MSI Weekly Bulletin - Week starting Monday 26 March, 2007

The Tutte-Whitney polynomial of a graph is a two-variable polynomial that contains a lot of interesting information about the graph. It includes, for example, the chromatic, flow and reliability polynomials of a graph, the Ising and Potts model partition functions of statistical mechanics, the weight enumerator of a linear code, and the Jones polynomial of an alternating link. This talk is a survey of this polynomial, including a generalisation to arbitrary real-valued functions on the power set of a set. BIO: http://www. csse.monash.edu.au/~gfarr/

Since the famous T(1) theorem of David and Journe, the leading paradigm in the study of various generalizations of Calderon-Zygmund singular integral operators has been the separation of the "canceling part" and the "principal part". While the former behaves somewhat like a classical convolution operator, the latter is very different. In the original T(1) theorem, the principal part is represented by paraproducts, whose boundedness is equivalent to Carleson's inequality for Carleson measures. The same paradigm is applicable in the Banach space -valued extension of the theory, making it interesting to investigate vector-valued paraproducts and Carleson's inequalities. In this talk, I discuss some of the related ideas and applications, including my recent work with Alan McIntosh and Pierre Portal on Kato's square root problem in Banach spaces.

Derived categories were introduced in the 1960s during the development of Grothendieck duality in algebraic geometry, and are now widely used in algebra. In the last few years a fascinating theory of homotopy categories, the simpler cousins of the derived category, has begun to emerge. We review this development and give a new result interpreting Grothendieck duality as an equivalence of homotopy categories.

The finite element method is a widely used numerical method in science and engineering for finding solutions to differential and integral equations. This talk will present the principle components of its approach, from problem formulation to computation. And while the focus will be mainly on its use in solving boundary value problems of elliptic PDE's, I will also explore some of the more general reasons for its popularity.

In this talk, we introduce the Level-Set Method as a numerical technique for tracking the interface between two regions in space. We might want to do this to compute simulations of oil and water interacting, the propagation of a flame or the collapse of a shape under its own curvature. We discuss the advantages and disadvantages of the level set method compared to other methods, and will see its use in simulating water flow over a step and other research. The talk should be accessible to anyone who has done calculus in several dimensions.

To be advised