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September 28 |
Peter Wear (UC San Diego) + pre-talk The Fargues-Fontaine curve is a fundamental object in p-adic Hodge theory. I will mention some of the important properties of the curve, then introduce the rings that are used to build the curve. These are the extended Robba rings; they share many properties with the one dimensional affinoid algebras of rigid analytic geometry. I will discuss these similarities, point out some key differences, and explain how to adapt some proofs from rigid geometry to bypass these differences. The pre-talk will give an overview of some of the basics of rigid analytic geometry. |
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October 5 |
NO MEETING |
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October 12 |
Hang Xue (Univ of Arizona) + pre-talk I will explain the arithmetic analogue of the Gan-Gross-Prasad conjecture for unitary groups. I will also explain how to use arithmetic theta lift to prove certain endoscopic cases of it. |
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October 19 |
Wei Ho (Michigan) + pre-talk We will discuss some geometric techniques used in proving "arithmetic statistics" results, primarily using the case of Selmer groups for families of elliptic curves as a motivating example. |
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October 26 |
Vladislav Petkov (UC San Diego) + pre-talk I will discuss the theory of theta representations for the degree n cover of GL(r) and in particular those distinguished ones that have unique Whittaker models. I will concentrate on the study of the known cuspidal distinguished representations and possible generalizations. Pre-talk: I will introduce the topic of non-integral weight modular forms and give a brief history of their study and the development of the Shimura correspondence. |
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November 2 |
Jeremy Booher (Univ of Arizona) + pre-talk Let G be a smooth group scheme over the p-adic integers with reductive generic fiber. We study the generic fiber of the universal lifting ring of a G-valued mod-p representation of the absolute Galois group of an l-adic field. In particular, we show that it admits an open dense regular locus, and is equidimensional of dimension dim G. This is joint work with Stefan Patrikis. |
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November 9 |
Kiran Kedlaya (UC San Diego) This is a continuation of my RTG colloquium lecture on November 8. In this lecture, we study the method of Birch in more detail, to see how it can be used to compute essentially arbitrary spaces of classical modular forms. This involves relating Birch's construction to orthogonal modular forms and Clifford algebras, and applying a form of the Jacquet-Langlands correspondence. We also report on some limited computational evidence that this method can also be applied to GSp(4) Siegel modular forms. A short computer demonstration using Sage may be included if time permits. Note: this is a report on the PhD thesis of Jeffery Hein, written under John Voight at Dartmouth in consultation with Gonzalo Tornaria. |
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November 16 |
Lauren Ruth (UC Riverside) + pre-talk In the pre-talk for graduate students, we will define discrete series representations and give examples for $SL(2,\mathbb{R})$ and $GL(2,F)$, where $F$ is a local non-archimedean field of characteristic $0$ with residue field of order not divisible by $2$. In the main talk, first, we will review how the multiplicities of discrete series representations of $SL(2,\mathbb{R})$ in $L^2(\Gamma \backslash SL(2,\mathbb{R}))$ are given by dimensions of spaces of holomorphic cusp forms for $\Gamma$; we will take a look at what happens if we try to use the formation of Poincar\'e series as an intertwiner; and we will summarize some of the methods available for guaranteeing occurrence of discrete series representations in $L^2(\Gamma \backslash G)$ when $G$ is a semisimple Lie group other than $SL(2,\mathbb{R})$. Second, we will compute the product of the formal dimension of two particular discrete series representations of $PGL(2,F)$ and the covolume of a torsion-free lattice $\Gamma$ in $PGL(2,F)$ by dealing carefully with Haar measure and applying standard facts from $\mathfrak{p}$-adic representation theory, thereby giving the first explicit computation of multiplicities of those two discrete series representations in $L^2(\Gamma \backslash PGL(2,F) )$; and we will say how the local Jacquet-Langlands correspondence and the work of Corwin, Moy, and Sally could be used to carry out similar calculations. (This material is part of our dissertation on representations of von Neumann algebras coming from lattices in $SL(2,\mathbb{R})$ and $PGL(2,F)$.) |
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November 23 |
NO MEETING (Thanksgiving) |
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November 30 |
Jesse Wolfson (UC Irvine) + pre-talk Resolvent degree is an invariant of a branched cover which quantifies how "hard" is it to specify a point in the cover given a point under it in the base. Historically, this was applied to the branched cover $\mathbb{P}^n/S_{n-1}\to \mathbb{P}^n/S_n$, from the moduli of degree n polynomials with a specified root to the moduli of degree n polynomials. Classical enumerative problems and congruence subgroups provide two additional sources of branched covers to which this invariant applies. In ongoing joint work with Benson Farb, we develop the theory of resolvent degree as an extension of Buhler and Reichstein's theory of essential dimension. We apply this theory to systematize an array of classical results and to relate the complexity of seemingly different problems such as finding roots of polynomials, lines on cubic surfaces, and level structures on intermediate Jacobians. |
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December 7 |
NO MEETING |