Graduate Colloquium: Fall 2007

Fall 2007

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.

**August 28**

**Speaker: ** Mike Purcell

**Title:** Sounds Like Math

**Abstract:**

**September 4**

**Speaker: Amber Smith**

**Title:** Presentations Using LaTEX: The Beamer Class

**Abstract:**

**September 11**

**Speaker: Zack Kilpatrick**

**Title:** Traveling pulses and wave propagation failure in inhomogenous neural media.

**Abstract:**

**September 18**

**Speaker: **Andrew Nelson

**Title:** Sparse Approximations and High Dimensional Geometry

**Abstract:**

**September 25**

**Speaker: ** Jason Preszler

**Title:** Leopoldt's Conjecture

**Abstract:**

**October 2**

**Speaker: Julian Chan**

**Title:** Sicherman Dice

**Abstract:**

**October 16**

**Speaker:** Bill Casselman

**Title:** Dirichlet Does Distributions

**Abstract:**

**October 23**

**Speaker: **Damon Toth

**Title:**Dynamics of Age-Structured Populations in a Chemostat

**Abstract:**

**October 30**

**Speaker: ** Dylan Zwick

**Title:** The Principle of "Least" Action

**Abstract:**

**November 6**

**Speaker:**Peter Trapa

**Title:** Flags Fixed by Matrices

**Abstract:**

**November 13**

**Speaker: ** Bobby Hanson

**Title:**A Survey of Geometric Constructions, and the Resulting Subfields of
$\mathbb{C}$

**Abstract:**

*
We are all familar with so-called Euclidean constructions (also called
Straightedge-and-Compass constructions). Recall that some
constructions are possible with a straightedge and compass, while others are
impossible. In particular, it is always possible to bisect a
given angle; while some angles are impossible to trisect. It is possible to
find the square-root of any previously constructed length,
but it is usually impossible to find a cube-root.
**
*

**November 27**

**Speaker: ** Liz Copene

**Title:** Potassium Coupled Oscillations

**Abstract:**

**December 4**

**Speaker: ** Nessy Tania

**Title:** Stochastic Simulation of Chemical Reactions

**Abstract:**

When first faced with the complexities of the western musical tradition,
many
people are understandably frustrated by the prevalence of (seemingly)
arbitrarily chosen structures. It is tempting to ask, for example, why is
the
piano keyboard designed like it is? Why are there only 12 notes? Why do we
spend so much time working with major scales and chords, and why do they
sound
"good"?

The answers to all of these questions (and many more) have their basis in the same physical phenomenon, the overtone series. In this talk we will see that the overtone series appears as a consequence of the general solution of the one-dimensional scalar wave equation and explore some of the acoustic consequences of these solutions (including a magic trick!).

We'll also take a look at some of the problems inherent in attempting to view all of music theory through the lens of the overtone series, and how modern music theorists are using topological techniques to bring a new perspective to their field of study.

This talk will NOT require any prior knowledge of music theory, and we'll take the time to develop any mathematical ideas that we'll need along the way. Therefore, the talk will be largely self contained.

LaTeX is one of the most powerful tools a scientist can use. Not only can
you write articles that correctly typeset mathematics, but you can use it
for slide presentations. Yes, your days of using powerpoint are over
(HALLELUJAH!)!! Using the Beamer class in LaTeX allows for production of
great beautiful presentations that will WOW your audience. It is easy to
use, automatically makes a PDF document, has sweet features for everyone
including using movies, makes navigating around your talk extremely smooth
(NO MORE flipping through every slide when someone asks a question!), and
gives a final product that your audience will remember.

We use averaging and homogenization theory to study the
propagation of traveling pulses in an inhomogeneous excitable neural
network. The network is modeled in terms of a nonlocal integro-differential
equation, in which the integral kernel represents the spatial distribution
of synaptic weights. We show how a spatially periodic modulation of
homogeneous synaptic connections leads to an effective reduction in the
excitability of the network and a corresponding reduction in the speed of a
traveling pulse. In the case of large amplitude modulations, the traveling
pulse represents the envelope of a multibump solution, in which individual
bumps are non-propagating and transient. The appearance (disappearance) of
bumps at the leading (trailing) edge of the pulse generates the coherent
propagation of the pulse. Wave propagation failure occurs when activity is
insufficient to maintain bumps at the leading edge.

By a lucky coincidence, most digital camera sensors are able to
have millions of sensors on a single silicon chip. Unfortunately, many of
these measurements are thrown away, as the image is then compressed to a
smaller size. It is then natural to ask whether or not we can take fewer
measurements and get a compressed image directly. The surprising answer is
yes, and that this sampling procedure is non-adaptive. The seminar today
will discuss this phenomenon and give some geometric arguments about why it
is true.

The idea of viewing a number field K as a subfield of the real or
complex numbers has resulted in some of the most fundamental results of
algebraic number theory. Embedding K into it's other completions should
yield results of the same, if not greater, importance; but in these p-adic
situations there is a plethora of unsolved problems. Leopoldt's conjecture
is the result of trying to transport the Dirichlet Unit Theorem into this
p-adic realm. After a discussion of these ideas and a description of
Leopoldt's conjecture we will conclude with a recent algorithm to verify
the conjecture.

