As part of the Mathematical Sciences Collaborative Diversity Initiatives, nine mathematics institutes (including ICERM) are pleased to host their annual pre-conference event, the 2013 Modern Math Workshop. This event precedes the SACNAS National Conference.
The Modern Math Workshop is intended to re-invigorate the focus of mathematics students and faculty at minority-serving institutions and the research careers of minority mathematicians.
The Tutte polynomial is a combinatorial polynomial that one associates to a set of vectors V. This polynomial knows an incredible amount about the geometric, algebraic, and combinatorial properties of V. I will define this polynomial and illustrate a few of its properties. This talk will be accesible to undergraduate students, and will require no previous knowledge of the topic.
About the Speaker
Federico Ardila is an Associate Professor of Mathematics at San Francisco State University. He studies objects in algebra, geometry, topology and applications by understanding their underlying discrete structure. He leads the SFSU-Colombia Combinatorics Initiative, a research and learning collaboration between students and researchers in the United States and Colombia. When he is not at work, you might find him on the fútbol field or treasure hunting in little record stores.
Dynamical Systems is the part of mathematics which studies the changes that occur as a system evolves over time. Some systems are easy to predict whereas others are not. Such difficult to predict systems are known as "chaotic systems" (for example, the stock market or the weather). It is a surprizing fact that very simple transformations can give rise to very complex (and beautiful !) behavior. In this mini-course we will start by describing the dynamics under iteration of a simple family of maps (real quadratic polynomials) and we will work our way through to the study of the dynamics of quadratic complex polynomials. We will tour the parameter space of complex quadratic polynomials, known as the Mandelbrot set, and will describe the Fatou and Julia sets for some parameters.
There are many beautiful identities involving positive integers. For example, Pythagoras knew 32 + 42 = 52 while Plato knew 33 + 43 + 53 = 63. Euler discovered 594 + 1584 = 1334 + 1344, and even a famous story involving G. H. Hardy and Srinivasa Ramanujan involves 13 + 123 = 93 + 103. But how does one find such identities? Around the third century, the Greek mathematician Diophantus of Alexandria introduced a systematic study of integer solutions to polynomial equations. In this session, we'll focus on various types of so-called Diophantine Equations, discussing such topics as the Postage Stamp Problem, Pythagorean Triples, Pell's Equations, Elliptic Curves, the ABC Conjecture and Fermat's Last Theorem.
