\mathrm{GL}_{n}\times \mathrm{GL}_{n+1}\) is essentially the only example where \(X\) is quasi-split but the Steinberg representation is not distinguished. As a corollary we obtain a case of relative local Langlands correspondence, we show that the Steinberg representation is distinguished if and only if its parameter factors through the dual group of \(X\).
Warmup talk for graduate students at 11:00pm-11:45pm in Krieger 204.
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October 21: Marco Sangiovanni, Columbia University
- Title: TBA
- Abstract: TBA
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November 4: Yingkun Li, Institute for Advanced Study
- Title: TBA
- Abstract: TBA
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November 11: Yu Luo
, MIT
- Title: TBA
- Abstract: TBA
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November 18: Manami Roy, Lafayette College
- Title: TBA
- Abstract: TBA
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December 2: David Roe , MIT
- Title: TBA
- Abstract: TBA