Undergraduate Colloquium

Fall 2012
Wednesdays 12:55 - 1:45
LCB 225

Pizza and discussion after each talk
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Past Colloquia

August 22 No Talk
August 29 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 5 Peter Alfeld: What is a sliderule?
Abstract: There was a time when calculators did not exist. That did not stop us from building the Boeing 747, or going to the moon. In those days engineers, scientists, and students used sliderules on a routine and daily basis in place of calculators. I will show several sliderules, explain how they work, and describe what kind of mathematical expressions can be evaluated with a slide rule. (There are tens of thousands.) We'll also have a drawing. The lucky winner will get a sliderule to keep.
September 12 Brendan Kelly: Fermat's Little Theorem: Over and Over Again
Abstract:This talk will present several proofs of Fermat's Little Theorem, a result in number theory. The statement is a now standard result with a notable application in the RSA cryptographic algorithm. By demonstrating an array of techniques to solve the problem we will investigate the question: if it only takes one proof to turn a conjecture into a theorem, why as mathematicians do we seek alternative solutions.
September 19 Dan Ciubotaru: Straightedge and Compass Constructions
Abstract: One of the most famous problems coming from the ancient Greeks is what geometric constructions are possible using only a straightedge and a compass. Is it possible to:

1) construct a cube with precisely twice the volume of a given cube?
2) trisect any given angle?
3) construct a square whose area equals the area of a given circle?

The answers to these questions are all negative, and this has been known for almost 200 years. In this talk, I will try to explain why this is the case, and also why certain regular polygons (like the 5-gon or the 17-gon) can be constructed with only a straightedge and compass, but not most of them.
September 26 Peter Trapa: Convergence of Difference Boxes
Abstract: Consider the following game. Draw a square and label each of the corners with some whole number. At the midpoint of each square write the (positive) difference between the numbers at the corresponding vertices. Then draw a new square though the midpoints and repeat the process. If you try a few examples, you'll likely end up at a square with all vertices labeled 0. Does this always happen? What if we allow the starting numbers to be any real numbers (not just integers)? We'll answer these question and pose some open ones. Along the way, we'll uncover a connection with the Tribonnaci Sequence (which begins 0, 1, 1, and whose successive terms are the sum of the previous three).
October 3 Sarah Cobb: Tourism in Higher Dimensions
Abstract: In Flatland, Edwin Abbott describes the life of A. Square in his two-dimensional home, Flatland. About halfway through the book, a sphere visits Flatland to teach Mr. Square about life in the mysterious third dimension. This talk will examine the experiences of Mr. Square in two and three dimensions, and how we might use these ideas to imagine our own tours in four (or more) dimensions.
October 10 No Talk - Fall Break
October 17 Nicos Georgiou: Infinite Queues and Random Growth Models
Abstract: Imagine the following hellish scenario. You are standing in line at the airport security gate, at which point you are the n-th person waiting to go through the security guard. As soon as you pass, you realize that the queue persists, because there's a second security point right after the first one! You go through that as well, and horrified you understand that there are many security points, one after the other and you need to go through all of them.

The important question at this point is to decide if you can make your flight. We will answer that. In the process, we will connect the above model with a disease propagation model, one lane highway traffic, and more realistic queuing models where the servers (or guards) get tired as the day progresses.
October 24 Stewart Ethier: The method of equal proportions, or why Utah is getting a fourth congressional seat
Abstract: After each decennial United States census, the 435 seats in the U.S. House of Representatives are apportioned among the 50 states in approximate proportion to their populations. A state's quota is its proportion of the population multiplied by 435. For example, the results of the 2000 census showed that North Carolina's quota was 12.470 seats, Utah's was 3.457 seats, and California's was 52.447 seats. Of course, the actual number of seats allotted to a state must be a positive integer. If each of these quotas (one for each of the 50 states) is rounded to the nearest positive integer, the total number of seats accounted for is 433. Who gets the two extra seats?

If you said North Carolina and Utah, you'd be using the Hamilton method, which was rejected by Congress in 1911 in part because it admits the Alabama Paradox. For the last 70 years, the method of choice has been the method of equal proportions, developed in the 1920s by mathematician E. V. Huntington of Harvard University. With this method, California and North Carolina got the two extra seats in 2002, with Utah missing out by a mere 856 people.