The regular dice are labeling one through 6. When you play with
two
dice there is a certain probability of landing a number. How would one
construct die with different numerical values on their faces that gives the
same
probability of landing a certain number as the regular dice.

In the {\sl Disquisiiones Arithmeticae},
Gauss mentioned that quadratic reciprocity followed from
an evaluation of the sign of certain quadratic Gauss sums.
At that time he had found a great
deal of empirical evidence for such an evaluation, but
was able only a few years later to prove his conjecture.
His proof was purely algebraic, but somewhat indirect,
and dealt separately with different cases.
The second evaluation of these signs
was by Dirichlet, many years later.
His very beautiful proof is much more direct
than Gauss', albeit much less elementary.
It relies on the theory of Fourier series,
which as rigorous mathematics was due entirely to
him, and Fresnel integrals, which had been introduced
only recently in the theory of diffraction.
I shall present a modern version of Dirichlet's
proof, which seems to be new.

A chemostat is a laboratory apparatus that provides a controlled,
resource-limited environment for microorganisms. Chemostat models are
of
considerable interest to theoretical ecologists because they can
provide
predictions with broad ecological ramifications that are testable in
a
laboratory. I will present the research I did at the University of
Washington
on chemostat models: a single-species chemostat model that couples a
PDE describing an age-structured population with a system of ODEs
describing a chemostat, and a periodically forced version of the
model that
displays strongly resonant and chaotic behavior.

One of the most amazing results in physics is the
principle of "least" (actually stationary) action and its application
to mechanics. In this talk I will discuss some classical problems,
including refraction and the Brachistochrone, and their history and
how the method for solving these problems led to a subject in
analysis called the calculus of variations. I will then go over some
of the basic results in the calculus of variations, and discuss their
application to mechanics. Time permitting I will even construct a
reformulation of Newtonian mechanics based on this principle of
"least" action. Nothing too advanced, and the talk should be
comprehensible to any graduate student, but if you've never seen
this before it's definitely worth knowing.

This talk is about an elementary construction from linear algebra
which, perhaps surprisingly, turns out be very important in several areas of
mathematics. Start by considering an $n$-by-$n$ complex matrix, say $N$.
Consider ``flags'' of nested subspace $V_1 \subsetneq V_2 \subsetneq \cdots
\subsetneq V_n$ inside $\mathbb{C}^n$ with the property that if $v_i$ is a
vector in $V_i$, then $Nv_i$ is also in $V_i$. (In this sense the flag is
``fixed'' by $N$.) It turns out that space of such fixed flags is
incredibly interesting for all sorts of reasons. We'll uncover a few of
them.

*
Recall, also, that the lengths that are constructible in this fashion
(called the Euclidean numbers) form a subfield of $\mathbb{C}$. In fact,
it is the smallest subfield of $\mathbb{C}$ which is closed under taking
square-roots. For this survey, we will look at the set of axioms that
give us Euclidean constructions, and see that subsets of these axioms lead
to other (smaller) subfields of $\bbC$. We will also see that
all of the axioms of Euclidean constructions can be realized by merely
making folds in our paper version of $\mathbb{C}$, rather than using the
clumsy straightedge and compass. Moreover, an addional axiom can be added,
also realizable through folding, which allows such amazing
things as trisecting angles or computing cube-roots!
*

**November 20**

**Speaker: **

**Title:**

**Abstract:**

Cardiac cells are electrically coupled through gap junction channels, which allow ionic current
to spread intercellularly from one cell to the next. However, because the extracellular
junctional cleft space between neighboring cells is so narrow and tortuous, it might act as a
microdomain for ionic concentrations. In this microdomain, ionic concentrations (potassium in
particular) might vary drastically and rapidly enough to conduct an electrical signal from one
cell to the next. I will present a model of two cells coupled through junctional potassium and
discuss the resulting dynamics of the coupled cells. Even if your not into the physiology, the
dynamical system itself is pretty cool!

In living cells, biochemical reactions often involve only a small number of
molecules. As a result, a continuous deterministic approximation is often
inadequate and a full discrete stochastic formulation is required to study
the system. The Gillespie algorithm commonly used to numerically simulate
stochastic chemical reactions will be introduced. Although the method is
exact, it can be very expensive because it simulates every reaction event.
This is especially true when fast and slow reactions are occurring
simultaneously. The Gillespie algorithm will spend the majority of the time
simulating the fast events even though the slow reactions often have a
greater impact on the behavior of the system. Two approaches to accelerate
the simulation will be discussed. The first approach involves approximating
the number of reactions by a Poisson process. The second approach involves
taking quasi-steady-state approximations of the fast reactions. The
strengths and weaknesses of each method will be discussed. To elucidate the
concepts introduced in this talk, a simple reversible isomerization reaction
will be used as an example, and the basic result will be discussed.

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