Undergraduate Colloquium

Fall 2011
Wednesdays 12:55 - 1:45
LCB 225

Pizza and discussion after each talk
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Past Colloquia

August 24 No Talk
August 31 Fernando Guevara-Vasquez: How to Make Objects Invisible
Abstract: What does it mean for an object to be invisible? We will find together what are the requirements for a cloak that can make objects placed within the cloak invisible. Then we will actually design a cloak to hide an object from a static electric field using networks. This construction actually extends to other physical situations. You only need to know what a derivative is for this talk.
September 7 Aaron Bertram: From trigonometry to elliptic curves and beyond
Abstract: One way to think about algebraic geometry is as a machine for turning shapes into systems of polynomial equations. When we study the solutions to these equations in different number systems (complex, real, rational, clock,...) we find some remarkable relationships between geometry and arithmetic. The sin and cos functions of trigonometry, for example, turn the (unit) circle into the polynomial equation x^2 + y^2 = 1 whose solutions in the system of rational numbers are the Pythagorean triples. Similarly, the Weierstrass P function and its derivative turn the torus (doughnut) into the set of solutions to a cubic equation, i.e. an elliptic curve. What happens to more complicated surfaces? In higher dimensions?
September 14 Erika Meucci: Classification of Conics
Abstract: Conics are obtained by intersecting a cone with a plane. There are three types of conics: the hyperbola, the parabola, and the ellipse. They were named by Apollonius of Perga (200 BC) and they have been studied by Ancient Greek mathematicians. In this talk I will present the classification of conics and some of their features.
September 21 Tommaso de Fernex: The Lakes of Wada
Abstract: Every closed curve splits the plane in several regions. The curve is simple if it never crosses over itself. Consider the following two statements:

(A) Every simple curve C cuts out two regions, both having C as their boundary.

(B) Any set C that is the boundary of each of the regions it cuts out is a simple curve.

First convince yourself by trying some some examples that both statements should be true. Then come to hear about the Lakes of Wada, an example that shows that at least one of the two statements must be false, without however telling us which one. Which statement is the true one?
September 28 Justin Boyer: Ground Heat Exchangers
Abstract: Ground Heat Exchangers (GHE) are a component of Ground Source Heat Pumps (GSHP). GSHP's utilize the constant temperature located 2 meters below the earth's surface to heat and cool buildings. This "green" technology uses 1/4 of the energy that traditional heating methods use. In this talk I will discuss the equations governing these systems, and the methods used to uncover useful properties.
October 5 Dan Ciubotaru: Bernoulli Numbers
Abstract: We all know that the sum of the integers from 1 to n is 1+2+3+...+n = n(n+1)/2. You may have also seen that 1^2+2^2+...+n^2 = n(n+1)(2n+1)/6 or that 1^3+2^3+...+n^3 = [n(n+1)/2]^2, but is there a formula for 1^k+2^k+...+n^k for every positive integer k? Similarly, many times you've impressed your friends by knowing that the sum of the series 1+1/2^2+...+1/n^2+... is π^2/6. But what is the sum 1+1/2^8+...+1/n^8+... for example? The answers to the above questions have to do with the Bernoulli numbers.
October 12 No Talk - Fall Break: Abstract:
October 19 Firas Rassoul-Agha: What is entropy and why is it useful? or Copying Beethoven
Abstract: Broadly speaking, entropy in thermodynamics is the amount of order, disorder, and/or chaos in a system. In information theory entropy is an absolute limit on the best possible lossless compression of any communication. In probability theory entropy is related to the odds of improbable events occurring.

Are these really three different notions of entropy?

I will show how the answer is no. Then, I will mention a few applications using entropy such as:

- telling if a Shakespeare, a Mozart, or a Picasso is a fake
- faking a Dickens, a Beethoven, or a Rembrandt
- breaking codes
- getting back at spammers
October 26 Movie Day!: Abstract:
November 2 Aaron Wood: Analytic Continuation and Divergent Series
Abstract: In a 1913 letter from Srinivasa Ramanujan to G.H. Hardy, Ramanujan expressed the formula 1 + 2 + 3 + 4 + ... = -1/12 and mentioned that his methods of proof could not be adequately conveyed in a single letter. In this talk we will introduce analytic continuation and develop the continuation of the Riemann-zeta function. Our goal: to verify Ramanujan's formula in a single lecture.
November 9 Andrej Treibergs: Constructing Polyhedra
Abstract: A polyhedron in three space is a piecewise flat surface obtained by gluing together planar polygons along their edges. The outer surfaces of a cube or a tetrahedron are examples. Minkowski asked whether it is possible to construct a closed polyhedron knowing only the directions and areas of its faces.

It turns out that this can be done if and only if the directions and areas satisfy certain geometric conditions. In this talk I'll discuss a general method to solve such problems, called Alexandrov's Mapping Lemma. The method illustrates that often in mathematics, to prove existence one needs to know uniqueness.
November 16 Sarthok Sircar: What happens after we eat pizza at the colloquium? The biology and mathematics behind digestion
Abstract: In this talk, I will describe the biology behind how our stomach prepares itself for the propagation of food through the remainder of the digestive system. In particular, I will focus on the mucus lining of our stomach, its role in digestion and what happens in the case of indigestion. Later, we explore a mathematical model to describe this process.
November 23 No Talk: Abstract:
November 30 Anurag Singh: Cutting Squares into Triangles of Equal Area
Abstract: Can you cut a square into an even number of triangles of equal area? Sure! What about an odd number of triangles of equal area? The answer - due to Paul Monsky - uses a fun mix of topology and algebra.
December 7 Robbie Snellman: Interesting Theorems
Abstract: The topic of this talk will cover the proofs of some simple, yet important, theorems that one should encounter as an undergraduate. Background needed to understand this talk includes function notation and knowing how to add and multiply numbers.