Topological invariants of classification problems

AbstractThere is a general agreement that problems which are highly complex in any naive sense are also difficult from the computational point of view. It is therefore of great interest to find invariants and invariant structures which measure in some respect the complexity of the given problem. The question which we are going to consider in the following paper are classification problems, the “computations” are described by questionnaires [3, 10] or, as they are called nowadays, by “branching programs” [11]. The “complexity” of the problem is measured by classical topological invariants (Betti numbers, Euler-Poincaré characteristic) of topological structures (simplicial complexes, topological spaces)

A topological approach for protein classification

Protein function and dynamics are closely related to its sequence and structure. However prediction of protein function and dynamics from its sequence and structure is still a fundamental challenge in molecular biology. Protein classification, which is typically done through measuring the similarity be- tween proteins based on protein sequence or physical information, serves as a crucial step toward the understanding of protein function and dynamics. Persistent homology is a new branch of algebraic topology that has found its success in the topological data analysis in a variety of disciplines, including molecular biology. The present work explores the potential of using persistent homology as an indepen- dent tool for protein classification. To this end, we propose a molecular topological fingerprint based support vector machine (MTF-SVM) classifier. Specifically, we construct machine learning feature vectors solely from protein topological fingerprints, which are topological invariants generated during the filtration process. To validate the present MTF-SVM approach, we consider four types of problems. First, we study protein-drug binding by using the M2 channel protein of influenza A virus. We achieve 96% accuracy in discriminating drug bound and unbound M2 channels. Additionally, we examine the use of MTF-SVM for the classification of hemoglobin molecules in their relaxed and taut forms and obtain about 80% accuracy. The identification of all alpha, all beta, and alpha-beta protein domains is carried out in our next study using 900 proteins. We have found a 85% success in this identifica- tion. Finally, we apply the present technique to 55 classification tasks of protein superfamilies over 1357 samples. An average accuracy of 82% is attained. The present study establishes computational topology as an independent and effective alternative for protein classification

What do Topologists want from Seiberg--Witten theory? (A review of four-dimensional topology for physicists)

In 1983, Donaldson shocked the topology world by using instantons from physics to prove new theorems about four-dimensional manifolds, and he developed new topological invariants. In 1988, Witten showed how these invariants could be obtained by correlation functions for a twisted N=2 SUSY gauge theory. In 1994, Seiberg and Witten discovered dualities for such theories, and in particular, developed a new way of looking at four-dimensional manifolds that turns out to be easier, and is conjectured to be equivalent to, Donaldson theory. This review describes the development of this mathematical subject, and shows how the physics played a pivotal role in the current understanding of this area of topology.Comment: 51 pages, 10 figures, 8 postscript files. Submitted to International Journal of Modern Physics A, July 2002 Uses Latex 2e with class file ws-ijmpa.cls (included in tar file

The mathematical research of William Parry FRS

In this article we survey the mathematical research of the late William (Bill) Parry, FRS

The classification problem for automorphisms of C*-algebras

We present an overview of the recent developments in the study of the classification problem for automorphisms of C*-algebras from the perspective of Borel complexity theory.Comment: 21 page