# Undergraduate Colloquium

**Spring 2014**

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

**LCB 225**

Pizza and discussion after each talk

Receive credit for attending

Past Colloquia

- January 8 No Talk

- January 15 Andrejs Treibergs
**Deforming Surfaces**

*Abstract:*This talk is about the differential geometry of surfaces. Suppose that we are given a two dimensional surface in three space that is flexible but inextensible, that we might imagine as made of a material like plastic that can be bent but not stretched. How can it be deformed? The directions in which it can be deformed are called infinitesimal isometric deformations. Since rigid motion doesn't stretch a surface, the velocity field of a rigid motion is one kind of infinitesimal deformation that all surfaces possess. Closed convex surfaces, however, do not admit any infinitesimal isometric deformations other than those coming from rigid motion, thus are called rigid. In this talk I will develop some notions from differential geometry, formulate infinitesimal deformations and sketch a proof of the rigidity of closed convex surfaces.- January 22 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 29 Maggie Cummings
**Are you smarter than a middle schooler?**

*Abstract:*You might think middle school math is easy; after all how hard can fractions, proportions, slope, and basic geometry be? No doubt you can "do" it, but can you explain things like why the "invert and multiply" algorithm for dividing fractions works? How about explaining why a negative times a negative is a positive or give a concrete explanation for why the slopes of perpendicular lines are negative reciprocals of each other? These are the kinds of things 13 year olds are now expected to do with the new Core Curriculum State Standards. This talk will challenge you to think deeply about, and explain, procedures you've taken for granted since middle school.- February 5 Evelyn Lamb
**Visualizing the hyperbolic plane**

*Abstract:*Do you think every triangle has 180 degrees? Do you think that given a line L and a point P not on L, there is exactly one line through P that doesn't intersect L? Throw off the shackles of Euclid's parallel postulate and discover a brand new world of hyperbolic geometry! We will be guided by the beautiful visualizations of MC Escher, Daina Taimina, and Henry Segerman.- February 12 Sarah Cobb
**Connecting the Dots**

*Abstract:*Using shapes to visualize collections of numbers is a very old idea. Students of mathematics are familiar with the real numbers as a line and with the complex numbers as a plane. In this talk, we will develop a way to describe the "shape" of certain sets of matrices. We will also discuss what this shape can tell us about the algebraic structure of these sets.

This talk will be accessible to anyone familiar with matrix multiplication.- February 19 Radhika Gupta
**Braid Groups**

*Abstract:*Braid group on n strands, denoted by*B*, is a group which has an intuitive geometrical representation, and in some sense generalizes the symmetric group_{n}*S*. In this talk we will define Braid groups and look at possible applications to cryptography._{n}- February 26 Yi Zhu
**The Addition Law on Cubic Curves**

*Abstract:*A plane cubic curve is defined to be the zero locus of a smooth cubic polynomial in two variables. The main goal of this talk is to define the "addition" law on any plane cubic with the help of linear algebra. I will then survey various applications of the addition law both in number theory and algebraic geometry.- March 5 Morgan Cesa
**Algebra Through Geometry**

*Abstract:*You are all familiar with the standard "number line" picture of the integers. But did you also know that its geometry encodes algebraic information about the integers? In this talk, we will explore how geometry relates to algebra, specifically what types of information we can learn about a group by understanding the geometry of its actions on different spaces. The main examples we will consider are the integers, symmetry groups, and matrix groups.- March 12 No Talk - Spring Break

- March 19 Brendan Kelly
**Catalan Numbers**

*Abstract:*The Catalan numbers form a sequence of natural numbers that arise in various counting problems. This talk will give us a chance to explore Catalan numbers from multiple perspectives as well as provide an introduction for students to the fascinating field of combinatorics.- March 26 D. James Bjorkman
**Math Department Alumnus and Vice Consul - Speaking on Careers**

*Abstract:*D. James Bjorkman is a Vice Consul at the U.S. Embassy in Managua, Nicaragua where he works as Immigrant Visa Unit Chief. In this position, he is responsible for adjudicating the visas for all applicants who wish to permanently reside in the United States. James joined the State Department in 2008 as a Presidential Management Fellow (PMF) with the Office of State Programs, Operations, and Budget (RM/BP) where his team provided high-level budget analysis for Department principals, was the primary contact for budget matters for Congress, and oversaw the compilation and production of the Congressional Budget Justification. While in RM/BP, James's efforts were rewarded with a Franklin Award and a Meritorious Honor Award. He resigned his fellowship to join the Foreign Service in February 2010. His previous assignments were to Tijuana, Mexico as a political officer and to the Office of eDiplomacy (IRM/BMP/eDIP) in Washington, DC where he served as a business practice advisor on internal collaboration information systems.

James holds a Juris Doctor (with specialization in Public International Law) from McGeorge School of Law. During his studies James worked as an intern at the Utah Supreme Court and the UN's International Criminal Tribunal for Rwanda. James also earned a B.A. in Mathematics and a B.S. in Political Science (with specialization in International Relations) from the University of Utah. James is a movie buff who volunteers at the Sundance Film Festival each January.- April 2 Peter Trapa
**The Hat Problem**

*Abstract:*A group of prisoners is given an opportunity to play a game for their freedom. Each prisoner has a hat, either white or black (both equally likely), placed on their heads. They cannot see their own hat, but can see the hats of the others. They win when at least one prisoner guesses the color of his hat without any incorrect guesses being made. The prisoners may work together to devise a strategy before the game begins, but cannot communicate once the game starts. What is an optimal strategy for the prisoners to secure their freedom?- April 9 Matt Cecil
**Random Walks on Graphs**

*Abstract:*A directed graph is a set of vertices and directed edges (arrows pointing from one vertex to another). We can assign a probability to each edge and imagine randomly moving from vertex to vertex along these edges (a 'random walk'). We will discuss how to describe this mathematically, paying particular attention to the concept of a `steady state', which tells us where our random walker is likely to be found far into the future. We will discuss two neat applications: determining the most likely-to-be-landed-upon properties in Monopoly and a basic version of the Google page rank algorithm. A little familiarity with basic probability and linear algebra is all that is necessary to understand this discussion.- April 16 Martin Bridson, Oxford University
**From symmetries to spaces**

*Abstract:*No matter what sort of mathematics you study, the symmetries (automorphisms) of the objects that you encounter form a group. We'll explore this idea and think about how one might go about understanding the universe of all the groups that can be described with a finite amount of information. Given a group (i.e. an abstract system of symmetry), how might one go about building objects that have exactly the given type of symmetry? And how easy is it to recognize a familiar group that is presented to us in an unfamiliar fashion -- might it be provably impossible? Can geometry help?- April 23 Matthew Housley
**Knots, Braids, and the Jones Polynomial**

*Abstract:*In this talk, I will introduce a relationship between mathematical braids and knots. In particular, we will discuss a remarkable game played with braids that led Vaughn Jones to his celebrated polynomial knot invariant.

**For those taking Math 3000-1, papers are due in class on April 23**