Graduate Colloquium: Spring 2009

Spring 2009

Tuesdays, 4:35 - 5:35 PM, JWB 335

Math 6960-001

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

**Abstract:**

**January 27**

**Speaker: **Ben Trahan

**Title:** Godel's Incompleteness Theorem

**Abstract:**

**February 3**

**Speaker: **Scott Crofts

**Title:** The Peter-Weyl Theorem

**Abstract:**

**February 10**

**Speaker: **NONE

**Title:** Special Colloquium

**Abstract:**

**February 17**

**Speaker: **Aaron Bertram

**Title:** Counting Points on Curves

**Abstract:**

**February 24**

**Speaker: **Tim Carstens

**Title:** On the Set {p : p prime}

**Abstract:**

**March 6 (Friday)**

**Speaker: **Matt Housley

**Title:** The Axiom of Choice: Intuition and Paradox

**Abstract:**

**March 10**

**Speaker: **Will Malone

**Title:** Isometries of Products of Euclidean Metric Spaces are Reducible

**Abstract:**

**March 17**

**Speaker:**NONE

**Title:** Spring Break

**Abstract:**

**March 24**

**Speaker: **NONE

**Title:** Campus Closed

**Abstract:**

**March 31**

**Speaker: **Stefano Urbinati

**Title:** Some Easy Projective Geometry

**Abstract:**

**April 7**

**Speaker: **Chris Remien

**Title:** The Blurring Effect of Bundling Hair in Stable Isotope Studies: implications and inversion techniques

**Abstract:**

**April 14**

**Speaker: **Masaki Iino

**Title:** Lagrangian Mechanics and Calculus of Variations

**Abstract:**

**April 21**

**Speaker: **Peter Kim

**Title:** Modeling Imatinib-Treated Chronic Myelogenous Leukemia: Reducing the
Complexity of Agent-Based Models

**Abstract:**

**April 28**

**Speaker: ** Mike Purcell and Britt Bannish

**Title:** End of Year Party

**Abstract:**

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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.

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.

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.

Special Colloquium

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?

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.

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.

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.

Spring Break

Campus Closed

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.

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.

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.

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.

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.

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