The SPAMS Schedule - Fall 2006

In all the hoopla surrounding Grigori Perelman's recent refusal of the Field's medal, the other top prize winners at this year's International Congress of Mathematicians (ICM) were all but ignored. Among them was Jon Kleinberg, recipient of the 2006 Nevanlinna Prize. Kleinberg is a pioneer in the field of social networks- a discipline that seeks to understand the graph structure of human relationships. In this talk, we will give an introduction to the mathematics of social networks, and sketch Kleinberg's main result, which aims to explain the "small worlds phenomenon" observed by psychologist Stanley Milgram in 1967. We will see why the world is not only small, but also navigable, and why log2(n) makes more sense than 6.

Abstract: It is [1] to define them, [3-] to guess how many there are, and [4+] to prove it. You will hear the fascinating story of a conjecture that resisted the attempts of a plethora of combinatorialists until it was finally proved bruteforcely in 1996. The proof was n pages long, it contained lemmas, sublemmas, subsublemmas, ..., (sub^i)lemmas, and it took k referees and a computer to check all the details. That very year, however, another (beautiful, elegant, short) proof appeared, using insights from statistical mechanics and needing remarkably fewer pages, (sub^i)lemmas and referees.

Abstract: The past decade has seen an explosion in the use of parallel computing by the scientific community. I will present a general overview of this subject, starting by providing some examples from research where parallel computing has led to insights that would have been infeasible to do on a traditional computer. I will show some state-of-the-art parallel machines, and I will also discuss how to set up and buy your own parallel cluster, if you happen to have $20,000 to spend. I will introduce several parallel programming languages (such as Star-P, gridMathematica, Cilk, and MPI) and provide some examples of frequently-used parallel algorithms.

Benfords Law is a distribution which is remarkable in that it shows up in the oddest of places e.g. the first digit of the prices of goods sold in stores or the frequency of the different values of the first digit of 2^m. I will give a brief history of Benford's Law and discuss its unique properties and then prove carefully the latter example (2^m) via techniques from ergodic theory.

Abstract: Since antiquity numbers have been a universal object of focus for humanity. In this talk we adopt the approach of comparative religion to investigate the different numeric preoccupations of different world religions. Topics will include: Kabbalah, Pythagoras, Number of the Beast, Hindu Numerology, and many more.

A Markov Ramdom Fields(MRF) is an n-dimensional random variable defined on a graph such that the probability distribution of each node(assigned with a random variable) depends only on the value of its negihbors. Application of MRF includes Ising model in statistical physics, decoding in error correcting codes, statistical language modelling in linguistics and vision and graphics in computer science. In many cases, main algorithmic qustion of interest for a given MRF is fidning marginal prob. dist. for each nodes, or finding Maximum a-posteriori(MAP) assignment. Belief Propagation(BP) algorithm is a distributed iterative algorithm for these problems that has received much recent attention due to its simplicity and effectiveness in the context of MRFs with relatively simpler structure. In the talk, I will present some examples and uses of MRF and application of Belief Propagaion and its variants.

For centuries, mathematicians have sought a meaningful and coherent way to extend differentiation and integration to non-integer order. In this talk, we discuss the history of fractional calculus and review several different definitions of the fractional integral/derivative. The benefits and drawbacks of each definition will also be discussed. Believe it or not, fractional calculus has many applications in science. We will illustrate this using examples from mechanics, heat transfer, and random walks.