__ Monday January 14, 3:00-3:50pm, LCB 215:__ (note unusual day and time)

__ Wednesday February 20, 4:00-5:00pm, LCB 215:__ (note unusual day and time)

** February 22:**

**Speaker: ** Kyo Nishiyama, Aoyama Gakuin and MIT

**Title: ** Steinberg theory for symmetric pairs and generalized
Robinson-Schensted correspondence for partial permutations

**Abstract: **
Let G be a connected reductive group and B its Borel
subgroup. Steinberg considered a double flag variety X = G/B \times
G/B with diagonal G action, and got a correspondence between
the Weyl group W and nilpotent orbits in Lie(G)
in terms of moment maps. In type A, this correspondence amounts to
the famous RS correspondence between permutations and pairs of
standard Young tableaux.
In this talk, we generalize it to get an RS correspondence for partial
permutations. We also interprete it by using double flag varieties
for symmetruc pairs, and get a further generalization to obtain a
correspondence between partial permutations and signed Young diagrams
parametrizing nilpotent orbits for a symmetric pair.
This reveals an interesting interchange between geometry and
combinatorics.
The talk is based on the on-going joint work with Lucas Fresse at IECL
(France).

** March 8:**

**Speaker: ** Wei Ho, University of Michigan

**Title: **Integral points on elliptic curves

**Abstract: ** Elliptic curves are fundamental and well-studied objects in arithmetic geometry.
However, much is still not known about many basic properties, such as the number of rational points
on a "random" elliptic curve. We will discuss some conjectures and theorems about this "arithmetic statistics"
problem, and then show how they can be applied to answer a related question about the number of integral points
on elliptic curves over Q. In particular, we show that the second moment (and the average) for the number of
integral points on elliptic curves over Q is bounded (joint work with Levent Alpoge).

** March 22:**

**Speaker: ** Eric Sommers, University of Massachusetts

**Title: **
Generalized Bott-Samelson resolutions for Schubert varieties

**Abstract: **
A Schubert variety is the closure of a B-orbit on the flag
variety G/B of a reductive algebraic group G and is indexed by an
element w of the Weyl group of G. In general these varieties are
singular and some of the structure of their singularities can be
understood via the Bott-Samelson resolutions and the resulting interplay
between the intersection cohomology of the Schubert variety and the
ordinary cohomology of the fibers of these resolutions. Bott-Samelson
resolutions arise from any factorization of w into simple reflections
and they are iterated P^1 bundles over P^1.
This talk focuses on joint work with Jennifer Koonz, on generalized
Bott-Samelson resolutions, which are iterated G/Q-bundles (for different
parabolic subgroups Q of G). These resolutions also provide information
about the singularities of Schubert varieties. Special cases have
appeared in work of Wolper, Ryan, Polo, and Zelevinsky in the case where
G is the general linear group. They also appear in other types in the
work of Vanchinathan-Sankaran and Billey-Postnikov. After introducing
these resolutions and some of their properties, we will suggest a
possible best generalized resolution for a given Schubert variety. For
these best resolutions, there appears to be a connection with work of
Williamson on torsion in intersection cohomology. Finally, we'll discuss
a computer program that calculates local intersection cohomology (i.e.,
Kazhdan-Lustztig polynomials) using these resolutions.

** April 19:**

**Speaker: ** Yunqing Tang, Princeton University

**Title: **
Reductions of abelian surfaces over global function fields

**Abstract: **
For a non-isotrivial ordinary abelian surface $A$ over a global function
field with everywhere good reduction, under mild assumptions, we prove that there
are infinitely many places modulo which $A$ is geometrically isogenous to the
product of two elliptic curves. This result can be viewed as a generalization of a
theorem of Chai and Oort. This is joint work with Davesh Maulik and Ananth Shankar.

