Undergrad Colloquium Winter 2013

January 23, 2013

Projective Geometry and Combinatorics

Gabriel Martins, PhD Student, UCSC Mathematics Department

In this talk we're going to learn the axioms of projective geometry (which are very similar to the ones of Euclidian geometry) and see how we can use them to solve some interesting problems in combinatorics. The main problem we're going to discuss:"Suppose you have a building which has 7 elevators, each of them stopping at at most 6 different floors. Suppose it's always possible to get from one floor to the other using only one elevator. What's the maximum number of floors that this building can have?"

January 30, 2013

Fermat's Christmas Theorem

Dr. Martin Weissman, Associate Professor, UCSC Mathematics Department

In a letter dated December 25, 1640, Pierre de Fermat told Marin Mersenne about a wonderful result. This result connects two seemingly unrelated questions about a prime number p.
Question 1: What is the remainder after dividing p by 4?
Question 2: Can p be expressed as the sum of two squares?
I will discuss a proof of this theorem, based on Lagrange (1775), but using a modern visualization called Conway's topograph.

February 6, 2013

No Colloquium

February 13, 2013

Typesetting Mathematics with LaTeX

Christopher Toni, Graduate Student, UCSC Mathematics

LaTeX is a very useful and multi-purpose typesetting language used by many mathematics students, mathematicians and scientists. The first part of the talk will be an overview of the various types of documents one can create with LaTeX, where many examples of such types of documents will be provided. We will then shift our attention to understanding the components that make up a LaTeX document. The syntax for typesetting equations, creating theorem environments, creating tables and lists, inserting external images, etc. will be presented and discussed in detail. This talk should be accessible to everyone and no previous programming experience is required. Bringing your laptop to the talk is encouraged, but not required. If you don't have LaTeX installed on any of your computers (PC or Mac), information on where to download LaTeX distributions and editors will be provided.

February 20, 2013

An Introduction to the Fundamental Group

Felicia Tabing, Graduate Student, UCSC Mathematics

The fundamental group is a useful tool in algebraic topology that can be used to give an algebraic picture of a topological space. It can also be thought of as a hole detector of a space. In this talk we will discuss how the fundamental group can give information on the structure of a space, and problems that can be solved using the fundamental group.

February 27, 2013

A Friendly Introduction to Differential Forms

Eric Miles, Graduate Student, UCSC Mathematics

The typical Vector Calculus class crescendos with the integration theorems of Green, Stokes and Gauss. These theorems, along with the fundamental theorem of calculus allude to a deep relationship between between derivatives and boundaries. While the (classical) notation and language of vector calculus is helpful in building intuition, it obscures the connection among these theorems. As it turns out all the theorems mentioned are special cases of a single idea. In this talk, we will introduce the (modern) language and notation which allows for this idea to be expressed clearly: the language of differential forms. The goal of the talk is to live up to its name. We will avoid technical matters and proofs in favor of examples and accessibility.

March 6, 2013

Special values of Riemann's zeta function

Cameron Franc, PostDoc, UCSC Mathematics

In this talk we will recall the definition of Riemann's zeta function and study its special values. We will begin by discussing the divergence of the zeta function at s = 1, and then we will give several proofs of the fact that the value of zeta at 2 is pi^2 over 6. We will conclude by discussing how to make sense of the zeta function at real numbers s < 1. In particular, we will explain how to use the Riemann zeta function to make sense of the statements that 1 + 2 + 3 + 4 + ... = -1/12 and that infinity-factorial equals the square-root of 2*pi.

March 13, 2013

Mathematical Modeling of Cyber Threats:  Worms, Zombies and Botnets

Braden Soper, Applied Mathematics & Statistics department

Threats originating from the internet, such as viruses, data theft, espionage, and distributed denial of service attacks are all on the rise with no sign of slowing.  A more rigorous and scientific approach to the field of cyber security is needed to improve security measures against these cyber threats.  We'll present mathematical modeling tools, such as dynamical systems theory and game theory, and see how they are used to improve our understanding of various cyber threats.