Georgia Institute of Technology College of Sciences

We describe two computational frameworks for the assessment of contractileresponses of enzymatically dissociated adult and neonatal cardiac myocytes.The proposed methodologies are variants of mathematically sound andcomputationally robust algorithms very well established in the imageprocessing community. The physiologic applications of the methodologies areevaluated by assessing the contraction in enzymatically dissociated adultand neonatal rat cardiocytes. Our results demonstrate the effectiveness ofthe proposed approaches in characterizing the true 'shortening' in thecontraction process of the cardiocytes. The proposed method not onlyprovides a more comprehensive assessment of the myocyte contraction process,but can potentially eliminate historical concerns and sources of errorscaused by myocyte rotation or translation during contraction. Furthermore,the versatility of the image processing techniques makes the methodssuitable for determining myocyte shortening in cells that usually bend ormove during contraction. The proposed method can be utilized to evaluatechanges in contractile behavior resulting from drug intervention, diseasemodeling, transgeneity, or other common applications to mammaliancardiocytes.This is research is in collaboration with Carlos Bazan, David Torres, andPaul Paolini.

Sandro Levi and I have investigated variational strengthenings of uniform continuity and uniform convergence of nets or sequences of functions with respect to a family of subsets of the domain. Out of our theory comes an answer to this basic question: what is the weakest topology stronger than the topology of pointwise convergence in which continuity is preserved under taking limits? We argue that the classical theory constitues a misunderstanding of what is fundamentally a variational phenomenon.

I will talk about new approximation algorithms for the Asymmetric Traveling Salesman Problem (ATSP) when the costs satisfy the triangle inequality. Our approach is based on constructing a "thin" spanning tree from the solution of a classical linear programming relaxation of the problem and augmenting the tree to an Eulerian subgraph. I will talk about Goddyn's conjecture on the existence of such trees and its relations to nowhere-zero flows. I will present an O(log n/log log n) approximation algorithm that uses a new randomized rounding method. Our rounding method is based on sampling from a distribution and could be of independent interest. Also, I will talk about the special case where the underlying undirected graph of the LP relaxation of the problem has bounded genus. This is the case for example, when the distance functions are shortest paths in a city with few bridges and underpasses. We give a constant factor approximation algorithm in that case. The first result is a joint work with A. Asadpour, M. Goemans, A. Madry and S. Oveis Gharan, and the second result is a joint work with S. Oveis Gharan.

We study the topology of the space bd K^n of complete convex hypersurfaces of R^n which are homeomorphic to R^{n-1}. In particular, using Minkowski sums, we construct a deformation retraction of bd K^n onto the Grassmannian space of hyperplanes. So every hypersurface in bd K^n may be flattened in a canonical way. Further, the total curvature of each hypersurface evolves continuously and monotonically under this deformation. We also show that, modulo proper rotations, the subspaces of bd K^n consisting of smooth, strictly convex, or positively curved hypersurfaces are each contractible, which settles a question of H. Rosenberg.