** August 24:**

**Speaker: **

**Title: ** Organizational meeting, 4:10-4:30

**Abstract: **

** August 31:**

**Speaker: **

**Title: **

**Abstract: **

** September 7:**

**Speaker: ** Tony Feng, Stanford University

**Title: **The Artin-Tate pairing on the Brauer group of a surface

**Abstract: ** There is a canonical pairing on the Brauer group of a surface over a ﬁnite
ﬁeld, and an old conjecture of Tate's predicts that this pairing is alternating. In
this talk I will present a resolution to Tate’s conjecture. The key new ingredient
is a circle of ideas originating in algebraic topology, centered around the Steenrod
operations. The talk will advertise these new tools (while assuming minimal
background in algebraic topology).

** September 14:**

**Speaker: ** Adam Brown, The University of utah

**Title: **Arakawa-Suzuki functors for Whittaker modules

**Abstract: ** The category of Whittaker modules, as formulated by Milicic and Soergel,
is a category of Lie algebra representations which generalizes other
well-studied categories of representations, such as the
Bernstein-Gelfand-Gelfand category O. In this talk we will construct a
family of exact functors from the category of Whittaker modules to the
category of finite-dimensional graded affine Hecke algebra modules, for
type A_n. These algebraically defined functors provide us with a
representation theoretic analogue of certain geometric relationships,
observed independently by Zelevinsky and Lusztig, between the flag variety
and the variety of graded nilpotent classes. Using this geometric
perspective and the corresponding Kazhdan-Lusztig conjectures for each
category, we will prove that these functors map simple modules to simple
modules (or zero). Moreover, we will see that each simple module for the
graded affine Hecke algebra can be realized as the image of a simple
Whittaker module.

** September 21:**

**Speaker: ** Gil Moss, The University of Utah

**Title: **
Characterizing the mod-\ell local Langlands correspondence for GL(n) using
nilpotent gamma factors.

**Abstract: **
In the complex representation theory of GL_n of a p-adic field,
Rankin-Selberg gamma factors uniquely determine the supercuspidal support
of a generic representation. This is known as a "converse theorem." By
using a modified gamma factor that accounts for nilpotent elements in the
mod-\ell Bernstein center, we can establish a converse theorem in the
mod-\ell setting, addressing a conjecture of Vigneras. With an appropriate
modification of gamma factors on the Galois side, this would give a
characterization of Vigneras' mod-\ell local Langlands correspondence.

** September 28:**

**Speaker: ** Kei Yuen Chan, University of Georgia

**Title: **
Ext-branching law for reductive p-adic groups

**Abstract: **
The local Gan-Gross-Prasad conjecture predicts the branching law for
some tempered representations of classical reductive groups over real and p-adic
fields. Dipendra Prasad suggests homological variations of those branching
problems and in particular conjectures that there is no higher extensions between
a generic representation of GL(n+1,F) and a generic one of GL(n,F). In a joint
work with Gordan Savin, we give a positive answer to the conjecture. I shall
report the work and other relevant results in the talk.

** October 5:**

**Speaker: ** Stefan Patrikis, The University of Utah

**Title: **
Lifting Galois Representations

**Abstract: **
This talk will be an amply-motivated introduction to some of the different phenomena one encounters when trying to lift mod p representations of
the absolute Galois group of a number field to ("geometric") p-adic representations. I will emphasize examples rather than general theory. Time permitting,
I'll describe some recent joint work with N. Fakhruddin and C. Khare.

__ October 12:__ No Seminar (fall break)

** October 19:**

**Speaker: ** Sabine Lang, The University of Utah

**Title: ** Restriction of the oscillator representation using dual pairs

**Abstract: **
In branching problems, the Ext functors vanish if the
restriction of a representation is projective. We will study two
restrictions of the oscillator representation of a real symplectic
group: the restriction to a smaller symplectic subgroup and the
restriction to an orthogonal subgroup. We will present conditions on the
relative sizes of these groups under which these restrictions are
projective. The main tools used in our proof are the theory of dual
pairs and the theta correspondence, first introduced by Howe.

** October 26:**

**Speaker: **

**Title: **

**Abstract: **

** November 2:**

**Speaker: ** Christian Klevdal, The University of Utah

**Title: **
Recognizing Galois representations of K3 surfaces

**Abstract: **
This talk centers around the question of whether a `cohomological
object' (e.g. a Hodge structure or Galois representation) appears in the
cohomology of an algebraic variety. For varieties over the complex numbers,
Riemann's theorem that weight one Hodge structures appear in the cohomology
of abelian varieties and the surjectivity of the period map for K3 surfaces
provide positive answers. Over number fields, this is a very hard question
to answer. However using certain motivic conjectures, recent work of
Patrikis, Voloch and Zarhin provides an analogue of Riemann's theorem for
Galois representations. We will then discuss the analogue of surjectivity
of the K3 period map for Galois representations.

