Claremont Center for the Mathematical Sciences

In a typical extremal problem one wants to determine maximum cardinality of discrete structure with certain prescribed properties. Probably the earliest such result was obtained 100 years ago by Mantel who computed the maximum number of edges in a triangle free graph on n vertices. This was generalized by Turan for all complete graphs and became a starting point of Extremal Graph Theory. In this talk we
survey several classical problems and results in this area and present some interesting applications of Extremal Graph Theory to other areas of mathematics. We also describe a recent surprising generalization of Turan's theorem which was motivated by a question in Computational Complexity.

We'll start by investigating the combinatorial properties of certain special pairs of
polygons defined on a planar lattice. By reinterpreting these, after Isaac Newton, we
will relate them to algebraic equations. The variables in the algebraic equations are
naturally complex numbers and they describe a “complex torus”. The vanishing loci
of the algebraic equations are elliptic curves, whose basic geometric and topological
properties we will discuss. If time permits, we may also describe an application to
string theory.

TBA

Hosted by Francis Su, HMC

Cardiac cells have a surprisingly complex architecture, and dynamic instabilities within them may lead to ventricular brillation, the leading cause of sudden cardiac death. The principle contractile signal, calcium release, must rise and fall in a
controlled fashion, yet is a result of the random action of thousands of subcellular \Calcium Release Units" (CRUs). How does the cardiac cell produce an orderly signal from a seemingly random process, and what causes this system to break down? We will first examine the dynamics of a single CRU represented as a Birth-Death (Markov) Process with multiple \xed points". At a higher scale, we consider a network of such CRUs, encoding their properties into a Cellular Automata scheme.
We analyze the average (ensemble) behavior of the system with an iterated map function and find sufficient conditions under which calcium release undergoes a period-2 bifurcation to instability.

No Colloquium

Numerical integration in one dimension is easy, even the simpler methods like the trapezoidal rule or Simpson's rule suffice for most problems. Higher dimensions present a problem though. To approximate an n-dimensional integral using methods like Simpson's rule requires time exponential in the dimension, an effect known as "The Curse of Dimensionality". High dimensional problems are common in statistics and combinatorics, so methods are needed to approximately solve these problems. In this talk, I'll talk about a new approach I've developed called TPA that uses random choices to break the curse, and show several applications.

A common misconception is that raindrops take the form of teardrops. In fact, they tend to be nearly spherical due to surface tension forces. This is an example of how at small scales, fluid molecules' tendency to adhere to each other is the dominate effect driving a fluid's motion. In this talk we will explain how surface tension arises from intermolecular forces. We will also examine some examples of the behavior that can occur at small scales due to the balance between fluid-fluid and fluid-solid forces, with applications as varied as understanding how detergents help clean clothes to the design of fuel tanks in zero gravity environments.

The Internet and the World Wide Web have led to a massive increase in the amount of data publicly available for researchers to analyze. This has resulted in the growth of new fields of study like Data Mining and Machine Learning. A hot topic these days is the field of Text Mining. In this talk, I'll give an introduction to some ideas and techniques in Text Mining that span a variety of disciplines including the humanities (Literature and History) and the sciences (Functional Genomics and Bioinformatics).

The analysis of patterns in data has typically been a subject in statistics and engineering. Recently, however, fundamental mathematical theory in areas such as linear algebra and differential geometry have provided a new mathematical framework and insights for understanding large data sets residing in spaces of large ambient dimensions. In this talk, we will explore a wide range of applications that are natural under the linear algebra and differential geometry framework. In particular, applications in image compression, handwritten digit and face recognition, image reconstruction from noisy and missing data will be discussed