## University of Colorado at Boulder Department of Mathematics Tuesdays 3-5pm, Math 350

This is a seminar series intended to involve people in the Front Range interested in Algebra, GeoMEtry and Number Theory. The seminar will meet either at CSU (green) or at CU Boulder (gold); look on this page for specific information.

For more information, or to provide a speaker, contact Renzo, renzo AT math.colostate.edu, or Yano, casa AT math.colorado.edu.

NEW FRAGMENT PAGE HERE

## Schedule of Talks:

 Date Speaker At Title (click on title to view summary) August 21 David Zureick-BrownEmory University CSU Abelian varieties with big monodromy August 30 Chris Hall(University of Wyoming) CSU Hyperelliptic Curves with Big Monodromy September 13 Dusty Ross(CSU) CSU The Gerby Gopakumar-Marino-Vafa Formula (and the Riemann Hypothesis) September 182-4 PM Nicholas Proudfoot  (U. Oregon) Boulder Toric and hypertoric varieties September 27 Takashi Kimura(Boston University) CSU Operations on the cohomology and K-theory of manifolds with group actions October 4 Paul Johnson(Columbia U./ CSU) CSU Hurwitz numbers of orbifolds and quasimodular forms. October 11 Aaron Bertram(U.Utah) CSU Some Lowbrow Tropical Plane Geometry October 18   THURSDAY MATH 220 Boulder double headerJulia Gordon (UBC)Jonathan Wise (Boulder) Boulder How to get uniform in "p" boundsandWhat is a commutative ring? October 25 David Ayala(USC) CSU Excisive invariants of 3-manifolds, links, and other singular spaces November 1 Euclid(U. Alexandria) Parallel lines don't meet. Nor do we on math day. November 6 David Jensen(SUNY Stony Brook) Boulder The Hassett-Keel Program in genus four November 8    3-4 PM Olivia Dumitrescu(UC Davis) CSU On the F-Nef conjecture on $\bar M_{0,n}$ November 13   4-5 PM Katherine Stange(Boulder) Boulder The sensual Apollonian circle packing November 15 Korben Rusek CSU $A$-Discriminant Amoebae and the Horn UniformizationThe $A$-Discriminant Amoeba Structure November  29  3-4 PM 4-5 PM David Eklund - CSUBo-Hae ImChung-Ang U. / Indiana U. CSU EigenschemesQuadatic Twists of Elliptic Curves December 4 Bhargav Bhatt(IAS) Boulder Comparison theorems in p-adic Hodge theory December 13 Rob MaierUniversity of Arizona Boulder

## Abstracts:

August 21: David Zureick-Brown

Serre proved in 1972 that the image of the adelic Galois representation associated to an elliptic curve E without complex multiplication has open image; moreover, he also proved that for an elliptic curve over Q the index of the image is always divisible by 2 (and in particular never surjective). More recently, Greicius in his thesis gave criteria for surjectivity and gave an explicit example of an elliptic curve E over a number field K with surjective adelic representation. Soon after, Zywina, building on earlier work of Duke, Jones, and others, proved that the adelic image random'  elliptic curve is maximal.

In this talk I will explain recent joint work with David Zywina in which we generalize these theorems and prove that a random abelian variety in a family with big monodromy has maximal image of Galois. I'll explain what big monodromy and maximal mean an explain the analytic and geometric techniques used in previous work and the new geometric ideas -- in particular, Nori's method of semistable approximation-- needed to generalized to higher dimension.

August 30: Chris Hall

Let $A$ be a principally-polarized abelian variety of dimension $g$ over a field $K$.  The field extensions $K_\ell=K(A[\ell])$ obtained by adjoining $\ell$-torsion for each prime $\ell$, and we say $A$ has {\it big monodromy} if the Galois group of $K_\ell/K$ contains $\Sp_{2g}(\F_\ell)$ for almost all $\ell$.  For example, if $f(x)$ is a square-free polynomial with complex coefficients and of degree $2g$, then we will show that the Jacobian of the hyperelliptic curve $y^2=f(x)(x-t)$ over the function field $\C(t)$ has big monodromy.

The first half of this talk should be accessible to graduate students.

September 13: Dusty Ross

The Gopakumar-Marino-Vafa formula, proven almost 10 years
ago, evaluates certain triple Hodge integrals on the moduli space of
curves in terms of Schur functions.  It has since been realized the
the GMV formula is a special case of the
Gromov-Witten/Donaldson-Thomas correspondence.  We prove an orbifold
generalization of the GMV formula and use it to deduce the orbifold
GW/DT correspondence for local Z_n gerbes over the projective line.

