Spring 2010, Thursdays, Milner 216, 4:00-4:50 PM

Date: March 5

"  Special Session "
* Corey Irving

* Timothy Rainone


C. Irving: "Young Tableaux"

: When doing advanced abstract mathematics it is extremely important to stay organised. Young tableaux are a tool that can organise lots of abstract data into a simple diagram. Given some mathematical object one would like to perform operations on these objects such as addition, multiplication, intersection, etc. It is often possible to represent objects as a Young tableaux and impose rules for combining these tableaux that correspond to the operations on our original objects. This makes it easier to stay organised and makes things seem much more concrete. We will discuss ways one may do this when working with Schur polynomials, symmetric polynomials, representations of the symmetric and general linear groups, and Schubert varieties.

T. Rainone: "In the beginning was the matrix"

: The Spectral Theorem provides a fairly good understanding of bounded normal operators on Hilbert spaces. In the finite dimensional case, this reduces to the freshman fact that every normal matrix is diagonalisable. But life is not always 'normal', nor is it always non-'normal', but indeed almost 'normal'. The latter case describes the contractive operators, for which there is the von Neumann inequality providing such operators with a disk-algebra calculus. Wermer's proof for the matrix case as well as S-Nagy's proof for the general case will be presented. If time permits, the lemma of Schwarz-Pick for holomorphic functions on the unit disk will be proved as a simple Corollary.