Invited Speakers: Matt Baker
(GA Tech), Elena Fuchs (UC Berkeley), Christopher Skinner (Princeton University), Manjul Bhargava (Princeton University)
Location: All talks will
be held in the Cahill Building Hameetman Auditorium on the
Caltech campus. Click on campus map to locate the Cahill building (building number 17).
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Please register at the conference sign in table if you have not already registered online.
Schedule:
9:30
 10:00  Refreshments
10:00
 11:00  Matt Baker (Georgia Institute of Technology),
"RiemannRoch for Graphs and Applications"
11:00
 11:30  Break
11:30
 12:30  Elena Fuchs (University of California, Berkeley),
"Thin Monodromy Groups"
12:30
 2:30  Lunch Break
2:30
 3:30  Christopher Skinner (Princeton University),
"Elliptic curves of rank at most one and the ppart of the BSD conjecture"
3:30
 4:00  Break
4:00
 5:00  Manjul Bhargava (Princeton University),
"The average number of elements in the 5Selmer groups of elliptic curves, and applications"
5:30
 Dinner  see below
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Abstracts:

MATT BAKER (Georgia Institute of Technology)
RiemannRoch for Graphs and Applications (10AM)
We will begin by formulating the RiemannRoch theorem for graphs due to the speaker and Norine. We will then describe some refinements and connections with Berkovich analytic spaces. Applications include a new proof by Jensen and Payne of the GiesekerPetri theorem in algebraic geometry, a generalization by Amini and the speaker of the EisenbudHarris theory of limit linear series, and a new ChabautyColeman style bound for the number of rational points on an algebraic curve over the rationals, proved recently by Katz and ZureickBrown. 

ELENA FUCHS (University of California, Berkeley)
Thin Monodromy Groups (11:30 AM)
In recent years, it has become interesting from a numbertheoretic point of view to be able to determine whether a finitely generated subgroup of GL_n(Z) is a socalled thin group. In general, little is known as to how to approach this question. In this talk we discuss this question in the case of hypergeometric monodromy groups, which were studied in detail by Beukers and Heckman in 1989. We will convey what is known, explain some of the difficulties in answering the thinness question, and show how one can successfully answer it in many cases where the group in question acts on hyperbolic space. This work is joint with Meiri and Sarnak. 

CHRISTOPHER SKINNER (Princeton University)
Elliptic curves of rank at most one and the ppart of the BSD
conjecture (2:30 PM)
This talk will describe some padic and mod p criteria for an elliptic curve over the rationals to have algebraic and analytic rank either zero or one as well as connections with Iwasawa theory and the ppart of the BSD conjecture.


MANJUL BHARGAVA (Princeton University)
The average number of elements in the 5Selmer groups of elliptic
curves, and applications (4:00 PM)
We determine the average number of elements in the 5Selmer groups of elliptic curves. Together with an analysis of root numbers, this yields new upper bounds on the average rank of elliptic curves. (Joint work with Arul Shankar.) Using the above together with the results of the previous talk, we also describe new *lower* bounds on the average rank of elliptic curves. 
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Parking: Parking at Caltech is free on the
weekends. Please park in the lot located on California Blvd (lot 3) which can
be located on this map.
Directions: Directions to Caltech can
be found by following this link. A map of the Pasadena area can be
found here. If you need a hotel, the Saga Motor Hotel and Vagabond Inn are close and affordable
Dinner: Dinner after the seminar will be held at Saladang Song Restaurant. The charge will be $25 per person ($15 per person for students) and does not include beverages. Please pay by cash or check (made out to Caltech) the day of the dinner.
Contact
Organizers: Dinakar Ramakrishnan, Matthias Flach, Elena
Mantovan
Attendees:
Richard Hilario
Alice Silverberg
Myungjun Yu
Heng Su
ChanHo Kim
Rufei ren
Maria Nastasescu
Zhiping Rao
Cristian Popescu
Emad Nasrollahpoursammi
Haruzo Hida
Adebisi Agboola
Karl Rubin
Jordan Schettler
Claus Sorensen
Ashay Burungale
Brian Hwang
Lin Weng
Francesc Castella
Jay Daigle
Serin Hong
Gunther Cornelissen
David Krum
Bold = Attending dinner