In the first hour of this talk, I will give a broad overview of curve
counting invariants and the GW/DT correspondence, paying particular
attention to the toric case as that is where our motivation lies.  I
will introduce the main objects which appear in the gerby GMV formula:
loop Schur functions, abelian Hodge integrals, and generalized
symmetric groups, and give a statement of the theorem.  Given time, I
will conclude by showing how the formula can be used to deduce the
GW/DT correspondence on a certain class of orbifolds.  This hour
should be accessible to a broad audience.

In the second hour I will discuss some of the main ideas used in the
proof of the formula.

September 18: Nicholas Proudfoot

If you ask somebody what a toric variety is, that person might say "it's a variety that you associate to a polyhedron", or perhaps "it's a variety that you associate to a fan".  I will explain what's behind these two answers and how they are related.  I'll also talk about hypertoric varieties, in which polyhedra are replaced by hyperplane arrangements and fans are (conjecturally) replaced by pointed oriented matroids.

The last part is work in progress with Matthew Arbo.

September 27: Takashi Kimura

If X is a smooth variety with a proper action of an algebraic
group G (e.g. G is a finite group action on X) then the equivariant cohomology
$H^*_G(IX)$ of the inertia variety has an orbifold product introduced by Chen-Ruan
as a kind of classical limit of orbifold Gromov-Witten theory of $[X/G]$. The orbifold product is the
prototypical example of an inertial product on $H^*_G(IX)$. We will explain
how, under certain conditions, inertial products on equivariant K-theory have
compatible inertial versions of power operations, Chern classes, and the
Chern character. We then use these power operations to provide a simple
presentation of the virtual K-theory ring of P(1,2) and P(1,3) and compare the
results with the ordinary K-theory ring of a crepant resolution.

The first hour will give a  presentation of the main ideas and constructions aimed at beginning graduate students.

October 4: Paul Johnson

Enumerative geometry counts geometric things satisfying certain
conditions.  For example, Schubert calculus counts subspaces of vector
spaces; Hurwitz numbers count ramified covers of Riemann surfaces, or
alternatively certain sets of permutations.  What is of interest is
patterns among these counts, which often manifest themselves as nice
properties of their generating functions.

For example, if X is a variety with Euler characteristic zero, and
F(g,X) is the generating function counting genus g curves in X'',
then it is a general principle that F(g,X) should be a quasimodular
form.  We verify this principle when $X$ is an orbifold curve and the
counting problem is Hurwitz theory.  The main tool in the proof is the
representation theory of the symmetric group.  The majority of the
talk will be background and context; the whole talk should be

October 11: Aaron Bertram

Algebraic geometry begins to get interesting when one considers
nine points on an elliptic curve and asks whether or not the
curve is the unique elliptic curve to pass through those points. The answer
depends on whether the sum of the points is trivial'' in the Jacobian.
What happens tropically? Somewhat disappointingly, but interestingly, the condition for nine (general) points on a tropical elliptic curve to determine the curve is topological. This is based on joint work with Dylan Zwick, Lance Miller and Drew Johnson.

October 18: Julia Gordon

This talk is about a method, based on model theory,  that can be used
to get some uniform in "p" estimates for integrals over p-adic fields
in the cases when it is hard to see directly how such integrals behave
for different places p.

One example of such a situation is orbital integrals on p-adic groups.
Let  $G$ be a reductive p-adic group (such as the group $GL_n(\Q_p)$).
Harish-Chandra introduced the notion of an orbital integral on $G$,
and proved that the orbital integrals, normalized by the discriminant,
are bounded, in a certain sense.
However, it is not easy to see how this bound behaves if we let the
$p$-adic field vary (for example, if  the group $G$ is defined over a
number field $F$, and we consider the family of groups
$G_v=G(F_v)$, as v runs over the set of finite places of $F$).
Using a method based on model theory and motivic integration, we
prove that the bound on orbital integrals can be taken to be a fixed
power (depending on $G$) of the cardinality of the residue field. This
statement has an application to the recent work of S.-W. Shin and N.
Templier on counting zeroes of L-functions.
This project is joint work with R. Cluckers and I. Halupczok.

October 18: Jonathan Wise

I'll describe some categories arising from deformation theory
that behave in many ways like commutative rings.  Then we'll discuss how to
axiomatize the situation to arrive at a definition of "higher commutative
rings".  At the end, we'll talk about what this has to do with "derived
algebraic geometry".  I'll assume some familiarity with commutative algebra
and basic category theory, but not much else.

