Greenslopes Fall 2011
Colorado State University Department of Mathematics

Greenslopes Seminar, Fall 2011

Thursdays at 1:00 in Weber 223.

To accomodate speakers, variations may be made and annotated below. All abstracts are listed at the bottom of this page.

Previous Semester Information

NOTE: If you are interested in speaking, contact Eric Hanson or Steven Ihde to volunteer.

For the seminar attendance sheet, click here.
Date Speaker Title Advisor
August 25
Kenneth Monks
Finding Complements in Groups

Alexander Hulpke
September 1
Steven Ihde
Preconditioning Using H-Basis Computations

Dan Bates
September 8
Dustin Ross
Stacks?

Renzo Cavalieri
September 15
Anne Ho
A Brief Overview of Origami Numbers

Rachel Pries
September 22
Francis Motta
Persistent Homology and Circle Map Dynamics

Patrick Shipman
September 29
Dustin Ross
Counting like an algebraic geometer

Renzo Cavalieri
October 6
Cassie Williams
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Jeff Achter
October 13
Eric Hanson
Numerical Algebraic Geometry, Fiber Products, and Applications

Dan Bates
October 20
Hilary Smallwood
Endomorphisms of Abelian Varieties over Finite Fields and Characteristic Polynomials of Frobenius

Jeff Achter
October 27
Michael Mikucki
Unraveling R0: From Ecology to Mathematics

Simon Tavener
November 3
MATH DAY

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November 10
Anthony (Drew) Schwickerath

Markov Random Fields: Think Locally -----
November 17
Justin Hughes

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November 24
FALL RECESS

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December 1
Jaime Shinn

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December 8
OPEN

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Abstracts

August 25: Finding Complements in Groups - Kenneth M. Monks
The problem of group extension is very hard in general. However in the case where the subgroup is abelian, we can call upon the great noble Linear Algebra to save us. I'll explain how to do this as well as work out an example I needed for my research, complements to a subgroup of the base group of a wreath product!

September 1: Preconditioning Using H-Basis Computations - Steven Ihde
In homotopy continuation we often end up with large numbers of paths that diverge to infinity. We want to precondition a polynomial system in order to cut out these infinite paths. The method that we use involves computations using H-bases first discovered by Macaulay. We move to a dual space and find a nice basis in order to remove the extraneous information. We we dualize again, we move back into the original space and have removed the infinite paths. I will introduce homotopy continuation, H-bases, Macaulay matrices and dual spaces.

September 8: Stacks? - Dustin Ross
Stacks are fundamental (yet notoriously abstract) players in modern algebraic geometry. This talk will be an attempt to answer the two major questions regarding any mathematical object: what is it and why do we care? I will motivate stacks through an example driven discussion of moduli 'spaces'. We will soon discover that the classical notion of 'space' is insufficient for the task at hand. I will define stacks, give examples, and show that the natural object we were previously looking for is that of a moduli 'stack'. I will not assume any knowledge of algebraic geometry.

September 15: A Brief Overview of Origami Numbers - Anne Ho
With some basic group and field theory, we can prove that certain straightedge and compass constructions are possible whereas others are not. We can use the additional tool of origami to make some of these impossible constructions possible. I will introduce constructible and origami numbers, their respective fields, and I will discuss the classic example of trisecting an angle. Paper-folding skills useful but not required.

September 22: Persistent Homology and Circle Map Dynamics - Francis Motta
Understanding the topology of a set carved out by a collection of equations or prescribing some topological characteristics to a set of noisy data motivated the development of persistent homology. At its core this tool is a method of visualizing data which may lie elusively in dimensions greater than three and in particular when that data is only a coarse or noisy sampling of some underlying space. Here this tool is utilized for and beyond its original purpose; said roughly, to describe coarse topological features of data, which in our case is generated by dynamical systems.

By its design and intended application the output of persistent homology, that is the strategy of visualization it renders (called a barcode), tends to require interpretation. However we shall see through example that this tool can reveal, for appropriately chosen data, stunning regularity not provisional to the conclusions of a trained eye. In fact persistent homology led us to a rediscovery and new characterization of a well-known and beautiful observation of the distribution of points under irrational rotations of the circle.

September 29: Counting like an algebraic geometer - Dustin Ross
This talk will give an introduction into modern algebro-geometric techniques in enumerative geometry. Mostly it will be about the sequence 1, 1, 12, 620, 87304,...

October 6: (Title) - Cassie Williams
(Abstract)

October 13: Numerical Algebraic Geometry, Fiber Products, and Applications - Eric Hanson
Numerical Algebraic Geometry (NAG) takes as one of its primary goals the solving of large systems of polynomial equations. I will give some basic background on the field and some software that is used to solve such systems. The concept of a Fiber Product will also be introduced and explained by example on a variety, V(f), and the projection map. Lastly I will discuss an application of these methods to Kinematics.

October 20: Endomorphisms of Abelian Varieties over Finite Fields and Characteristic Polynomials of Frobenius - Hilary Smallwood
In 1966 John Tate published a paper ''Endomorphisms of Abelian Varieties over Finite Field''. The main result of this paper states that there is a bijection between k-homomorphisms between two abelian varieties and G-homomorphisms of their respective Tate modules. In my talk we will explore the proof of Tate's main theorem, applications of this result, and the special case of when the abelian varieties are elliptic curves.

October 27: Unraveling R0: From Ecology to Mathematics - Michael Mikucki
In disease ecology, researches want to know which parameters have the largest affect on the model. While forward sensitivity analysis describes this well, it can be increasingly difficult to compute as models become more complex. SENSAI is a software that automatically computes the forward sensitivity analysis of any equation in the model, including any user defined quantity of interest (QoI). One important QoI that often arises in disease ecology is the basic reproductive number R0. While this QoI has a nice biological meaning, it does not have a strong mathematical definition. This talk will provide a concrete mathematical definition for R0, allowing for automatic sensitivity computation through SENSAI.

November 10: Markov Random Fields: Think Locally - Anthony (Drew) Schwickerath
Whether because of noisy data or lack of knowledge, I regularly encounter applications involving randomness. Markov random fields provide a principled way of formulating problems involving both randomness and spacial locality. In this talk, I will present an introduction to Markov random fields motivated by a few applications in image analysis. This is a self contained talk, so very little knowledge of probability theory is required.

November 17: (Title) - Justin Hughes
I will discuss the initial goal of my research which was to create the concept of a group action on the homology of certain chain complexes generated from Cayley graphs. The first step towards this goal is to create the Cayley graph. From there we construct the neighborhood complex of the Cayley graph and then the corresponding chain complex. By acting with a group element on the Cayley graph we induce a action the homology of the corresponding chain complex.

Past "Speed Presentations": Spring 2011


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Original page design by Holger Kley.
Additional thanks to Cassie Williams and Lori Ziegelmeier for passing the torch.

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