**Oberseminar**

This is an activity run by the
AGANT group at UGA (AGANT= Algebra, Algebraic
Geometry, and Arithmetic
Geometry/Number Theory).

During the academic year, the AGANT group runs three weekly seminars,

in Algebra
(Mon 3:30-4:30),

in Algebraic
Geometry (Wed 2:30-3:30), and

in Arithmetic
Geometry/Number Theory (Wed 3:45-5:15).

We further link the three fields of AGANT with the development of our Oberseminar, which started in the fall of 2011. The Oberseminar takes place twice a semester, in general at the
time of one of the current Wednesday seminars, and has the following goals:

(1) A place where an incoming postdoc or an incoming tenure-track faculty could
give a talk in their first semester at UGA, addressing the whole AGANT group,
and thus giving a talk that would be less specialized that if given in one of
the separate seminars that we hold. It is important that incoming people meet
and interact with the current members of the group as soon as possible after
they arrive at UGA. To give such a talk gives an incoming member the
opportunity to explain their research interests to a wider audience. Such a
talk, for a postdoc, could also be construed as putting in place the foundation
of the job talk that he or she will need to prepare for their next job
application.

(2) The Oberseminar is a place where big research
ideas are discussed rather than a focus on technical details. A place where
connections between the areas of AGANT are sought, or explained. A place where
the latest major achievements in one of the AGANT areas are explained with a
twist: how can these results be used, interpreted, viewed, in the other AGANT
areas. A place where fertilization and cross-pollination can easily occur.

*Oberseminar** welcoming new postdocs Kei Yuen Chan, Nikon Kurnosov,
Chun-Ju Lai, Ben Lund, Scott Mullane,
and Alex Stathis, *Wed, August 30, 2017, 2:30 – 5:00, Boyd, Room 304

*Oberseminar** welcoming new postdocs Kei Yuen Chan, Nikon Kurnosov,
Chun-Ju Lai, Ben Lund, Scott Mullane,
and Alex Stathis, *Wed, August 23, 2017, 2:30 – 5:00, Boyd, Room 304

*Oberseminar**: Paul Pollack, *Tu April 25, 2017, 3:30
– 4:30, Boyd, Room 23

Title:
**Arithmetic functions:
old and new**

I will survey some of what is known (and still unknown) about the value
distribution of classical arithmetic functions. The problems discussed have in
common that they owe their origin, in one way or another, to the fascination of
the ancients with sums of divisors.

*Oberseminar** welcoming new postdocs *Asilata Bapat, Anand Deopurkar,
Andrew Niles, and Michael Schuster*, *Wed, August 24, 2016,
2:30 – 5:00, Boyd, Room 304

Anand Deopurkar,
2:30: **The
algebra of canonical curves and the geometry of their moduli space**

Every non-hyperelliptic curve of genus g canonically embeds in the
projective space of dimension (g-1). There are fascinating connections between
the algebra of the corresponding homogeneous ideal and the geometry of the
curve. Going further, it seems that understanding the algebra of homogeneous
ideals will shed light on the birational geometry of
the moduli space of all curves. I will discuss an ongoing project to understand
this connection (partly joint with Fedorchuk and Swinarski).

Asilata Bapat, 3:00: **Calogero****-Moser space and GIT**

The Calogero-Moser
space is a symplectic algebraic variety that deforms
the Hilbert scheme of points on a plane. It can be interpreted in many ways,
for example as the parameter space of irreducible representations of a Cherednik algebra, or as a Nakajima quiver variety. It has
a partial compactification that can be described combinatorially
using Schubert cells in a Grassmannian. The aim of my
talk is to introduce the Calogero-Moser space, and some
work in progress towards constructing another partial compactification using
Geometric Invariant Theory (GIT).

Tea and Social, 3:30-4:00

Andrew
Niles, 4:00: **The Picard Groups of
Certain Moduli Problems**

The Picard
group of the stack M_{1,1} of elliptic curves, over an
algebraically closed field of characteristic coprime to 6, was computed in 1965
by Mumford. However, the Picard group of M_{1,1} over
more general base schemes (such as over the integers) was not known until it
was computed in 2010 by Fulton and Olsson; their result holds over an arbitrary
reduced base scheme or an arbitrary base scheme on which 2 is invertible. We
present a partial generalization of the result of Fulton-Olsson, computing the
Picard groups of the stacks Y_0(2) and Y_0(3) over any base scheme on which 6
is invertible.

Michael Schuster, 4:30:
**The
multiplicative eigenvalue polytope**

The multiplicative
eigenvalue problem asks the following: for which sets of eigenvalues do there
exist special unitary matrices A_1,...,A_n having those
eigenvalues, that when multiplied A_1*A_2*...*A_n give
you the identity? The set of such eigenvalues forms a convex polytope called
the multiplicative eigenvalue polytope, which is connected to a number of
important objects and spaces in representation theory and algebraic geometry.
In this talk I will discuss the multiplicative polytope and its connections
with quantum cohomology, conformal blocks, moduli
spaces of parabolic bundles, and moduli spaces of curves, time permitting.

