Undergraduate Colloquium Fall 2008
August 27     No Talk

September 3     Mark Zajac
Depolymerization Can Drive Cell Migration
Abstract: Cell migration promotes wound healing yet allows cancer to spread. I will present a two-phase model for the solid cytoskeleton and fluid cytosol inside a crawling nematode spermatozoon. This model demonstrates that disassembly of the cytoskeletal polymer network can provide most of the force that drives a cell forward. The drag force exerted by cytosolic fluid also plays a significant role. The model also shows that cytoskeletal anisotropy is required.

The model uses level set methods to provide an implicit representation of the cell surface. Testing the model requires image processing, which can be cast as a minimization problem, leading to a differential equation. Tracking cytoskeletal features makes use of correlations.

September 10     Dylan Zwick
NP or Not NP. That's a million dollar question!
Abstract: In this talk I introduce the concept of algorithmic complexity and give some examples. We'll then examine two classes of problems, P and NP, and discuss some ideas about them such as NP completeness. We'll end with a statement of one of the seven millennium problems, that by the end of the talk you should understand.

September 17     Jim White, Career Services
Career Services is for YOU!
Abstract: Will you be graduating soon? Are you a freshman? Somewhere in between? Then Career Services has something for you! Come find out what Career Services has to offer, how to prepare for the upcoming Technical Career Fair (Wednesday, October 8), and why you'll want to take advantage of these services long before you're ready to graduate.

September 24    Peter Alfeld
Infinity is Different
Abstract: There are as many prime numbers as there are natural numbers, and there are as many natural numbers as there are rational numbers. There are more real numbers (points) in an interval than there are rational numbers anywhere, but there are no more points in a square than there are in an interval. No matter how big infinity is, you can make it bigger. How can all that be? Come to this talk and find out.

October 1     Andrejs Treibergs
Integral Geometry and Geometric Probability
Abstract: Integral Geometry deals with integrals over groups of motions of a space. If we're given a function that assigns a number to a set in the plane, such as the area or the length of the boundary or the number of components, then we can find the average of such a function applied to all possible intersections of a fixed set with a mobile set over all the positions in which the mobile set overlaps the fixed set. I shall discuss such formulas due to Cauchy and Crofton along with integral geometric solutions to some problems. One is the Buffon Needle Problem: what is the probability that a needle randomly dropped on a wooden floor touches a crack? Another is the Isoperimetric Problem: show that among all simple closed curves in the plane bounding a fixed area, the circle is the unique one with smallest length.

October 8    Jimmy Dillies
Dessins d'enfants
Abstract: One of the key groups in arithmetic is the absolute Galois group. So far, it has been eluding the mathematical community. In 1984, Grothendieck suggested to understand this group by looking at its action on dessins d'efants. Dessins d'efants (literally 'children's drawings') are graphs drawn on Riemann surfaces, such that two neighboring vertices have opposite colors. In this talk we will define what the absolute Galois group is and see how it acts on those naive objects that are dessins.

October 15    No Talk - Fall Break

October 22    Dan Ciubotaru
Solving Equations by Radicals
Abstract: We all know how to solve a quadratic equation ax^2+bx+c=0. The solutions are given by the quadratic formula. In other words, there exists a general formula involving the coefficients of the equation and only addition/subtraction, multiplication, division, and taking radicals. If this is the case, we say that the equation is solvable by radicals. The quadratic formula has been known since 1500 BC, but a formula by radicals for the cubic equation and the quartic equation were only discovered in the 16th century. For more than 250 years after that, there were many unsuccessful attempts for finding a general formula that solves by radicals any quintic equation. Then around 1800, Abel and Ruffini and then later Galois, explained why one cannot solve by radicals an equation like x^5-x-1=0, and more generally, that there is no general solution by radicals for equations of degree five or greater. This is a beautiful application of what is known as "Galois theory."

October 29    Jeff Blanchard
What is a Wavelet?
Abstract: In the information age, we deal with extremely high dimensional data sets. Due to natural physical constraints, the information contained in this high dimensional data is concentrated in fantastically lower dimensional subspaces. Wavelets are a tool to find ways to represent the information efficiently. We will learn what a wavelet is via an example, the Haar wavelet. In this presentation we will see the basics of the mathematical theory of wavelets and then see an application of the Haar wavelet to image compression. If you know the Riemann integral (the integral from Calculus I) you know everything you need. If you don't know the Riemann Integral, come anyway!

November 5    Milena Hering
Pick's Theorem
Abstract: Pick's theorem relates the area of a lattice polygon to the number of interior lattice points and the number of lattice points on the boundary of the polygon. I will give a proof of this theorem, and discuss some applications.

November 12   Mike Purcell
The Central Limit Theorem: A Thermodynamic Perspective
Abstract: The Strong Law of Large Numbers and the Central Limit Theorem are sometimes referred to as the Fundamental Theorems of Probability/Statistics. The Strong Law, while somewhat difficult to prove in full generality, is in some sense intuitive and one can give hueristic arguments for why it should be true. In contrast, The Central Limit Theorem is mysterious and is often stated without serious proof or intuitive justification. In particular, the question "Why is the limiting distribution in the Central Limit theorem normal instead of something else?" is usually very carefully avoided. In this talk, I will give a simple, intuitive, and rather non-rigorous answer to this question and an explanation of why the Central Limit Theorem should be true using the well known Second Law of Thermodynamics.

November 19    Hal Daumé III
What is Truth? Incompleteness in Mathematical Logic
Abstract: Math is the field that most closely studies what it means for something to be true or false. This raises the question: can every claim be proved true or proved false? That is, are there some mathematical questions that we simply cannot solve? Can we prove that this is or is not the case? I'll survey the origins of these questions, talk about their resolutions and the techniques that were developed to answer them, and consider some of the consequences.

November 26    Movie Day
The Right Spin
Abstract: The story of a dramatic rescue in space and the mathematics behind it, as told by astronaut Michael Foale.

December 3    Andrej Cherkaev
Optimal Designs and Optimal Structures
Abstract: An optimal design problem asks for an optimal layout of several materials in a domain. The domain is subject to an external loading, the goal is to maximize its overall response. We will see that this problem typically has no solution, and discuss the ways to approach problems without solutions. The approach will bring us to fractal-type minimizers, materials with microstructures, and nonclassical approaches to Calculus of Variations.

December 10    Dan Ciubotaru
Discrete Mathematics: an advertisement
Abstract: One of the newest courses offered by the Mathematics department is Discrete Math 2200. The purpose of the course is two-fold: on the one hand, it provides a rigorous introduction to proofs, on the other, it touches upon many interesting topics that one usually doesn't encounter in the calculus sequence: logic, sets, counting and combinatorics, induction, residues, prime numbers, cryptography, and probability. I will try to convince you that this is a class worth taking, by presenting some of my favorite problems. Here are three examples:

- The Birthday problem: What is the minimum number of people who need to be in a room so that it is more likely that there exist two of them with the same birthday than not?
- The Hatcheck problem: A new employee checks the hats of n people at a restaurant, forgetting to put claim check numbers on the hats. When customers return for their hats, the checker gives them back hats chosen at random from the remaining hats. What is the probability that no one receives the correct hat?
- The Pigeonhole principle: At any party, there must be two people who know the same number of other people there (assuming "knowledge" is symmetric).