April 3 
Stephan Garcia (Pomona College) The theory of supercharacters, which generalizes classical
character theory, was recently developed in an axiomatic fashion by P. Diaconis
and I.M. Isaacs, based upon earlier work of C. Andre. When this machinery is applied
to abelian groups, a wide variety of applications emerge. In particular, we develop a
generalization of the discrete Fourier transform along with several combinatorial tools.
This perspective illuminates several classes of exponential sums (e.g., Gauss, Kloosterman,
and Ramanujan sums) that are of interest in number theory. We also consider certain exponential
sums that produce visually striking patterns of great complexity and subtlety. (Partially
supported by NSF Grants DMS1265973, DMS1001614, and the Fletcher Jones Foundation.)

April 10 
Claus Sorensen (UCSD) In their attempt to mimic the proof of Fermat's Last Theorem for GL(n),
Clozel, Harris, and Taylor, were led to a conjectural analogue of Ihara's lemma  which is
still open for n>2. In this talk we will revisit their conjecture from a more modern point
of view, and reformulate it in terms of local Langlands in families, as currently being developed
by Emerton and Helm. At the end, we hope to sketch how this can be used to obtain a factorization of
completed cohomology for U(2). [The last part is joint work with P. Chojecki.]

April 17 
Liang Xiao (UC Irvine) 23 in 7421 Let F be a totally real number field, p a prime number, and M the (splitting model of) Hilbert modular variety for F (of some fixed level) defined over a finite field of characteristic p. I will explain how exploiting the geometry of M, and in particular the existence of the partial Hasse invariants, one can attach Galois representations to Hecke eigensystems occurring in the coherent cohomology of M. This is a joint work with Matthew Emerton and Davide Reduzzi.
ChanHo Kim (UC Irvine) 34 in 7421? We construct anticyclotomic padic Lfunctions of Hida families
in a more controlled way using a multiplicity one result arising from arithmetic of
Shimura curves. Using this construction, we can calculate the difference of Iwasawa invariants
of padic Lfunctions of congruent modular forms in different weights. As an application, we
can see the equivalence of the main conjectures of two congruent forms under certain conditions.
This is joint work in progress with Francesc Castella and Matteo Longo.

April 24 
Kiran Kedlaya (UCSD) The functor of ptypical Witt vectors is most well known for lifting
perfect fields of characteristic p into complete discrete valuation
rings. However, it is a welldefined functor on arbitrary rings; we will
indicate how applying this functor to local rings of mixed
characteristic gives some new perspectives on padic Hodge theory. We
will also touch briefly upon some mysterious links to algebraic Ktheory
coming from the work of Hesselholt. Based on joint papers with Chris
Davis (Copenhagen).

May 1 
Cristian Popescu (UCSD) First, I will describe how our results (joint with Greither)
on the BrumerStark conjecture lead to a new construction of Hecke characters
for CM number fields, generalizing A. Weil's Jacobi
sum Hecke characters. Second, I will show how the values of these
characters can be used to construct special elements in
the even Kgroups of CM and totally real number fields. Several
applications ensue: a general construction of Euler Systems in the
even Ktheory of CM and totally real number fields;
a Ktheoretic reformulation (and potential proof strategy)
of a classical and wide open conjecture of Iwasawa on class groups of
cyclotomic fields; potential new insights into Hilbert's 12th problem for CM
number fields etc. Time permitting, I will touch upon some of these
applications as well. This is based on joint work with G. Banaszak
(Poland).

May 8 
Cristian Popescu (UCSD) I will briefly review the material covered in last week's
lecture (May 1) and will continue with a more detailed
description of the Ktheoretic constructions and their arithmetic
applications mentioned in last week's abstract. Joint work with G.
Banaszak (Poland.)

May 15 


May 22 


May 29 
Ryan Rodriguez (UCSD) It is desirable for the spectrum of a uniform Banach algebra to have a structure sheaf.
This happens when the algebra is acyclic. We will discuss what it means for a uniform Banach algebra to be acyclic.
I will explain how to show preperfectoid algebras are acyclic.

June 5 
TBA () TBA 
June 12 
TBA() TBA 