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   FRAGMENT                

 Spring 2013                

 



Colorado State University 
Department of Mathematics Thursdays 3-5pm, Weber 201

 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.


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 Whitney-deRham 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 Non-principal Ideals
 Mar 11   2-3 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 16-23rd



 Boulder Spring Break is March 25-29      
 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 4-5pm
(Monday!)
 Seyfi Turkelli
(Western Illinois)
 CSU Lefschetz numbers of Bianchi groups and the dimension of their cohomology
 April 16 3-4pm
(Tuesday!)
 Cassie Williams
James Madison University
 CSU  Abelian varieties with complex multiplication by non-maximal 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

Logistics:

·        CU Boulder Campus map

·        CSU Campus map


 

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 Whitney-deRham 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
one-way and two-way 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 F-injectivity. 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 non-principal 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 wall-crossing techniques to study the birational geometry of a moduli space M of Gieseker-stable 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/Hassett-Tschinkel/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 hands-on, 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 l-adic cohomology, essentially reducing many
questions about the variety to problems in linear algebra. In particular, it says that the rank of the Neron-Severi group of X can
 be computed as the dimension of the largest sub-space of H^2(X) on which the Galois group acts via the l-adic 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 Kuga-Satake 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 Brill-Noether 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 scheme-theoretic 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 T-valued 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.