# Undergraduate Colloquium

**Spring 2012**

**Wednesdays 12:55 - 1:45**

**LCB 222**

Pizza and discussion after each talk

Receive credit for attending

Past Colloquia

- January 11 No Talk
- January 18 Nelson Beebe
**Newcomb, Benford, Pareto, Heaps, and Zipf**

Are arbitrary numbers random?

*Abstract:*An arbitrary collection of measured numbers from various sources ought, it seems, to be random, but the surprising answer is quite different.

The implications of this discovery are astonishingly broad, from mathematical curiosity, to accounting, criminology, demographics, fraud, linguistics, nuclear decay, taxation, terrorism, Web searching, and even the rise of Fascism in the early Twentieth Century, and the recent Greek debt crisis.

Come and learn something of this fascinating subject that can broaden your view of what mathematics is good for, and perhaps even turn your planned career in another direction.- January 25 Fernando Guevara-Vasquez
**The Fast Fourier Transform**

*Abstract:*In 1965 Cooley and Tukey discovered a revolutionary algorithm: the Fast Fourier Transform (FFT). It reduces the number of operations required to do frequency analysis of a signal of length N from N^2 to about N log (N). We will look under the hood of this ubiquitous algorithm, and explore a few applications including noise reduction, image compression (JPEG) and sound compression (MP3).- February 1 Peter Alfeld
**Hotel Infinity**

*Abstract:*You are the owner of Hotel Infinity. It has infinitely many rooms, and it's full. A new guest arrives and insists you give her a room. How do you accommodate her? The next day, a family with infinitely many members arrives, each of whom wants a private room. The next day infinitely many families, each with infinitely many members, arrive. Each family member insists on a private room. You can do it! Infinity is different.- February 8 Sarah Cobb
**Triples (Almost) Without Triangles**

*Abstract:*The ancient Greeks, always interested in geometry, were among the first to discover that the square on the hypotenuse of a right triangle is equal to the sum of the squares on the legs, a formula familiar to most of us from our high school years by the simpler statement a^2 + b^2 = c^2. Even forgetting the triangle, it is interesting to study trios of numbers that satisfy this equation. In this talk we will explore questions about these triples: how many are there? how do we find them? which ones are "interesting"? what things do they all have in common?- February 15 Jack Jeffries
**Gröbner Bases and Integer Programming**

*Abstract:*In this talk we consider the problem of choosing the best combination of coins to make change. Curiously enough, this simple (?) problem will lead us to a vast generalization of polynomial long division. No background is required.- February 22 Stefano Urbinati
**A First Look at Projective Geometry**

*Abstract:*Leonardo said about his notes: "let no one who has not studied mathematics read these books". Why does an artist care about math?

Some ideas of for studying perspective are the base for projective geometry. I'll try to explain some basics of it, with focusing on the projective plane and drawing a lot of non-artistic pictures.- February 29 Morgan Cesa
**The Geometry of Origami**

*Abstract:*This talk will focus on the underlying geometry of paper-folding. We will explore some interesting and surprising results about traditional origami, followed by a discussion of rigid and modular origami, and conclude by working together to build some polyhedra from smaller modules.- March 7 Mladen Bestvina
**Topology of Numbers**

*Abstract:*The goal of the lecture is to explain how to answer questions such as:

Does the equation`x`^{2}-2`y`^{2}=5 have a solution in integers`x`,`y`?

Such questions were considered in antiquity (cf. the famous Cattle Problem of Archimedes), and the modern understanding was accomplished with the works of Legendre and Gauss around 1800. We offer a ``geometric'' way of answering such questions, due to Conway.

The answer will be found by doodling in the ``topograph''. This is an infinite tree drawn in the plane with every vertex of valence 3. In addition, each complementary region is labeled by a reduced fraction.- March 14 No Talk - Spring Break
- March 21 Jingyi Zhu
**Do Leveraged ETFs Really Generate Double or Triple Returns?**

*Abstract:*ETFs (exchange traded funds) are like mutual funds that track certain indices without requiring you to collect a number of stocks in a portfolio. Unlike mutual funds, you don't just get a daily net asset value (NAV), instead you can trade the fund any time just like a regular stock. Leveraged ETFs are another kind of financial products that a return two or three times that of a targeted index is generated using leveraging, thus provide investors opportunities to take more risk. In this talk, we will use elementary tools to show how the leverage is executed, and investigate whether the promised return ratio can be realized over a given span of time. Towards the end of the talk, we will show that one of the most celebrated formulas in stochastic analysis can directly explain and quantify the gap between what is promised and what is realized, and that the decisive quantity that determines this gap is the so-called volatility of the underlying ETF.- March 28 Christel Hohenegger
**The Scallop Theorem: How to Swim in Honey**

*Abstract:*Have you ever wondered what would happen if you try to swim in honey or why bacteria have such a long and flexible flagella? Drag is the resistance to movement of a body through a fluid and we will see how nature figured out a way to overcome the drag for creatures whose inertia is negligible. The answer lies in the way these little creatures use their shape to effectively swim in water or break the symmetry. Mathematically, this fact can be very simply derived from the form of the famous Stokes equations describing the motion of fluid.- April 4 Jason Albright
**The Search for the Perfect Wave**

*Abstract:*What makes the perfect wave? Many characteristics of waves in the ocean can be described by a class of PDEs known as Hyperbolic Conservation Laws. I will start with a very simple example, the linear advection equation, to highlight several of its wider-reaching features. I will tie this together with some of the numerical techniques designed to solve a variety of related problems. Specifically, a current research topic, the 2-D Shallow Water equations which model ocean waves, including tsunamis.- April 11 Howard Masur
**Topology and Geometry of Surfaces**

*Abstract:*In this lecture I will consider surfaces sitting inside 3 dimensional space from different points of view. One of which concerns how you can decompose the surface into triangles and leads to what is called the Euler characteristic. Another coming from calculus is to ask whether there is a vector field on the surface and a third coming from geometry is to ask about the curvature of the surface sitting inside 3 space. Remarkably all of these concepts are related by some fundamental theorems in mathematics. I will attempt to describe all of these concepts and their relationships to each other.- April 18 Elena Cherkaev
**Fractals**

*Abstract:*Fractals are fascinating geometric structures with very unusual properties, such as non-integer dimension or infinite "distance" between any two points. Starting with a non-differentiable Weierstrass function and Peano's space-filling curve, for some time these objects were regarded as "pathological monsters" or as a mathematical curiosity. Currently, the number of applications of fractals is overwhelming - we see them everywhere: in natural shapes of clouds and coastlines, mountain ranges and Saturn rings, in shapes of galaxies, distribution of earthquakes, diffusion of chemicals, etc. The talk will discuss characteristic features of fractal structures, their relation to strange attractors, and a way of creating fractals using an iteration method.- April 25 Movie Day!