February 2019
February 1  Karol Koziol (Alberta)
Serre weight conjectures for unitary groups Note: This talk will take place in CAB 563 at 3:00pm. In the 1970s, Serre formulated
his remarkable conjecture that every twodimensional
mod$p$ Galois representation of the absolute Galois
group of $\mathbb{Q}$, which is odd and irreducible,
should come from a modular form. He later refined
his conjecture, giving a precise recipe for the
weight and level of the modular form. Both the "weak
form" and "strong form" of Serre's conjecture are
now theorems, due to the work of many mathematicians
(KhareWintenberger, Kisin, Edixhoven, Ribet, and
others). In this talk, we will discuss how to
generalize Serre's weight recipe when the Galois
representation is replaced by a homomorphism from an
absolute Galois group to the Langlands dual of a
rank 2 unitary group. This is joint work with
Stefano Morra.

February 8  No seminar

February 15  No Seminar

February 22  No Seminar 
March 2019
March 1  Jamie
Juul (UBC) Arboreal Galois Representations The main questions in arithmetic
dynamics are motivated by analogous classical
problems in arithmetic geometry, especially the
theory of elliptic curves. We study one such
question, which is an analogue of Serre's open image
theorem regarding $\ell$adic Galois representations
arising from elliptic curves. We consider the action
of the absolute Galois group of a field on
preimages of a point $\alpha$ under iterates of a
rational map $f$ (points that eventually map to
$\alpha$ as we apply $f$ repeatedly). These points
can be given the structure of a rooted tree in a
natural way. This determines a homomorphism from the
absolute Galois group of the field to the
automorphism group of this tree, called an arboreal
Galois representation. As in Serre's open image
theorem, we expect the image of this representation
to have finite index in the automorphism group
except in certain cases.

March 8  No Seminar 
March 14 Note the date and time change! 
Khoa
Nguyen (Calgary) Dfiniteness, rationality, and height Note: This talk will take place in Cameron 310 at 3:30pm. Let $K$ be a field of
characteristic $0$ and let $m\geq 1$. A power series
$f(x_1,\ldots,x_m)\in K[[x_1,\ldots,x_m]]$ is said
to be Dfinite if all the partial derivatives of $f$
span a finite dimensional vector space over the
field $K(x_1,\ldots,x_m)$. Examples in the univariate
case include rational functions, algebraic functions
such as $\sqrt{1+x}$, the exponential function, and
$\log(1+x)$. The logarithmic Weil height is a
function $h:\ \bar{\mathbb{Q}}\rightarrow
\mathbb{R}^{+}$ that plays an important role in
diophantine geometry. Motivated by earlier work of
van der PoortenShparlinski and BellChen, we study
problems related to the growth of the height of
coefficients of Dfinite power series in
$\bar{\mathbb{Q}}[[x_1,\ldots,x_m]]$. This is joint
work in progress with Jason Bell and Umberto
Zannier.

March 22  Clifton
Cunningham (Calgary) The geometry of Arthur packets This talk uses two examples to
show how Arthur parameters determine a category of
equivariant perverse sheaves and how the microlocal
perspective on this category reveals the Arthur
packet attached to the Arthur parameter. The full
picture is explained in Arthur packets for
padic groups by way of microlocal vanishing
cycles, with examples, joint with Andrew
Fiori, Ahmed Moussaoui, James Mracek and Bin Xu, to
appear in Memoirs of the American Mathematical
Society. The talk includes some remarks on an
emerging categorical local Langlands correspondence.

March 29  No Seminar 
April 2019
April 8 Note the date and time change! 
Hadi
Salmasian (Ottawa) The minimal faithful dimension of finite $p$groups: an application of the orbit method to the essential dimension Note: This talk will take place in CAB 657 at 2:00pm. The essential dimension of a
finite group over a field $K$, defined by Buhler and
Reichstein, is the smallest dimension of a
linearizable $G$variety over $K$ with a faithful
$G$action. By a result of Karpenko and Merkurjev,
if $G$ is a finite $p$group and $K$ contains a
primitive $p$th root of unity, then this number is
equal to the smallest dimension of a faithful
$K$representation of $G$. Using the orbit method we
prove qualitative and quantitative results on the
essential dimensions of finite $p$groups. This talk
is based on a joint project with M. Bardestani and
K. M. Karai.

April 12  Adrian
Diaconu (Minnesota) Moments and multiple Dirichlet series In the 1980's the idea emerged
that it could be useful to tie together a family of
related $L$functions to create a double, or
multiple, Dirichlet series, which could be used to
study the average behavior of the original family of
$L$functions. The local structure of these multiple
Dirichlet series shows a rich connection to the
theory of automorphic forms (i.e., Whittaker
functions on $p$adic groups and their covers are
the fundamental objects), representation theory, and
as noticed recently, to algebraic geometry.
In this talk, I will focus on the most important case, namely the multiple Dirichlet series associated to moments of $L$functions. I will discuss the connection between the local parts of these series and the compactifications of certain moduli spaces of curves, and how this information can be combined with the (conjectural in general) analytic continuation of the multiple Dirichlet series to obtain precise asymptotics for moments, for example, of the classical family of quadratic Dirichlet $L$functions. 
April 19  TBA (TBA) TBA 
April 26  Monica
Nevins (Ottawa) Towards proving the unicity of types for supercuspidal representations Recent work has shown that to each
irreducible smooth complex representation $\pi$ of a
$p$adic group $G$ we may associate a type,
which is a pair $(K,\rho)$ (for $K$ a compact open
subgroup and $\rho$ an irreducible representation
$K$) from which one may recover the (Bernstein
component of) $\pi$. These pairs are not unique,
even up to conjugacy. For the case of supercuspidal
representations, the conjecture of "unicity of
types" originally proposed that the restriction of
$\pi$ to a compact open subgroup could contain at
most one type.
Using tools from BruhatTits theory, we show this version of unicity fails, yet deduce from our methods a strategy for proving a refined version of the conjecture. This is ongoing joint work with Peter Latham of King's College, London. 
May 2019
May 3  Claus
Sorensen (UCSD) Koszul duality for Iwasawa algebras modulo $p$ A result of Bernstein from the 80s
gives an equivalence between modules over the
IwahoriHecke algebra and the category of smooth
$G$representations generated by their Iwahorifixed
vectors. Here $G$ is a $p$adic Lie group and the
coefficient field is $\Bbb{C}$. This equivalence
usually fails if we take a coefficient field $k$ of
characteristic $p$, as in the mod $p$ local
Langlands correspondence. Schneider found a way to
remedy the situation by instead relating the derived
category $D(G)$ to modules over a certain
differential graded variant of the (pro$p$)
Iwahori Hecke algebra. The goal of the talk is to
interpret Schneider's equivalence in the case $G=I$
as a version of Koszul duality for the Iwasawa
$k$algebra $\Omega(G)$. This makes use of Lazard's
theory of $p$valuations, which played a key role in
his proof that $\Omega(G)$ is Noetherian. The rough
idea is that the filtration on $\Omega(G)$ somehow
corresponds to an $A_{\infty}$structure on the
Yoneda extension algebra. When $G$ is abelian the
$A_{\infty}$structure is trivial (the converse was
recently shown by Carl WangErickson); this
generalizes a result of Schneider for $\Bbb{Z}_p$.
