The Fall 2006 TAP Seminar is organized by Rob Kusner, meeting at GANG
(1535 LGRT) every Monday around noon to discuss topics of current
interest in mathematics with a small group of talented young
undergraduates. Students are asked to keep notebooks in which they
will record a paragraph about the topic of the week and a paragraph
about their reaction to the topic. Each week's topic will be led by
the person listed in parentheses:
09/11: 3-manifolds according to Grisha Perelman (Rob Kusner)
In its popular "border patrol version" the Poincare' Conjecture
asserts that a 3-dimensional alien which can deform itself to slip out
of any lasso is actually just a deformed 3-ball. Thus will begin a
G-rated version of the Thurston Geometrization and the Poincare'
Conjecture for 3-manifolds, working up from the situation for 1- and
2-manifolds. Using Ricci-flow, a diffusion process which averages a
Riemannian metric according to how (Ricci) curved it is, and careful
surgery arguments to deal with singularities in the flow, it now
appears Grisha Perelman has effectively completed Richard Hamilton's
program to geometrize compact 3-manifolds (and thereby earned one of
the $1,000,000.00US Clay "Millenium" Prizes)!
09/18: Coding Theory (Farshid Hajir) ***Special time: 4PM***
Without the theory of error-correcting codes, compact discs, DVDs,
satellite communications, and regular old telephones would be a lot
less efficient if not downright impractical. I'll try to explain what
error-correcting codes are through some basic examples.
09/25: Ramanujan graphs (Paul Gunnells)
Graphs are mathematical gadgets used in many mathematical
applications, from designing telecommunications networks to analyzing
large-scale software systems. When a graph models a communications
network, a number called the "expansion constant" encodes how
efficient the network is in propagating information. In this talk
I'll describe how to use sophisticated mathematical tools to build
graphs with high expansion constants.
Exercise: Recall that the expansion constant h(G) of a graph G is the
minimum over all partitions G = A \union B of E(A,B)/M where M is the
smaller of #A or #B and E(A,B)=#edges between A and B. Show that for
the sequence G_m of "cyclic graphs with diagonals" (that is, G_m is
made from the cyclic graph C_{2m} by connecting vertex i to vertex i+m
(mod 2m) that Paul discussed in the lecture) the expansion constant
h(G_m) goes to zero.
10/02: [no meeting!]
10/11: Lines in space (Peter Norman) ***special Wednesday meeting***
How many lines pass through a given line, or through 2 or 3 or more
given lines? Can 3-space be filled with straight lines, no two of
which are parallel? Questions like this will be raised (and
answered with your help...)!
10/16: [no meeting!]
10/23: Pricing stock options (Mike Sullivan)
We'll discuss a hot topic in financial math: How much should a stock
option cost? With a few reasonable options, we'll figure this out and
come up with some seemingly unintuitive but important facts.
10/30: Modeling Traffic via Stochastic Dynamics (Alexandros Sopasakis)
We employ a novel energy driven stochastic model in order to describe
the complex traffic flow phases which arise for different
concentration regimes. A major advantage of the proposed stochastic
model is that it lends itself to analysis and, most importantly, no ad
hoc noise is introduced. Kinetic Monte Carlo simulations are used to
validate the proposed model against experimental data.
11/06: [no meeting!] PLEASE DON'T FORGET TO VOTE TOMORROW!!!
11/13: Root Finding, Linear Systems and Badly-conditioned
Problems (Nate Whitaker) ***Special time: 4PM***
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I will discuss some ill-behaved problems relating to
finding the roots of polynomials and solving matrix systems.
In addition, I will discuss ways to analyze ill-conditioned
problems. Examples will be given to illustrate the interaction
between the theory and the computations.
11/20: Modulational Instability (Panos Kevrekidis)
The real title is "Modulational Instability: A Case Example where
Mathematical Analysis meets Numerical Computation and Physical
Experiment", but it didn't fit on the line. We'll try to build (from
close to scratch) a theoretical understanding of a mechanism
(modulational instability) that is responsible for the formation of
large amplitude waves (such as the ones we see in deep ocean) in a
simple yet physically relevant model: the nonlinear Schrodinger
equation. In this partial differential equation, we'll seek the
simplest possible solutions, examine whether these solutions are
"stable" (introducing the notion of "stability" in the process) and
finally, see our results "in action": testing what we will get from
mathematical theory against computational simulations and real-life
experiments for the propagation of light in what we will call layered
optical media. The aim is to show a case example of what the value of
truly "Applied Mathematics" is in providing us with an understanding
of observed physical phenomena.
11/27: A Model of Angiogenesis [Tumor Growth] (Nate Whitaker)
*****NOTE NEW TOPIC***
I will present a model to s[t]imulate and inhibit tumor growth. A
tumor releases chemical stimuli attracting endothelial cells towards
it which eventually form a network of blood vessels. This network
provides it with oxygen and nutrients which allows it to grow as well
as providing a means of transport to other parts of the body. The
mathematical model will simulate the proliferation of this network.
12/04: How much can you "hear" about the shape of a drum? (Andrea Nahmod)
The pure tones - in fact, the base note alone - of an idealized violin
string let us determine its geometry - i.e., its length - completely!
What can the pure tones of a drumhead tell us about its 2-dimensional
shape? And what about higher dimensional "drums"?
12/11: Folding and unfolding planar polygons (Louis Theran)
Let P be a simple polygon in the plane. The carpenter's rule problem
asks whether there is a smooth motion of the vertices that maintains
all the edge lengths of P and puts P into convex position. Streinu
and Connelly, Demaine, and Rote answered this question affirmatively
in 2001.
12/18: Binary number system (Chris)
"There are 10 types of people: those who understand binary and those
who don't!" We'll cover binary arithmetic (addition, subtraction,
squaring, multiplication...), how to change from one number system to
another, and so forth.
The Monty Hall paradox (Nicole)
We'll talk about the old game show problem: You are given a choice of
three doors, one holds a car, and two hold goats; once you've picked a
door, one of the other doors is opened, revealing a goat; then you are
given the option of changing your choice or keeping your original. Is
it statistically better to change doors? We'll also talk about some
twists on the scenario.