Friday, March 16, 2007
Speaker: Naeem Muhammad Ahmad

Title: A topology valentine

Abstract: We will talk about and appreciate the intricate but elegant work of two Fields medalists Milnor and Thom. Both in their cognate works kind of played the role of topology Valentine. More precisely they tied up the cobordism and stable homotopy in a sort of "connubial relationship"!. Cobordism is a geometric notion of differential topology while homotopy an algebraic one of algebraic topology. This relationship entices and piques the interest of both algebraic and differential topologists and plays an important roles in various areas of topology. An attempt will be made to contrue the stuff systematically to make it some how intelligible to graduate students.
Naeem

Title: Sheaf Cohomology

Abstract: In this talk, the definition of the cohomology $H^1(X,F)$ and $H^{01}(X,F)$ will be introduced, where $F$ is a sheaf of abelian groups on a topological space $X$.

Title: Numerical Solution of some ill-posed Problems using Dynamical Systems Method

Abstract: Some numerical experiments on solving ill-posed linear algebraic system are shown in this work. The numerical experiments are based on a new method called DSM (Dynamical System Methods) which has been justified in [1]. DSM gives a new view in dealing with the ill-posed problems. This method consists of solving an ordinary differential equation instead of solving the non-linear discrepancy principle equation as in the variational regularization. Recently many numerical methods for solving ordinary differential equations (ODE) have been established and justified such as family of Runge-Kutta methods, see [2], which make the DSM method can be easily implemented numerically. Some care have to be made in applying the numerical method of ODE in order to minimize the computation time. A strategy in reducing the computation time is proposed in this Thesis. The numerical results of DSM method are compared with the variational regularization method.

[1.]A.G. Ramm,Dynamical systems method for solving linear ill-posed problems,Elsvier, Amsterdam, 2007.
[2.]L.F. Shampine,Numerical Solution of Ordinary Differential Equations, Chapman and Hall,Newyork,1993.

Title: Categorification

Abstract: Louis Crane coined the term "categorification" in the 1990s. In this talk I will explain the philosophy of categorification and, as an example, I will work out a categorification of the natural numbers. The notion of a weak monoidal category will be motivated and developed through this exercise. The philosophy of categorification has proven to be a fertile one. It has facilitated cutting edge research in algebraic topology which relates n-categories to homotopy theory.

The main reference for this talk is the paper "Categorification" by Baez and Dolan. (http://arxiv.org/abs/math.QA/9802029)

As a secondary reference, I use the book "Categories for the Working Mathematician" by Mac Lane (mainly to look up definitions not spelt out in the Baez - Dolan paper).

A third reference, which I'm not really using but it's probably worth looking at, is the paper "On Categorification" by Lucian Ionescu (a recent PhD graduate of KSU and former student of Crane). (http://arxiv.org/abs/math/9906038)