**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: Dino Lorenzini, *Wed Nov 1, 2017, 3:45
– 4:45, Boyd, Room 304

Title:
**The Direct Summand
Conjecture**

The
Direct Summand Conjecture, a 1973 conjecture of Hochster, was recently proved
by Andre. This conjecture is one of the many related Homological Conjectures,
and this talk will be a brief survey on this topic for non-experts.

*Oberseminar welcoming new postdocs Chun-Ju
Lai, Nikon Kurnosov, and Alex Stathis*,
Wed August
30, 2017, 2:30 – 4:30, Boyd, Room 304

*Chun-Ju Lai*, 2:30: **From Schur-Weyl duality to quantum
symmetric pairs**

*The famous Schur-Weyl duality is a double
centralizer property between the symmetric groups and the general linear
groups/Lie algebras bypassing Schur algebras. It plays a fundamental role in
early development of representation theory around a century ago, and there are
still novel mathematical ideas that can be drawn from a Schur-type duality.
Around 1985, Jimbo introduced a quantized duality between the Hecke algebras
and the quantum groups via the q-Schur algebras. In contrast, it is made
possible to construct quantum groups from the family of q-Schur algebras by
Beilinson, Lusztig and MacPherson(BLM). In a seemingly unrelated direction,
Letzter and Kolb developed a theory quantizing the symmetric pairs consisting
of a Lie algebra and its fixed-point subalgebra. The objects constructed are
called the quantum symmetric pairs, including examples arising from the
reflection equations, the Onsager algebras from Ising model, the (twisted)
Yangians. In this talk I will provide examples of certain Schur-type dualities
beyond type A, and exhibit a new family of quantum symmetric pairs in terms of
the algebras constructed a la BLM. I will summarize with applications in
representation theory.*

*Nikon Kurnosov, 3:00*: **Hypekahler manifolds: restrictions
and subvarieties**

*A triple of complex Kahler structures gives
us a hyperkahler manifold. And a number of questions arise naturally - what are
examples of hyperkahler manifolds and their "good" subvarieties? I
will introduce my interest in this questions.*

*Tea and Social: 3:30-4:00*

*Alex Stathis*, 4:00: **The Hilbert Scheme of Points in the
Projective Plane and its Intersection Theory**

I
will introduce the Hilbert scheme of points, give the background and motivation
for the work conducted in my thesis, and finish by stating my results. This talk
is intended to be accessible to nonalgebraic geometers.

*Oberseminar welcoming new postdocs Ben Lund,
Scott Mullane, and Kei Yuen Chan, *Wed, August 23, 2017, 2:30 – 4:30, Boyd, Room 304

*Ben Lund*, 2:30: **Flats determined by points**

*Start with a set of n points in the real
plane, and draw a line through each pair. How many lines have you drawn? In
1948, de Bruijn and Erdos showed that this number is either 1, or at least n.
In 1983, Beck showed that either nearly all of the points lie on a single line,
or the number of lines is a constant fraction of n^2. I will discuss these
results, along with their generalization to higher dimensions.*

*Scott Mullane, 3:00*: **Flat geometry, the strata of abelian
differentials and the birational geometry of M_g,n.**

*An abelian differential defines a flat metric
with singularities at its zeros and poles, such that the underlying Riemann
surface can be realized as a polygon whose edges are identified pairwise via
translation. A number of questions about geometry and dynamics on Riemann
surfaces reduce to studying the strata of abelian differentials with prescribed
number and multiplicities of zeros and poles. After introducing abelian
differentials or flat surfaces, we'll discuss how flat surfaces degenerate and
my interest in how flat geometry informs the birational geometry of the
underlying moduli spaces of Riemann surfaces.*

*Tea and Social: 3:30-4:00*

*Kei Yuen Chan*, 4:00: **Dirac cohomology versus homological
properties for graded affine Hecke algebras**

Dirac operator has its
origin in the study of quantum mechanics. It has been applied in the
representation theory of reductive groups to realize discrete series by the
work of Parthasarathy and Schmid. The notion of Dirac cohomology was introduced by Vogan
along with a deep conjecture relating to the infinitesimal character of
Harish-Chandra modules. The conjecture has been later proved by Huang-Pandzic.

Graded affine Hecke
algebras have been a useful tool in the study of the representation theory of
p-adic groups. Motivated from analogies between real groups and p-adic groups,
Barbasch-Ciubotaru-Trapa generalized the notion of Dirac cohomology to the
setting of graded affine Hecke algebras. In this talk, I shall explore
connections between the Dirac cohomology and homological properties for the
modules of graded Hecke algebras, centraling around some of my results.

*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.