Undergrad Colloquium Spring 2013

Here are the dates for upcoming talks. Check this page for updates regarding speakers and abstracts.

April 3, 2013

Stephen Hernandez (Earth and Planetary Sciences Department)

Mathematical methods in the geophysical sciences: Applications to assessing seismic hazard at subduction zones

 Subduction zones are the sites of some of the largest and most destructive earthquakes known to man. Quantifying the hazard associated with subduction zones benefits from detailed timing and locations of various seismic signals such as Low frequency and Very Low Frequency Earthquakes (LFEs and VLFEs). Here I review some important methods in signal processing and mathematical modeling that aid in such efforts. Principally, I will review the benefits of applying Fast Fourier Transforms in the design of correlation detectors to search for repeating seismic events. The concepts of moment tensors, Green's functions, and the generation of synthetic seismograms allow an accurate determination of the sense of slip of these exotic events.

April 10, 2013

Ed Fisher
 
Constructions with the Double Straightedge
 
This talk will begin with a brief introduction to projective geometry, assuming only a freshman physics knowledge of vector spaces.  Although most of the constructions with double straightedge (an arbitrarily extendible ruler without markings) will be straightforward, a crucial basis for many will use the projective concept of "harmonic points".  The last part of the talk will indicate, using a little algebra of complex numbers, how close one can get to the classical constructions with compasses and straightedge.

April 17, 2013

Rob Laber, UCSC

Elliptic Curves and Cryptography

Many modern cryptosystems rely on the so-called "discrete log problem" which arises in the setting of finite groups.  One can exploit this phenomenon using any finite group, but those groups arising from elliptic curves are particularly well-suited for cryptographic purposes.  In this talk, I will introduce the idea of an elliptic curve defined over the field of real numbers and over finite fields.  We will see how to turn such objects into abelian groups using algebraic and, in the case of the real numbers, geometric descriptions of the group law.  I then show how to use such a group to construct the El Gamal cryptosystem.

April 24, 2013

TBA

May 1, 2013

Eric Miles, UCSC

Julia sets and their connection to the Mandelbrot set

May 8, 2013

David Hoffman, Stanford

Hanging with the Catenoid

The curve formed by a hanging chain is called a catenary.  A surface of revolution
formed from a catenary is called a catenoid.   In this talk I will discuss these shapes
and some of their remarkable properties. Along the way we will discuss minimal surfaces.
calculus of variations (both of which will be defined in the lecture)
 and the use of mathematically defined surfaces in architecture.

May 15, 2013

Debra Lewis, UCSC

Hares, lynx, and investment bankers - modeling predator-prey dynamics

Mathematical modeling of species interactions is one of the oldest and best known applications of dynamical systems theory outside the physical sciences. The Lotka-Volterra model, often illustrated using the classic Hudson Bay Company snowshoe hare-lynx data, is a simple, elegant model demonstrating the cyclic behavior of idealized predator-prey ecosystems. But... reality isn't always that simple. It's rarely even close to being that simple. The day-to-day functioning of an ecosystem is too complex to 'render' in full detail, the data collection process can heavily influence the actual or perceived dynamics, and even highly sophisticated models can't predict game-changing events outside the box. What's a mathematician to do? We'll do a quick tour of Lotka-Volterra and some new predator-prey models, and see how they perform with the Hudson Bay Co. data (1845-1935) and the Isle Royale moose-wolf (1959-2012) data. 

May 22, 2013

Chris Toni, UCSC

A Brief Introduction to Creating Graphics in LaTeX using Tikz.

Tikz is a powerful package in LaTeX that allows people to create graphics and diagrams.  In this talk, we’ll focus primarily on 2-D graphics and some very basic 3-D graphics. We will first introduce the audience to the syntax of creating simple and basic graphics using lines, paths, and predefined shapes (such as rectangles, circles, ellipses, etc.); we will then talk about various ways to color and shade graphics, how to use for loops to simplify some of the work, and how to introduce text and symbolic objects into the graphic using nodes.  Finally, we will end with how to plot graphs of functions (this will rely on using a program called gnuplot).

May 29, 2013

Joel Langer, Case Western Reserve University

A Short History of Length

 A handy old device called a waywiser|basically a wheel and axle mounted on a handle|may be used to measure the length of a path, straight or curved.
If the wheel is one meter in circumference, the waywiser measures the length of the path in meters by counting revolutions of the wheel as it is walked from beginning to end of the path.

 It works well enough in practice|but does it also work in theory? In fact, the waywiser and the concept of arc length may be used to illustrate both successes of ancient geometers and some of the struggles faced by subsequent mathematicians and philosophers in coming to terms with infinity, infinite processes and associated computations.

The story of arc length alternates between geometry and the theory of numbers, between the continuous and the discrete, over two thousand years.

June 5, 2013

Joel Langer, Case Western Reserve University

Bernoulli's Remarkable Lemniscate

A plane curve which resembles the symbol for infinity has been hiding in plain sight for two thousand years; from the spiric sections of Perseus, to the planetary orbits of Cassini, to mechanical linkages of Watt and others, not to mention Viviani's Temple.  But it was only after Jakob Bernoulli's investigations on mechanics and elasticity that this curve made its mark on mathematical history.  Now the famous "lemniscate of Bernoulli" is tied to the birth of elliptic functions and the theory of equations and numbers through the discoveries of Count Fagnano, Euler, Gauss, Abel and others.

All of this predates the fuller view of the lemniscate as a Riemann surface of genus zero---a sphere sitting in the complex projective plane. In this setting, the elegant lemniscate turns out to have octahedral symmetry. In fact, I will attempt to provide some impression of the lemniscate as a disdyakis dodecahedron.