*Oberseminar**: Dan Nakano, *Wed May 4, 2016, 2:30 – 3:30, Boyd, Room 304

Title:
**Irreducibility of Weyl
modules over fields of arbitrary characteristic
**

In the
representation theory of split reductive algebraic groups, the following is a
well-known fact: for every minuscule weight, the Weyl module with that highest
weight is irreducible over every field. The adjoint
representation of E_8 is also irreducible over every field. Recently,
Benedict Gross conjectured that these two examples should be the only cases
where the Weyl modules are irreducible over arbitrary fields. In this talk I
will present our proof of Gross' suggested converse to these statements, i.e.,
that if a Weyl module is irreducible over every field, it must be either one of
these, or trivially constructed from one of these. My coauthors will be
revealed during my talk.

*Oberseminar**: Paul Pollack, *Wed March 30, 2016, 2:30 – 3:30, Boyd, Room 304

Title:
**A survey of recent
work on gaps between primes**

I will present
an overview of the spectacular progress from the past few years towards the
(in)famous twin prime conjecture. At the conclusion of the talk, I will discuss
a very recently discovered (just this month!) "repulsion phenomenon"
for consecutive primes in residue classes.

*Oberseminar**: Eric Katz (Waterloo), *Wed December 2, 2015, 3:45 – 4:45, Boyd, Room 304

Title:
**Hodge Theory on Matroids**

The
chromatic polynomial of a graph counts its proper colorings. This polynomial's coefficients were
conjectured to form a unimodal sequence by Read in 1968. This conjecture was extended by Rota in
his 1970 address to assert the log-concavity of the characteristic polynomial
of matroids which are the common generalizations of
graphs and linear subspaces. We
discuss the resolution of this conjecture which is joint work with Karim Adiprasito and June Huh. The solution draws on ideas from the
theory of algebraic varieties, specifically Hodge theory, showing how a
question about graph theory leads to a solution involving Grothendieck's
standard conjectures.

*Oberseminar**: Valery Alexeev and Elham
Izadi, *Wed
November 4, 2015, 3:30 – 4:30, Boyd, Room 304

Title: **What is an abelian
6-fold?**

Abelian
varieties, higher-dimensional generalizations of elliptic curves, are basic
objects in algebraic geometry, arithmetic geometry, and number theory. Over the
complex numbers, they are quotients of vector spaces by lattices. Classically,
(principally polarized) abelian varieties of low dimension g have a very
special description: for g up to 3 they are Jacobians of curves, and for g up
to 5 they are Pryms associated to curves with
involution. This implies that moduli spaces of abelian varieties for g up to 5
are unirational: they can be rationally parameterized
by g(g+1)/2 parameters. On the other hand,
Harris-Mumford proved that for g \ge 7 the moduli
spaces are of general type, which is on the opposite side of the spectrum. The
situation for g=6 has been open since the 1980s. In this work, joint with Donagi, Farkas, Ortega, we prove
a beautiful conjecture of Kanev, describing a general
abelian 6-fold as a "Prym-Tyurin" variety
for a 27:1 cover of curves with the same symmetry as the 27 lines on a cubic
surface in P3. We also make a big advance towards determining the birational type of the moduli of abelian 6-folds.

*Oberseminar** welcoming new postdocs Julian Rosen, Reza Seyyedali,
and Paul Sobaje, *Wed, August 26, 2015, 2:30 – 4:30, Boyd, Room 304

Paul
Sobaje, 2:30: **Modular representation of algebraic groups**

Let G be a
linear algebraic group over a field of positive characteristic.
We'll look at questions and methods which arise from studying the
representation theory of G by restriction to its various finite
subgroups (and subgroup schemes), in particular focusing on the theory of
support varieties for modules.

Julian
Rosen, 3:00: **Periods and multiple
zeta values **

A period is complex number
that, roughly speaking, arises as an integral of a rationally defined function
over a rationally defined region. Although periods are often transcendental,
they have lots of algebraic structure, including a (largely conjectural) Galois
theory. The multiple zeta values are a particular class of periods that arise
in many areas of pure an applied math. These periods can also be described be
infinite series, and finite truncations of these series are rational numbers
with interesting arithmetic properties. This talk will be an introduction to
periods, multiple zeta values, and their finite truncations.

