NTS Spring 2012/Abstracts: Difference between revisions

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== November 8 ==
== March 29 ==


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| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Christelle Vincent''' (Madison)
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''David P. Roberts''' (U. Minnesota Morris)
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| bgcolor="#BCD2EE"  align="center" | Title: On the construction of rational points on elliptic curves
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Abstract: If E is an elliptic curve defined over a number field K, Mordell's Theorem asserts that the group E(K) of points of E defined over the field K is abelian and finitely generated. While the torsion part of this group is considered to be well-understood, the rank of the infinite part is very difficult to compute in general. In an effort to understand this quantity better, Darmon has proposed a conjectural construction of so-called Stark-Heegner points. We will begin by presenting a prototypical example of such a construction found in the work of Gross and Zagier, which will be accessible to undergraduate students. Building on this framework, we will explain how this can be generalized to construct points on a larger class of elliptic curves.
Abstract: tba


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== November 10 ==
== April 12 ==


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| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Luanlei Zhao''' (Madison)
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Chenyan Wu''' (Minnesota)
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| bgcolor="#BCD2EE"  align="center" | Title: Integral of Borcherds forms of orthogonal type
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Abstract: Following S. Kudla, J. Bruinier and T. Yang, we compute the integral of an automorphic Green functions coming from vector valued harmonic Maass forms for the dual pair (O(''n'');&nbsp;Sp(1)) over the negative 2-planes with signature (''r'',&nbsp;2) for
Abstract: tba
0&nbsp;<&nbsp;''r''&nbsp;<&nbsp;''n'', which is of interest in Arakelov geometry. We connect this integral of Borcherds form with the derivative of a Rankin–Selberg ''L''-function.


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== November 17 ==
== April 19 ==


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| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Robert Harron''' (Madison)
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Robert Guralnick''' (U. Southern California)
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| bgcolor="#BCD2EE"  align="center" | Title: ''L''-invariants of symmetric powers of modular forms
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Abstract: A fruitful way to study the arithmetic significance of special values of ''L''-functions is via their interpolation by ''p''-adic ''L''-functions. In this talk, we will discuss the phenomenon of ''L''-invariants, which arise when the interpolation property provides no immediate information. Specifically, the value of the ''p''-adic ''L''-function may vanish even when the value of the original ''L''-function does not. Beginning with the work of Mazur–Tate–Teitelbaum on a ''p''-adic Birch–Swinnerton-Dyer conjecture, it has been conjectured that the value of the ''derivative'' of the ''p''-adic ''L''-function should relate to the original''L''-value, up to the introduction of a new factor: the ''L''-invariant. We will discuss some of the known cases of this phenomenon, as well as ongoing work on the study of ''L''-invariants of symmetric powers of modular forms.
Abstract: tba
 
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== December 1 ==
== April 26 ==


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| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Andrei Caldararu''' (Madison)
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Frank Thorne''' (U. South Carolina)
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| bgcolor="#BCD2EE"  align="center" | Title: The Hodge theorem as a derived self intersection, part 2
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Abstract: The Hodge theorem on the decomposition of de Rham cohomology into ''H''<sup>&thinsp;''p'',''q''</sup>-pieces was phrased by Deligne–Illusie as the splitting of a complex in the derived category, which is then proved using positive characteristic methods.
Abstract: tba
 
A similar splitting result was obtained by Arinkin and myself for a certain complex associated to a closed embedding. In my earlier talk in the algebraic geometry seminar I explained how to recast the Deligne–Illusie so that it can be seen as a particular case of the Arinkin–Caldararu result.
 
In my current talk I shall quickly review this story, and then provide some of the details that were skipped in my earlier talk. This is joint work with Dima Arinkin and Marton Hablicsek.


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== December 8 ==
== May 3 ==


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| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Xinwen Zhu''' (Harvard)
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| bgcolor="#BCD2EE"  align="center" | Title: Local models of Shimura varieties
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Abstract: I will report some recent progress in the study of local
Abstract: tba
models of Shimura varieties, including the proof of the coherence
conjecture of Pappas–Rapoport and the Kottwitz conjecture.


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== December 15 ==
== May 10 ==


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| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Shamgar Gurevich''' (Madison)
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Samit Dasgupta''' (UC Santa Cruz)
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| bgcolor="#BCD2EE"  align="center" | Title: Quadratic reciprocity and the sign of the Gauss sum via the finite Weil representation
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Abstract: I will show how to encode in the language of the finite Weil representation, two basic results of number theory - the quadratic reciprocity, and the sign of the Gauss sum. This will enables us to use tools from group representation theory to give new proofs for these two results.
Abstract: tba    
 
I will assume knowledge of basic linear algebra.
 
Joint work with  Ronny Hadani (Austin), and Roger Howe (Yale).    


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== Organizer contact information ==
== Organizer contact information ==

Revision as of 03:54, 26 January 2012

February 2

Evan Dummit (Madison)
Title: tba

Abstract: tba


March 29

David P. Roberts (U. Minnesota Morris)
Title: tba

Abstract: tba



April 12

Chenyan Wu (Minnesota)
Title: tba

Abstract: tba


April 19

Robert Guralnick (U. Southern California)
Title: tba

Abstract: tba


April 26

Frank Thorne (U. South Carolina)
Title: tba

Abstract: tba


May 3

?
Title: tba

Abstract: tba


May 10

Samit Dasgupta (UC Santa Cruz)
Title: tba

Abstract: tba


Organizer contact information

Shamgar Gurevich

Robert Harron

Zev Klagsbrun

Melanie Matchett Wood


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