Undergraduate Colloquium

Spring 2013
Wednesdays 12:55 - 1:45
LCB 222

Pizza and discussion after each talk
Receive credit for attending
Past Colloquia

January 9 No Talk
January 16 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).
January 23 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.
January 30 Dan Ciubotaru: Certain irrational numbers: √2, e, π
Abstract: In this talk, I will present elementary arguments (algebraic, geometric, or based on calculus) that show the irrationality of the familiar numbers √2, e and π.
February 6 Stewart Ethier: The Mathematics of Texas Hold'em
Abstract: Texas hold'em is the most widely played form of poker today. Each player receives two hole cards face down. Then three community cards are dealt face up (the flop), then a fourth one (the turn), and finally a fifth one (the river). There are four betting rounds, pre- flop, post-flop, post-turn, and post-river. A showdown follows, with the best five-card poker hand winning. As they say on television, "It takes a minute to learn and a lifetime to master."
We begin with some elementary considerations and then turn to the difficult question of how to rank the 169 distinct initial hands. We then use this information to illustrate how a particular hand might be played by a mathematically inclined player.
February 13 Frank Stenger: The Planar Index, and Its Computation
Abstract: The index of a closed curve about a point is the number of times the curve winds counterclockwise around that point. It is a quantity with many applications in mathematics. In this talk I will derive a simple combinatorial formula for the index of a closed curve which uses only the signs of the parametric description of the curve at a finite number of points on the curve.
February 20 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.
February 27 Aaron Bertram: 8 Points in the Plane
Abstract: The vector space of polynomials of degree $\le d$ in one variable has dimension $d+1$. When you specify values for those polynomials at points of the line, then each such specification drops the dimension by exactly one, leaving us ultimately with a single polynomial of degree $d$ with specified values at $d+1$ points. When you try to do this in the plane, where polynomials of degree $d$ in two variables have dimension ${{d+2} \choose 2}$, geometry begins to intrude, and it matters, for example, whether lots of points lie on a line or not. The case of 8 points in the plane is particularly interesting, as we shall see.
March 6 Brendan Kelly: Guards, graphs, and geography - oh my!
Abstract: This talk will begin by considering the geographer's daily dilemma: how do you go about coloring a map with a minimum number of colors? The problem will be translated to the language of graphs creating a mathematical framework. The basic notions of color-ability will then be applied to the security of an art museum, answering the question how many guards do you need to keep the art safe?
March 13 No Talk - Spring Break
March 20 Fred Adler: Length-biased sampling
Abstract: It would seem easy to estimate how long a particular illness lasts: the average of the time between when it started and when it ended. But what if the start date is unknown, as is typical of diseases like cancers? Finding this average becomes much more challenging and fraught with biases. By generating data from the group, I'll try to show how length-biased sampling always produces an overestimate of disease duration, how this can be partially corrected, and the important implications for evaluating the effects of testing and therapy.
March 27 Yi Zhu: Geometry of Conics
Abstract: A conic is a curve obtained as the intersection of a cone with a plane. In analytic geometry, a conic may be defined as a plane algebraic curve of degree 2. In this talk, we will use a little bit of linear algebra to explore several classical results, e.g., the classification theory of conics, Pascal's "Mystic hexagon" and Brianchon's theorem.
April 3 Yekaterina Epshteyn: Shallow water models: From bathtub waves to ocean waves
Abstract: System of shallow water equations is used to model water waves in rivers, lakes, coastal areas. The development of robust and accurate numerical methods for the computation of its solutions is an important and challenging problem. The shallow water equations are a set of hyperbolic partial differential equations (PDEs) which are derived from the principles of conservation of mass and momentum.
I will start my lecture with linear advection equation - one of the simplest mathematical models that produces traveling waves. Using this example I will introduce basic ideas about hyperbolic PDEs and about the existing numerical techniques to solve these and related problems. In particular, numerical approximation of 2D Saint-Venant system of shallow water equations will be the main topic of the talk.
April 10 Rex Butler: Counting Lattice Paths
Abstract: Given a rectangular grid, the number of direct paths from one corner to its opposite can be found using binomial coefficients. This follows from a "proof by picture" which also demonstrates the idea of a "proof by bijection". Examining these lattice paths provides yet another way to see many of the basic properties of binomial coefficients, including Vandermonde's identity.
April 17 Christel Hohenegger: The Scallop Theorem: What to do when viscous forces dominate
Abstract: Viscous drag is the resistance of a body to displacement in a fluid, while inertia is the resistance of a body to change in its state. Small creatures like bacteria have negligible inertia (we'll see what this means) and live in a drag dominated world. How to move or swim in such an environment is quite challenging and these little creatures use their shape, like the long flexible flagella of a bacteria, to achieve forward motion. This is the result of the Scallop Theorem. Mathematically, it can be derived from the Navier Stokes equations describing the motion of the fluid in the particular regime of negligible inertia.
April 24 No Talk