
FRAGMENT
Spring 2013


Colorado State University
Department of Mathematics Thursdays 35pm, Weber 201

University of Colorado at Boulder
Department of Mathematics
Tuesdays 35pm, 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.
Schedule of Talks:
Date

Speaker

At

Title (scroll down to view summary)

January 22 
Bryce Chriestenson
(University of Colorado)

Boulder 
The Real homotopy type of singular spaces via the WhitneydeRham complex


Feb 7 
Patrick Ingram
Renzo Cavalieri

CSU

The filled Julia set of a Drinfeld module
What the...is mirror symmetry? 
Feb 14 
Elizabeth Gross
Jose Rodriguez 
CSU 
Maximum likelihood degree of the variance components mode
Maximum Likelihood Estimation on Determinantal Varieties 
Feb 28 
Lance Miller
(U.Utah)

CSU 
Deformation of singularities 
Mar 7 2pm (note unusual time) 
Kevin Tucker
(Princeton/MSRI)

CSU 
Test Ideals of Nonprincipal Ideals 
Mar 11 23 pm
(note unusual date and time) 
Emanuele Macri
(Ohio State University)

CSU 
Minimal Model Program for moduli spaces of sheaves on K3 surfaces and Cone Conjectures 
CSU springbreak is
March 1623rd 



Boulder Spring Break is March 2529 



April 4 
Eric Miles
(CSU) 
CSU 
Quivers, Line Bundles and Bridgeland Moduli 
April 11

Jacob Tsimerman
(Harvard)

CSU 
On a Conjecture of Mazur regarding Classifying Elliptic curves over Q by their Torsion 
April 15 45pm (Monday!)
 Seyfi Turkelli
(Western Illinois)
 CSU  Lefschetz numbers of Bianchi groups and the dimension of their cohomology 
April 16 34pm
(Tuesday!)

Cassie Williams
James Madison University

CSU 
Abelian varieties with complex multiplication by nonmaximal orders 
April 25

Keerthi Madapusi Pera
(Harvard)

CSU 
The Tate conjecture for K3 surfaces in odd characteristic 
April 30 
Brian Osserman
(UC Davis)