The 2010 census gave Utah a quota of 3.898, easily qualifying it for a fourth congressional seat in 2012, as we will see. In this talk we discuss the history and mathematics of the apportionment problem.
October 31 Nelson Beebe: Cryptography, Freedom, and Democracy:
How Basic Science Affects Everyone
Abstract: To most people, research in basic science seems irrelevant, and consequently, citizens, legislators, government funding agencies, and corporations are disinclined to support it.

Nevertheless, basic science can have deep impacts on our lives. This talk examines two developments in basic science in the Twentieth Century. The first of them, Albert Einstein's work in 1905, changed the field of physics, and the course of history. The second, the invention of public-key cryptography in 1975, has important consequences for secure communications.

Many of mankind's discoveries have potential for both good and bad. The talk concludes with a discussion of some recent uses of technology that pose the very serious risk of our complete loss of privacy, freedom, and democracy.
November 7 Andrejs Treibergs: The Geometry of Bending
Abstract: A sheet of paper may be rolled up into a cylinder or a cone. Can it be bent into other shapes? It turns out that there is only one other basic possibility, the tangent developable surface. All smooth bendings of flat pieces of paper are made up of these three types glued together. More generally, given a surface in Euclidean three space, which is assumed to be flexible but inextensible, to which other surfaces can it be deformed so as to preserve lengths of curves on the surface? Such deformations are called local isometries. To study such problems, I will discuss the differential geometry of surfaces. Properties that depend on lengths of curves, such as angles, areas and the Gaussian Curvature are called intrinsic, and are shared by all locally isometric surfaces.
November 14 Ivan Sudakov: Arctic Armageddon, More Mathematics
Abstract: Over the past few decades, Earth's warming climate has received great attention. In particular, it is interesting to estimate the effects connected with greenhouse gas emissions and propagation in the atmosphere. For example, methane is a dangerous greenhouse gas and the problem of methane emission from the Siberian tundra has been a subject of significant interest. An interesting idea, proposed recently, is connected with the so-called "methane hydrate gun" in the Arctic shelf. It was suggested that the famous Permian catastrophe was a result of fast methane emissions that perhaps occurred many years ago. We propose to discuss a new approach that makes this problem mathematically tractable and allows us to describe catastrophic bifurcations in the atmosphere induced by soil greenhouse gas sources.
November 21 Alessandro Gondolo: The Eikonal Equation
Abstract: The Eikonal equation is a non-linear partial differential equation (PDE) arising in many physical situations where the goal is to find paths of minimal cost. We explain how we solve this PDE numerically using a particular advection (or transport) PDE. We will then show applications of the Eikonal equation: modeling fire propagation in a forest, finding the first time of arrival of a wave and even solving mazes.
November 28 Steffen Marcus: Deforming geometry: moduli spaces in action.
Abstract: Moduli spaces are geometric objects that parameterize a collection of other mathematical objects. In this talk I will discuss how moduli space can be constructed and how a mathematician might use them. We will go through some explicit examples.
December 5 Chris Kocs: Holiday Logic
Abstract: Alone in a cave on the island of misfit toys, there are 50 plastic robot toys which all possess a serious defect--if one of these robots discovers its own eye color, it will explode within 24 hours! Fortunately, there are no reflective surfaces in the cave, and the robots, being rather simple in design, have no means of communicating with each other. All robots are aware of their shared defect and also know that the other 49 robots have red eyes. However, since the robots don't have enough information to logically conclude that they also have red eyes, they continue merrily coexisting and jostling each other in the dark until, one day, an armless teddy bear stumbles into their cave. The teddy makes the single following remark addressed to all of the robots (and none in particular): "I see a pair of red eyes." What are the exact consequences of this seemingly innocuous comment? Why?

To solve this riddle, I'll introduce a simple and very useful tool that all mathematicians should have at their disposal. Then we'll consider other riddles and apply the same approach. To find out what this tool is (and the answer to the above riddle), you'll have to attend the talk!