** November 9:**

**Speaker: **

**Title: **

**Abstract: **

** November 16:**

**Speaker: ** Claus Sorensen, University of California, San Diego

**Title: **
The local Langlands correspondence on eigenvarieties

**Abstract: **
In this talk we will consider eigenvarieties for definite unitary groups, which are
rigid analytic spaces parametrizing systems of Hecke eigenvalues appearing in spaces
of finite slope p-adic modular forms. These varieties carry a natural coherent
sheaf whose fibers interpolate the local Langlands correspondence for GL(n) at
places away from p. To show this we consider the action on the sheaf itself of certain Bernstein
center elements which appeared in works of Chenevier, and more recently in Scholze's
proof of local Langlands. This bears a strong resemblance to "local Langlands in
families" as formulated by Emerton, Helm, and Moss. In order to control the generic
constituents of the fibers we make critical use of a newly established genericity
criterion of Chan and Savin. This is a report on joint work with Christian Johansson
and James Newton.

__ November 23:__ No Seminar (Thanksgiving break)

** Tuesday December 4 (note unusual time), 3:30-4:30pm LCB 222:**

**Speaker: ** Nivedita Bhaskhar, UCLA

**Title: ** Reduced Whitehead groups of algebras over p-adic curves

**Abstract: **
Any central simple algebra A over a field K is a form of a matrix
algebra. Further A/K comes equipped with a reduced norm map which is
obtained by twisting the determinant function. Every element in the
commutator subgroup [A*, A*] has reduced norm 1 and hence lies in SL_1(A),
the group of reduced norm one elements of A. Whether the reverse inclusion
holds was formulated as a question in 1943 by Tannaka and Artin in terms of
the triviality of the reduced Whitehead group SK_1(A) := SL_1(A)/[A*,A*].

Platonov negatively settled the Tannaka-Artin question by giving a counter
example over a cohomological dimension (cd) 4 base field. In the same paper
however, the triviality of SK_1(A) was shown for all algebras over cd at
most 2 fields. In this talk, we investigate the situation for l-torsion
algebras over a class of cd 3 fields of some arithmetic flavour, namely
function fields of p-adic curves where l is any prime not equal to p. We
partially answer a question of Suslin by proving the triviality of the
reduced Whitehead group for these algebras. The proof relies on the
techniques of patching as developed by Harbater-Hartmann-Krashen and
exploits the arithmetic of these fields.

** December 7:**

**Speaker: ** Moshe Adrian, Queens College, CUNY

**Title: **On the sections of the Weyl group

**Abstract: ** Let G be a connected reductive group over an algebraically closed
field and W the Weyl group of a split maximal torus. In 1966, Tits defined
a representative n_\alpha, in G, for each simple reflection s_\alpha in the
Weyl group. These representatives satisfy the braid relations, and
therefore define a "cross section" of W. This section has been used
extensively ever since, in many works.
On the other hand, there exist sections of the Weyl group that are not
Tits' section. For example, in some cases (for example type B_n adjoint
and even simpler, GL_2), W embeds into G, but the Tits section does not
realize such an embedding. More importantly for our purposes (in the
theory of p-adic groups), Tits' section does not realize a splitting of the
Kottwitz homomorphism.
Motivated by this question about the Kottwitz homomorphism, we have
computed and classified all sections of the Weyl group that satisfy the
braid relations, in types A through G. It turns out that these sections
form a partially ordered set, with a unique maximal element. We will
describe this poset, which is interesting in and of its own right. The
maximal element in this poset is the "most homomorphic" of all of the
sections. This "maximal" section answers our question from p-adic groups:
it splits the Kottwitz homomorphism. Moreover, essentially by
construction, it always realizes an embedding of W in G, whenever such an
embedding exists.