Boulder 
Refinements and generalizations of determinantal loci 
May 2  Aaron Bertram (Utah)  CSU  
May 9 
Noah Giansiracusa
(UC Berkeley) 
CSU 
Tropical scheme theory 
Abstracts:
January 22: Bryce Chriestenson
It has long been known that one can determine the rational
homotopy type of a smooth manifold from the deRham complex. The goal of
this talk is to generalize this result to singular algebraic or analytic
sets over the real or complex numbers. I will start with a quick review of
homotopy theory, then explain what rational homotopy theory is and how it is
different than homotopy theory. This will be followed by the definition of
the WhitneydeRham complex, which will be used to achieve the desired
results.
Feb 7: Patrick Ingram
I will survey, with lots of background, a paper I wrote last year on the filled Julia set of a Drinfeld module over a local function field. Briefly, a Drinfeld module is an object arising in the arithmetic geometry of function fields of positive characteristic, and resembles in some ways an elliptic curve in the number field case. Over a local function field, one can consider the maximal compact submodule of a given Drinfeld module, which resembles in some ways both the filled Julia set of a dynamical system, and the special fibre of the Neron model of an elliptic curve. The talk will not be as inaccessible as the title makes it out to be.
Feb 7: Renzo Cavalieri
Mirror symmetry is a mysterious area straddling complex, algebraic and symplectic geomety. It is about some mathematically nontrivial equivalences of geometric invariants of algebraic/symplectic spaces that have been discovered as consequences of dualities in string theory. This talk will be vastly expository, attempting to give a layman map to at least "observe" some part of the subject. Hopefully at the very end there we'll zoom into some recent work with Andrea Brini (Imperial College) and Dusty Ross.
Feb 14: Jose Rodriguez
Maximum likelihood estimation is a fundamental computational task in
statistics and it also involves some beautiful mathematics. We discuss
this task for determinantal varieties (matrices with rank constraints)
and show how numerical algebraic geometry can be used to maximize the
likelihood function. Our computational results with the software Bertini
led to surprising duality conjectures and theorems.
This is joint work with Bernd Sturmfels and Jon Hauenstein, and joint work
with Jan Draisma.
Feb 14: Elizabeth Gross
Given a statistical model, the maximum likelihood degree (ML degree) is the
degree of the variety defined by the likelihood equations. The ML degree
measures the algebraic complexity of the maximum likelihood estimation
problem. For most models, little is known about the ML degree. In this talk,
we will introduce the variance components model, discuss the ML degree for
oneway and twoway layouts, and explain the role algebraic geometry and
computational algebra has in this investigation. We will end by describing
recent results on the problem and some questions that remain open. This is
joint work with Mathias Drton and Sonja Petrovic.
Feb 28: Lance Miller
In deformation theory, it is important to determine how the geometry of the special fiber of a fibration relates to that of the total space. In this talk, we discuss some singularity types for varieties over positive characteristic fields defined by Frobenius and how their singularities "deform"; in particular Finjectivity. This is joint work with Kazuma Shimomoto and Jun Horiuchi.
Mar 7: Kevin Tucker
Test ideals are a measure of singularities in positive characteristic, and are analogs of multiplier ideals from characteristic zero. In this talk, I will describe some recent joint work with Karl Schwede on the test ideals of nonprincipal ideals. In particular, time permitting I will discuss a description of test ideals using regular alterations, as well as positive characteristic global division theorem for test ideals.
Mar 11: Emanuele Macri
We report on joint work with A. Bayer on how one can use wallcrossing techniques to study the birational geometry of a moduli space M of Giesekerstable sheaves on a K3 surface X. In particular: () We will give a "modular interpretation" for all minimal models of M. () We will describe the nef cone, the movable cone, and the effective cone of M in terms of the algebraic Mukai lattice of X. () We will establish the so called Tyurin/Bogomolov/HassettTschinkel/Huybrechts/Sawon Conjecture on the existence of Lagrangian fibrations on M.
Apr 4: Eric Miles
Bridgeland Stability Conditions (BSC's) give a notion of stability for objects in derived categories. In general they can be difficult to construct, but there exists a class of BSC's given by representations of quivers  very handson, combinatorial gadgets. We'll use these to introduce stability and prove nice properties for a separate (but related) class of stability conditions on P^2. We'll consider the stability of line bundles for these BSC's as well as extensions of these ideas to other surfaces. This is joint work with Daniele Arcara.
Apr 11: Jacob Tsimerman
For $E$ an elliptic curve over $\C$, the torsion subgroup of E is known to be the direct sum of two copies of $\Q/\Z$. However, if $E$ is defined over $\Q$, then we additionally get an action of the absolute Galois group $\GQ$, which allows for a much richer study. Mazur conjectured that for a sufficiently large prime $p$, the torsion group $E[p]$ together with the Galois action classifies E up to isogeny. More succinctly put: isogeny classes of Elliptic curves over $\Q$ are classified by their $l$torsion group schemes. We will survey what is known about torsion in elliptic curves over $\Q$ and explain recent results regarding this conjecture , and in particular prove an analogue of it over function fields. This is joint work with B.Bakker.
Apr 15: Seyfi Turkelli
Let K be an imaginary quadratic field and O be its ring of integers. A Bianchi group is a congruence subgroup of SL(2,O).
Let G be a Bianchi group acting on a finite dimensional vector space
(over complex numbers) M. Then, by a theorem of Franke, the cohomology
group H^1(G,M) can be considered as the space of certain automorphic
forms associated to Res_{K/Q} SL_2.
Very little is known about these cohomology groups, which are in fact
finite dimensional complex vector spaces themselves. A fundamental open
question is to get a closed formula for the dimension of the spaces in
terms of the "level" of the congruence subgroup G and the "weight" of
the module M.
In this talk, after defining the necessary notation, I will survey the
results on the dimension of Bianchi groups. And, I will talk about a
result where we produce lower bounds for the dimensions of the
cohomology groups using an idea due to Harder; namely, by calculating
the corresponding Lefschetz numbers.
Apr 25: Keerthi Madapusi Pera
The Tate conjecture predicts that, given a smooth, projective variety X
over a finitely generated field (a number field, finite field, or
function field over such), we can deduce a great deal about it by
studying its ladic cohomology, essentially reducing many questions
about the variety to problems in linear algebra. In particular, it says
that the rank of the NeronSeveri group of X can be computed as the
dimension of the largest subspace of H^2(X) on which the Galois group
acts via the ladic cyclotomic character. When X is an abelian variety
over a finite field, Tate proved this last assertion in the 60s, by
showing that it has an endomorphism ring of the expected dimension. We
will show how his proof (or a reformulation of it) can be leveraged to
prove the Tate conjecture for divisors on K3 surfaces. The key input is a
miraculous construction (called the KugaSatake construction) that
attaches to any K3 surface X, an abelian variety A such that divisors on
X correspond to certain endomorphisms on A.
April 30: Brian Osserman
We present some results on classical determinantal loci which describe their behavior in terms of kernels of the associated maps. This suggests that there may be an alternative definition of determinantal loci expressed in terms of kernels rather than determinants, although thus far such a definition is elusive. We use these ideas to develop a generalization of determinantal loci to pushforwards of sheaves under proper morphisms, and sketch how this can in turn be used in BrillNoether theory.
May 8: Noah Giansiracusa
I'll discuss joint work with Jeff Giansiracusa (U. Swansea) in which we describe a framework for producing/studying equations cutting out tropical varieties. This entails working with an extension of scheme theory based on semirings rather than rings, developed by various authors in the context of the "field with one element". We construct a schemetheoretic tropicalization functor sending closed subschemes of a toric variety over a valued field to closed subschemes of the corresponding tropical toric variety. Upon restricting to Tvalued points this recovers Payne's tropicalization functor. We show that for projective subschemes the Hilbert function is preserved under tropicalization, thereby revealing a hidden flatness in the degeneration sending a variety to its polyhedral skeleton.
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 , Spring 11 here and Fall 11 here! And here go Spring 12 and Fall 12.