Tea and Social, 3:30-4:00

Reza Seyyedali,
4:00: **Chow stability of ruled manifolds**

In 2001, Donaldson proved that
the existence of cscK metrics on a polarized
manifold (X,L) with discrete automorphism
group implies that (X,L^k) is Chow stable for k large
enough. We show that if E is a simple stable bundle over a polarized
manifold (X,L), (X,L) admits cscK
metric and have discrete automorphism group, then
(PE^*, \O(d) \otimes L^k)
is Chow stable for k large enough.

Oberseminar: Amber Russell, Mon, April 27, 2015,
3:30 – 4:30, Room 302, Boyd

Title: **Perverse Sheaves: Powerful results and related constructions**

The focus of
this talk will be on results whose proofs rely on perverse sheaves. For
example, the Fundamental Lemma and Deligne's proof of
the Weil Conjecture. I will also discuss the properties of perverse
sheaves which have made them so useful, and if time permits, I will discuss
briefly perverse coherent sheaves and parity sheaves, focusing on their
relation to perverse sheaves and the new results associated to them.

Oberseminar: Dino Lorenzini,
Wed, Feb 18, 2015, 3:45 – 4:45, Room 304, Boyd

Title: The Mathematics of Alexander Grothendieck (1928-2014)

Alexander
Grothendieck (1928-2014) died last November. He wrote
a thesis on topological vector spaces under Laurent Schwartz (Fields medal
1950) in which he completely solved a series of 14 problems published in a
paper by Dieudonné and Schwartz in 1949. Sixty
years ago in 1955, Grothendieck completely changed
his field of research and began a revolution in algebra, algebraic geometry and
number theory. I plan to discuss Grothendieck's
mathematics starting with a letter of his dated February 18, 1955.

*Oberseminar** welcoming new faculty Noah Giansiracusa, *Wed, November 19, 2014,
3:45pm – 4:45pm, Boyd, Room 304

Title:
Berkovich analytification
and the universal tropicalization

I'll
discuss joint work with my brother in which we use our theory of tropical
schemes to develop a purely algebraic (based on semirings)
view of non-archimedean analytification

Oberseminar welcoming new postdocs Abbey Bourdon,
and Patricio Gallardo, Wed, September 3, 2014, 2:30pm
– 3:30pm, Boyd 304

Abbey Bourdon: A Uniform Version of a
Finiteness Conjecture for CM Elliptic Curves

One
approach to studying the absolute Galois group is to examine its action on
other objects, such as the algebraic fundamental group of $\mathbb{P}^1_{

Patricio
Gallardo: On the parameter space of elliptic quartics
in the projective space. How can we compactify the
open set that parametrizes smooth elliptic quartics
in the projective space?

In this talk, we discuss several answers to this question, open problems, and partial new results. This is an ongoing project with C. Lozano-Huertas and B. Schmidt.

Oberseminar welcoming new postdoc Anna Kazanova,
Wed,
October 2, 2013, 3:45pm – 4:45pm, Room 303, Boyd

Title:
Degenerations
of surfaces of general type and vector bundles.

We will describe a relation between some boundary components of the moduli space of stable surfaces of general type and certain vector bundles.

Oberseminar welcoming new postdoc Joseph Vandehey, Wed, September 4, 2013,
2:30pm – 3:30pm

Title: Digit patterns in the number
of prime divisors function

The function omega(n) counts the number of distinct prime factors of n. It is well-known that omega(n) acts similar to a random variable with mean and variance log log n; or, roughly speaking, given a random n, we can guess the first half of the digits of n with high probability of being correct. But what about the rest of the digits? What can we say about them, if anything? The answer will take us from ergodic theory, through analytic and elementary number theory, asymptotic and Fourier analysis.

Oberseminar: Robert Varley, Wed, April 3, 2013,
3:30pm – 4:30pm, Room 304, Boyd

Title: Operad actions on configurations
and cohomology

The operad
concept basically took off in 1963 with J. Stasheff's
criterion for a topological space to have the homotopy
type of a loop space. His analysis of associativity up to homotopy
was expressed in terms of an A-infinity space in topology and an A-infinity
algebra on the cohomology level. Then Boardman and
Vogt used E-infinity (for "homotopy
everything") to characterize infinite loop spaces, and in 1972 J.P. May
formalized the definition of operad exactly as it
stands now, at least in the topological category. Following an introduction I
will discuss some self-contained and interesting known examples of operad actions (or compositional structures) related to (1)
moduli of Riemann surfaces with marked points, and (2) cohomology
operations. I am not planning to give the general definition of an operad in a symmetric monoidal category; the best short
introduction is probably Stasheff's piece "What
is ... an operad?" in the 2004 Notices.

Oberseminar: Dino Lorenzini,
Wed, February 6, 2013, 2:30pm – 3:30pm, Room 304 in Boyd

Title: On Kim's approach to Faltings' theorem and other diophantine
problems using fundamental groups

Abstract: A general talk
accessible to graduate students in algebra, number theory, and algebraic
geometry.

