In this lecture, we will explore an age-old question, which is still lingering despite efforts by mathematicians through the centuries. How to decide/prove a real number is rational or not? This problem leads to interesting cross roads: analytical, combinatorial, special functions and of course classical number theory. The content of the talk is readily accessible to participants of MSRI-UP 2012.

Dr. James Propp, University of Massachusetts, Lowell

Negative Numbers in Combinatorics: Geometrical and Algebraic Perspectives

How many subsets of size 3 does a set with -2 elements have?" The question might seem nonsensical, but there's a very real sense in which the answer is -4. One tool for making sense of such questions is the concept ofEuler measure. Another is the trick of encoding combinatorial objects as multivariate polynomials satisfying recurrence relations. Both approaches will be used to make sense of the sequence 1,1,0,1,-1,2,-3,5,-8, … obtained by running the Fibonacci recurrence in reverse.

Prof. Persi Diaconis, Stanford University

Shuffling cards and adding numbers

When several numbers are added in the usual way, 'carries' occur. It is natural to ask, how many carries and how are they distributed? This turns out to be closely related to the question 'how many times must a deck of cards be shuffled to mix it up?' One link is the combinatorics of descents in permutations and Eulerian polynomials.

Prof. Ivelisse Rubio, University of Puerto Rico, Rio Piedras

Using the covering method to compute the p-divisibility of exponentialsums and applications to coding theory

Divisibility of exponential sums has been used over the years to determine important properties of error correcting codes such as the covering radius and the minimum distance. On the other hand, applications to coding theory have provided insight and motivated the use and development of methods to simplify proofs and improve results in exponential sums.

In 1994 Moreno-Moreno introduced a combinatorial method, called the covering method, which provides an elementary way to estimate the divisibility of exponential sums over the binary field. In 2010 Castro-Randriam-Rubio-Mattson generalized the use of the covering method to any finite field providing an elementary approach to estimate the p-divisibility of exponential sums of polynomials over prime fields and obtaining improvements of some results.

Recently, Castro-Medina-Rubio used the covering method to compute the exact 2-divisibility of exponential sums of certain Boolean functions. In this talk we give an overview of some of the connections between error control codes and the divisibility of exponential sums, present the covering method and explore its possible use for the computation of exact p-divisibility of exponential sums.

Prof. David Eisenbud, University of California, Berkeley

Ehrhart Polynomials, Hilbert Functions and Free Resolutions

The Ehrhart polynomial of a polytope is a special case of a much older notion: the Hilbert polynomial. First described by Hilbert to make sense out of phenomena observed in 19th century invariant theory, the Hilbert polynomial is one of the most important invariants in algebraic geometry. I'll describe these invariants and explain the closely related notion of a free resolution -- a subject of very active current research.