# Undergraduate Colloquium

**Fall 2013**

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

**LCB 225**

Pizza and discussion after each talk

Receive credit for attending

Past Colloquia

- August 28 No Talk

- September 4 Andrejs Treibergs
**Bending Polyhedra**

*Abstract:*A polyhedron is a two dimensional closed surface made by gluing together planar polygons along their edges. A rigid motion of three space will move the polyhedron without distorting the polygonal faces, but is there any other deformation that bends the polyhedron along edges but preserves its faces? For example, the octahedron pictured, found by Bricard in 1897, admits such a deformation although its faces intersect each other. We discuss the notion of infinitesimal rigidity of a polyhedron, that the only motions that preserve the faces up to first order are just the rigid motions of three space. We shall prove the 1916 theorem of Max Dehn, that strictly convex polyhedral are infinitesimally rigid.

Slides from this talk- September 11 Peter Alfeld
**What is a slide rule?**

*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 slide rules on a routine and daily basis in place of calculators. I will show several slide rules, 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 slide rule to keep.- September 18 Francine Mahak & Amy Jackson from Career Services
**Career Services: For Math Majors**

*Abstract:*Will you be graduating soon? Are you a freshman? Somewhere in between? Then Career Services has something for you! Come find out about potential career opportunities available to those with degrees in mathematics. We'll also discuss the upcoming Career Fair (Tuesday, September 24th) and what you can be doing now to prepare for your future career - even before you graduate.- September 25 Peter Trapa
**Simply Laced Dynkin Diagrams**

*Abstract:*The diagrams are called simply laced Dynkin diagrams. They turn up in an unexpectedly large number of seemingly unrelated classification problems (compact Lie groups, reflection groups, finite subgroups of SU(2), quivers of finite type, surface singularities, and so on) and suggests deep and still poorly understood connections between these problems. This talk will look at a few appearances of Dynkin digrams and offer some hints at the mysterious connections between them.- October 2 Ivan Sudakov
**Mathematics of Our Ice Dependent World**

*Abstract:*The Cryosphere has melted faster in last 20 years than in the last 10,000. For example, if the rising temperature affects glaciers and icebergs, could the polar ice caps be in danger of melting and causing the oceans to rise? This could happen, but no one knows when it might happen. The global climate models predict a nearly ice-free Arctic as early as 2020. The prophets prophesy: "Around 2033 - all of the polar ice caps melt!". However mathematics can help to the study of these complicated phenomena. In this lecture we propose to discuss: "How the Stefan problem can be applied to the study of Arctic melt ponds? "; "Why the phase transition theory improves our knowledge about climate tipping points?"; "What is connection between permafrost and superconductors?"; "Which is the worst, compost bomb or nuclear bomb?".- October 9 Kenneth Bromberg - Director of Graduate Studies
**Applying for and attending graduate school**

*Abstract:*This weeks Undergraduate Colloquium will be aimed at helping undergraduates answer the following questions:- Should I apply to graduate school?
- How do I apply to graduate school?
- What will it be like when I'm in graduate school?

- October 16 No Talk - Fall Break

- October 23 Spencer Phippen
**Solving differential equations numerically, and the Difference Potentials Method**

*Abstract:*Mathematical models from many fields rely on the solution of systems of differential equations (DE) with complicated structure. In this talk, we will first explore some structure-related complications by deriving a simple DE method and watching it fail on difficult problems. Then, we will introduce the Difference Potentials Method, a flexible method that is able to solve more complicated systems with ease. Finally, we'll discuss the image for this talk, a frowny face, and its relevance to mathematical modeling with differential equations. You only need to know what a derivative is for this talk.- October 30 Stewart Ethier
**How to gamble when you have the advantage**

*Abstract:*Suppose you can bet on the toss of a biased coin, winning the amount of your bet with probability 3/5 and losing the amount of your bet otherwise, and further that you can do this as often as you like. How much of your capital should you bet at each trial? One possible answer is to bet all of it, thereby maximizing expected wealth. But clearly this is a poor strategy that will eventually lead to ruin. In 1956, John L. Kelly, Jr., then a physicist at Bell Labs, found the definitive answer: You should maximize not expected wealth but rather expected log wealth. In our example, you would bet one fifth of your capital at each trial. In this talk we will review properties of the Kelly system, which has been used by hedge fund managers, sports bettors, and blackjack players. In 1961 Breiman conjectured that the Kelly system minimizes the expected number of trials needed to achieve a goal, at least up to some portion of the goal, beyond which one should bet enough to achieve the goal in one trial. Breiman's conjecture was disproved in 1998. In 2011 a new and perhaps simpler counter-example was found, and we will explain it.- November 6 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.- November 13 Drew Johnson
**Burnside's Lemma, or How to Count Things**

*Abstract:*In many situations, it is natural to want to count the number of possible arrangements while taking symmetry into account. For example, how many ways are there to fill a tic-tac-toe board with five X's and four O's, but where we consider configurations obtained by reflections and rotations to be the same? For example, we considerX X O X O X X O O to be essentially the same as O X X X O X O O X

Slides from this talk- November 20 Firas Rassoul-Agha
**Spot the Math!**

*Abstract:*Spot It! is a pattern recognition game. There is a deck of 55 cards. Each card shows eight different symbols. Any two cards have exactly one symbol in common. I will explain the rules of the basic version of the game during the talk, then I will answer the following two questions:

1) How can we generate such a deck?

2) In particular, what's the total number of symbols in the whole deck?

Off the bat, both questions are pretty difficult to answer. However, they become quite simple when using a little bit of geometry. Come and see how! The only prerequisite for this lecture is to know that the equation of a line is y=a+bx.- November 27 Andrej Cherkaev
**Variational principles and optimality in nature**

*Abstract:*Is our world the best of all possible worlds? This question puzzled scientists and philosophers for centuries. Particularly, is the inertial motion the most "economic", and if yes, what quantity does it minimize? A mathematical approach of this problem contributed to the development of Calculus of Variations. The story was knotty: Lagrange suggested the minimizing functional - action, then Jacobi proved that the action is not minimized, and finally Einstein in special relativity restored the action minimizing. Today, search for extremal principles in physics and evolutional biology continues. The talk will introduce variational principles and discuss what does it mean to be optimal.- December 4 Christopher Hacon
**How to Turn the Sphere Inside Out**

*Abstract:*We will discuss some mathematical ideas that arise when one tries to turn the sphere inside out and we will illustrate a particularly simple way to visualize a sphere eversion.- December 11 No Talk
**For those taking Math 3000-1, papers are due**