Oberseminar welcoming new postdoc Jie Wang, Wed, November 7, 2012, 3:45pm – 4:45pm,
Room 302 in Boyd

Title: Generic vanishing results on
certain Koszul cohomology
groups

Abstract: A central problem in
curve theory is to describe algebraic curves in a given projective space with
fixed genus and degree. One wants to know the extrinsic geometry of the curve,
i.e., information on the equations defining the curve. Koszul
cohomology groups in some sense carry 'everything one
wants to know' about the extrinsic geometry of curves in projective space: the
number of equations of each degree needed to define the curve, the relations
between the equations, etc. In this talk, I will present a new method using
deformation theory to study Koszul cohomology of general curves. Using this method, I will
describe a way to determine number of defining equations of a general curve in
some special degree range (but for any genus).

Oberseminar welcoming new postdoc Amber Russell,
Wed, October 3, 2012, 3:45pm – 4:45pm, Room 302 in Boyd

Title: Perverse Sheaves and the
Springer Correspondence

Abstract: Perverse sheaves were
first defined in the early 1980's, and they arise largely out of the theory of
intersection homology. They have been instrumental in results in multiple areas
of mathematics, but particularly in representation theory. In this talk, we
will begin by discussing briefly the usefulness of these objects, and then
focus on Borho and MacPherson's particular
application to the Springer Correspondence, a Lie theoretic result relating
representations of a Lie algebra's Weyl group to its nilpotent orbits.

Oberseminar welcoming new postdoc Lola Thompson,
Wed, September 5, 2012, 3:45pm – 5:15pm, Room 302 in Boyd

Title: Products of distinct cyclotomic polynomials

Abstract: A polynomial is a
product of distinct cyclotomic polynomials if and
only if it is a divisor over Z[x] of x^n-1 for some positive integer n. In this
talk, we will examine two natural questions concerning the divisors of x^n-1:
"For a given n, how large can the coefficients of divisors of x^n-1
be?" and "How often does x^n-1 have a divisor of every degree between
1 and n?" We will consider the latter question when x^n-1 is factored in
both Z[x] and F_p[x].

Oberseminar: Valery Alexeev,
Wed, February 15, 2012, 2:30pm – 4:00pm, Boyd 328

Title: Moonshine

Abstract: "All you ever
wanted to know about... Moonshine", or: "Mathieu groups, K3 surfaces,
and moonshine" This is going to be a general-audience talk about some
fascinating, mysterious and largely unexplained connections between algebra
(sporadic simple groups and their representations), number theory (elliptic
curves and modular curves), algebraic geometry (elliptic curves, K3 surfaces),
and physics (conformal field theory). The "Monstrous moonshine" was a
1979 conjecture of Conway and Norton concerning a totally unexpected connection
between the monster group (the largest simple group, of order about 10^54) and
modular functions. It was proved by Borcherds.
Mathieu group M24 is another sporadic simple group, and has order about 10^8.
It is a subgroup of S24 preserving the binary Golay
code (an error-correcting code used in digital communications) and the Witt
design, otherwise known as the Steiner system S(5,8,24).
Its definition is intimately connected with the 24 24-dimensional unimodular Niemeier lattices,
which include the fabulous Leech lattice. Subgroups of M24 stabilizing 1, resp.
2 points are called Mathieu groups M23 and M22. The connection with algebraic
geometry was discovered by Mukai in a famous 1988 paper whose main result is
that G is a finite group of symplectic automorphisms of a K3 surface iff
G is a subgroup of M23 with at least 5 orbits. Recently, physicists found the
"Mathieu moonshine", a mysterious connection between M24, elliptic
genus of a K3 surface, and mock theta functions (and black holes, of course).
Currently, no explanation for this moonshine exists.

Oberseminar:
Brian Boe and Angela Gibney, Wed, November 30,
2011, 2:30pm – 4:00pm, Boyd 303

Title: Conformal blocks

Oberseminar:
Pete Clark, Daniel Nakano, and Dino Lorenzini,
Wed, October 26, 2011, 2:30pm – 4:00pm

Title: On characteristic p problems.

Oberseminar:
Danny Krashen and William Graham, Wed,
September 21, 2011, 2:30pm – 3:30pm, Boyd 323

Title: On Langlands
Program

Abstract: The Langlands program relates geometry, number theory and
representation theory. The relation arises because given an algebraic variety,
one can define certain functions ($L$-functions) which are related to number
theory (to the number of solutions to the equations defining the variety with
coefficients in a finite field). These functions turn out to be related to
functions which are ``automorphic", i.e., functions which are (almost)
invariant under some group action; more generally, ``automorphic
representations" of the group appear. In this talk we will attempt to
explain some of the motivation and ideas involved in this relationship. The
talk should be accessible to graduate students in algebra, number theory and
algebraic geometry.