October 23: David Ayala

Various interesting examples of invariants of 3-manifolds and links satisfy a local-to-global expression.
In this talk, we will axiomatize invariants satisfying such an expression and classify them in terms of algebraic data.
Through this language there is a refinement of Poincar\'e duality which leads to some interesting conclusions.
A few examples will be explicated.
This is a report on joint work with John Francis and Hiro Tanaka.

November 6: David Jensen

The Hassett-Keel program aims to give modular interpretations of log canonical models for the moduli spaces of curves.  The program, while relatively new, has attracted the attention of a number of researchers, and has rapidly become one of the most active areas of research concerning the moduli of curves.  In the genus four case, we construct several of these models as GIT quotients of a single, elementary space.  After discussing our construction, we will consider how we expect these ideas to generalize to the general genus case.

November 8: Olivia Dumitrescu

We present the connection between Fulton's conjecture on the cone of nef divisors for $\bar M_{0,n}$ and

Harbourne-Hirshowitz's conjecture for $\bar M_{0,n}$. In particular, we reduce the F-nef conjecture for $\bar M_{g}$ to a combinatorial question.

November 13: Katherine Stange

The curvatures of the circles in integral Apollonian circle
packings, named for Apollonius of Perga (262-190 BC), form an infinite
collection of integers whose Diophantine properties have recently seen
a surge in interest. Here, we give a new description of Apollonian
circle packings built upon the study of the collection of bases of
Z[i]^2, inspired by, and intimately related to, the sensual quadratic
form' of Conway.
November 15: Korben Rusek

Talk 1
$A$-Discriminant Amoebae and the Horn Uniformization

Let $A\subset\mathbb{Z}^n$ be a collection of $n+m+1$ exponent vectors. The
$A$-discriminant variety is the collection of polynomials with support $A$ with
degenerate roots. The $A$-discriminant amoeba is the component-wise logarithm of
this variety. This variety (as well as the resulting amoeba) admits an elegant
and computationally efficient parametrization called the Horn uniformization
map. We also give various examples of recent applications of the $A$-discriminant
variety from real polynomial solving to satellite orbits.

Talk 2
The $A$-Discriminant Amoeba Structure

We discuss the structure of $A$-discriminant amoebae in more depth. We focus on
the case where $A$ is a collection of $n+3$ points in $\mathbb{Z}^n$ in general
position. We will see how these discriminant amoebae have surprisingly simple
structure. A few simple observations will lead to new and tight lower and upper
bounds on the number of connected components in the complement of the amoeba. We
will also discuss tight bounds on the number of cusps in such Amoebae.

Both talks should be accessible to graduate students.

November 29: David Eklund

This talk is about some aspects of linear algebra seen from a non-linear algebra perspective. Eigenvectors and generalized eigenvectors to square matrices with complex entries can be packed into subschemes of projective space in a natural way. These schemes have an interesting geometry reflecting the underlying linear algebra. In this talk I will explain an application of this formalism, namely a numerical method to compute the Jordan normal form of matrices. The method operates via numerical homotopies and their associated deformations, which provides a link to intersection theory in algebraic geometry. Joint work with Chris Peterson.

November 29: Bo-Hae Im

Title: Quadratic twists of elliptic curves
(Bo-Hae Im, Chung-Ang University / Indiana University)
Abstract: First of all, if we have three elliptic curves $E_i$ over a number field $K$, we prove that there exists a number field $L$ over $K$ there exist infinitely many $d\in L$ such that the rank of quadratic twists by $d$ of each $E_i$ are positive.
Secondly, as a joint work with Michael Larsen, we talk about elliptic curves over a  number field $K$ with the property that for any products $D$ of $d_i$ in a finite set of $d_i$ in $K$, the rank of the quadratic twist by $D$ of $E$ is positive.

December 4: Bhargav Bhatt

A basic theorem in Hodge theory is the isomorphism between de Rham and

Betti cohomology for complex manifolds; this follows directly from the

Poincare lemma. The p-adic analogue of this comparison lies deeper,

and was the subject of a series of extremely influential conjectures

made by Fontaine in the early 80s (which have since been established

by various mathematicians). In my talk, I will first discuss the

geometric motivation behind Fontaine’s conjectures, and then explain a

simple new proof based on general principles in derived algebraic

geometry — specifically, derived de Rham cohomology — and some

classical geometry with curve fibrations. This work builds on ideas of

Beilinson who proved the de Rham comparison conjecture this way.

In previous semesters the seminar page was maintained by Renzo Cavalieri, Rachel Pries, Jeff Achter and Yano Casalaina. You can find the Fall 09 page here, and the Spring 10 here, Fall 10 here  and Spring 11 here! Then there is Fall 11 and Spring 12.