Upcoming seminars

Parallel computing has been used for scientific computing applications since the 1960s, when the first supercomputers were developed. However, only recently have these programming paradigms become useful for software running on desktop and notebook computers. In this dissertation we demonstrate the advantage of exploiting modern computer architectures in scientific computing with multithreaded programming in Java for applications in image processing. A significant contribution of this work is an open source, multithreaded high performance scientific computing Java library called Parallel Colt. In addition, on top of Parallel Colt, we have implemented six ImageJ plugins for deconvolution, super resolution, fast Fourier transforms and image cropping. Hence, we are able to provide software to solve important problems in real image processing applications, and which can effectively make use of multi-core CPUs available on affordable desktop and notebook computers.

We outline the general development of a theory of symmetric homology of algebras, an analog of cyclic homology where the cyclic groups are replaced by symmetric groups. This theory is developed using the framework of crossed simplicial groups and the homological algebra of module-valued functors. The symmetric homology of group algebras is related to stable homotopy theory. Two spectral sequences for computing symmetric homology are constructed. The $E_1$ term of one of these relates to a new class of geometric complexes for Coxeter groups. Many open questions remain.

A pseudo-reductive group is a smooth connected affine algebraic group over a field k which does not contain any nontrivial smooth connected normal unipotent subgroups defined over k. Such groups arise naturally as the quotient of any smooth connected affine algebraic k-group by the maximal smooth connected normal unipotent subgroup defined over k. For study of general affine algebraic groups it is important to know the structure and classification of pseudo-reductive groups. In a joint work with Brian Conrad and Ofer Gabber we have determined the structure and classification of these groups. In my talk I will explain the classification, and also mention group theoretic and arithmetic applications.

It is a classical problem in geometry to find the densest sphere packing in n-dimensional Euclidean space. Analogous problems of packing among lattices, or on compact spaces such as the sphere or Hamming space, are widely studied in number theory, discrete geometry, coding theory and combinatorics. I will talk about some recent work which puts these problems in the framework of potential energy minimization. This leads to experimental and theoretical techniques to approach these optimization problems (and their inverse problems), and leads to some surprising new results in high dimensions.

I'll spend a fair amount of time discussing the mathematical models behind the CT (computed tomography) and PET (positron emission tomography) imaging modalities, with the goal of bringing the audience along for as much of 50 minutes as possible. At some point, however, I'll dive into my own research focusing on the PET imaging problem, which includes iterative methods, as well as the choice of the regularization function and parameter.\\ \\ Anyone curious about how these standard medical imaging techniques work is encouraged to attend.