Bay Area Algebraic Number Theory and Arithmetic Geometry Day 4
Saturday, March 3, 2012
Adebisi Agboola, University of California, Santa Barbara
Elena Fuchs, University of California, Berkeley
Ralph Greenberg, University of Washington
Martin Weissman, University of California, Santa Cruz
All talks will be in the Stanford math department, room 380-C. Refreshments (including the morning coffee)
will be in the 2nd floor common room.
|Dinner, to be announced
Please RSVP to sdasgup2 (at) ucsc (dot) edu
Titles and Abstracts:
Bisi Agboola, "Statistical Galois Structure of Rings of Integers"
Let K be a global field, and G a finite abelian group. I
shall discuss the asymptotic behavior of the number of tamely ramified
G-extensions of K with ring of integers of fixed realisable class as a
Galois module. The answers that one obtains are rather surprising, in that
they very much depend upon how one counts the number of rings of integers
of a given realisable class.
Elena Fuchs, "Counting Integers Represented by Binary Quadratic Forms"
There are many classical results about the number of integers, primes, shifted primes, etc. less than X represented by a given binary quadratic form. However, given a family of binary forms whose discriminants vary with respect to X, it is quite difficult to produce uniform bounds on the number of integers less than X represented by forms in the family. Unexpectedly, this problem has something to do with counting the integers which come up as curvatures of circles in Apollonian packings: it turns out that these curvatures contain integers represented by a family of shifted binary forms of arbitrarily large discriminants. In this talk, we discuss how this relationship between packings and binary forms arises, and what goes into counting these curvatures. Furthermore, we discuss the more difficult problem of extending this method to counting circles of prime curvature in a given Apollonian packing. This work is joint with Jean Bourgain.
Ralph Greenberg, "Galois Representations With Open Image"
Martin Weissman, "Incorporating metaplectic groups in the Langlands program"
The Langlands conjectures give many predictions linking
smooth representations of p-adic groups, Galois representations, and
L-functions. After surveying some of these conjectures, I will
identify a few weak points -- places where the conjectures have little
supporting evidence, places where the conjectures have awkward issues
of normalization, and places where there should be a precise
conjecture but there is only rough speculation. One such area where
there is no precise conjecture relates to "metaplectic groups",
interpreted broadly to include many (nonalgebraic) central extensions
of reductive p-adic groups by finite groups. I will explain why these
groups should be included in the Langlands conjectures, recent efforts
to incorporate them, and evidence for the resulting metaplectic local
Parking is free and plentiful in the Oval and surrounding lots (on Roth Way and Lausen St) on weekends.
Here is a campus map, with the math building (380) labelled "Math Corner."
There is no formal registration, but if you plan to attend, we would appreciate an email to sdasgup2 at ucsc dot edu to
help plan the event, especially if you plan to attend the dinner afterwards.
There will be a dinner following the conference at 6pm in downtown Palo Alto. Please send an email to sdasgup2 at ucsc dot edu if you plan to attend.