Department of Mathematics
Graduate Colloquium: Spring 2009

Graduate Colloquium
Spring 2009
Tuesdays, 4:35 - 5:35 PM, JWB 335
Math 6960-001
(credit hours available!)

GSAC Home | Past Graduate Colloquia

The goal of this Colloquium is to encourage interaction among graduate students, specifically between graduate students who are actively researching a problem and those who have not yet started their research. Speakers will discuss their research or a related introductory topic on a level which should be accessible to nonspecialists. The discussions will be geared toward graduate students in the beginning of their program, but all are invited to attend. This invitation explicitly includes undergraduate students.

January 20
Speaker: Kevin Wortman
Title: The Large-scale Geometry of Groups
There is an ongoing program in geometric group theory to classify all finitely generated groups using metric spaces that are inherently associated with them. I'll explain what this metric is, along with why the program is of current interest in mathematics.

January 27
Speaker: Ben Trahan
Title: Godel's Incompleteness Theorem
One of the major problems of the 19th and early 20th centuries -- indeed, it was Hilbert's Second Problem -- was to enumerate a set of axioms which described mathematics both completely and consistently. In 1931, Godel proved that it couldn't be done -- any non-trivial, consistent formal system will include sentences that cannot be proven or disproven. He then took the result further and proved that, in fact, no consistent system can prove its own consistency. In this talk, I will describe the proof of this surprising but extremely accessible result.

February 3
Speaker: Scott Crofts
Title: The Peter-Weyl Theorem
In this talk, I will introduce some of the basic ideas in the representation theory of compact Lie groups. In particular, I will discuss the Peter-Weyl theorem which relates a (canonical) space of L^2 functions on a group to its representation theory. Properly interpreted, the Peter-Weyl theorem gives a generalization of the classical theory of fourier series.

February 10
Speaker: NONE
Title: Special Colloquium
Special Colloquium

February 17
Speaker: Aaron Bertram
Title: Counting Points on Curves
I will talk about curves in the plane of the form y^2 = x^d + n, where d and n are integers. How many points are there with rational coordinates? How many points are there with (mod p) coordinates? What do the real points look like (easy!)? What do the complex points look like? How are they all related? How does all this depend upon d and n? What happens when we throw in more variables and more terms?

February 24
Speaker: Tim Carstens
Title: On the Set {p : p prime}
A good deal of number theory is concerned with the following heuristic conjecture: everything which isn't obviously forbidden should happen in the primes. We'll look at some strange examples of this principle in action and talk about ways in which the principle can be rigorized in the form of modern conjectures.

March 6 (Friday)
Speaker: Matt Housley
Title: The Axiom of Choice: Intuition and Paradox
The Axiom of Choice and the Zermelo~VFraenkel axioms of set theory form the generally accepted logical foundation of modern mathematics. In this talk, I will discuss some of the Axiom's convenient and disturbing consequences along with the strengths and perils of some alternative set theories.

March 10
Speaker: Will Malone
Title: Isometries of Products of Euclidean Metric Spaces are Reducible
Give a product P of Euclidean metric spaces the sup metric. There are two obvious types of isometries from P to itself namely a product of isometries and a reindexing. In this talk we will show two things. The first is that the number of Euclidean spaces in P is an isometry invariant. The second is that isometries from P to any product of Euclidean spaces with the sup metric is a composition of the two obvious isometries.

March 17
Title: Spring Break
Spring Break

March 24
Speaker: NONE
Title: Campus Closed
Campus Closed

March 31
Speaker: Stefano Urbinati
Title: Some Easy Projective Geometry
I will consider geometry on the projective plane. To this end, I'll introduce different definitions and explain why it is better to work with projective objects whenever you need to talk about intersection theory.

April 7
Speaker: Chris Remien
Title: The Blurring Effect of Bundling Hair in Stable Isotope Studies: implications and inversion techniques
Stable isotope ratios in body tissues can be used to provide insight into an animal's position in a trophic web, diet, movement and migration, and other ecological parameters. The basic idea is that you are what you eat and drink. Through serial sampling of directionally growing tissues, it is possible to obtain time dependent information. The hair of many animals is not thick enough to sequentially sample individual hair strands, thus bundling multiple strands is necessary to obtain high-resolution time dependent isotopic data. If each hair within the bundle does not grow at exactly the same rate, an alignment problem arises with regard to the isotopic signal. By formulating this as an integral equation, we can quantify the effect of bundling multiple hairs when sampling. We then present an inverse method to reconstruct the input signal given the bundled hair signal.

April 14
Speaker: Masaki Iino
Title: Lagrangian Mechanics and Calculus of Variations
In undergraduate physics classes, classical mechanics formulated by Issac Newton is studied. The Newtonian mechanics is mathematically very straightforward and applied to a wide variety of problems, but this is not a unique formulation of mechanics. We will discuss and compare other common alternative formulations of mechanics: Lagrangian mechanics and Hamiltonian mechanics.

After the brief comparison of these formulations of mechanics, we will give a closer look at the Lagrangian mechanics by defining action so as to answer the following fundamental question: "Why does the nature like to minimize the action?"

In order to actually minimize the action and solve other important problems such as the Brachistochrone problem (i.e., find the curve along which a particle takes the least time to travel between two points) and isoperimetric problem (i.e., determine a plane figure of the largest possible area whose boundary has a specified length), the idea of Calculus of Variations has been developed. In this talk, we will derive the central equation of Calculus of Variations--the Euler-Lagrange Equation (a necessary condition for a functional to have an extremal) and illustrate it with an example of a Catenary curve problem by using the method of Lagrange Multipliers.

April 21
Speaker: Peter Kim
Title: Modeling Imatinib-Treated Chronic Myelogenous Leukemia: Reducing the Complexity of Agent-Based Models
We develop a model for describing the dynamics of imatinib-treated chronic myelogenous leukemia. Our model is based on replacing the recent agent-based model of Roeder et al. (2006) by a system of deterministic difference equations. These difference equations describe the time-evolution of clusters of individual agents that are grouped by discretizing the state space. Hence, unlike standard agent-base models, the complexity of our model is independent of the number of agents, which allows to conduct simulation studies with a realistic number of cells. This approach also allows to directly evaluate the expected steady states of the system. The results of our numerical simulations show that our model replicates the averaged behavior of the original Roeder model with a significantly reduced computational cost. Our general approach can be used to simplify other similar agent-based models. In particular, due to the reduced computational complexity of our technique, one can use it to conduct sensitivity studies of the parameters in large agent-based systems.

April 28
Speaker: Mike Purcell and Britt Bannish
Title: End of Year Party

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