With this article I want to take the opportunity to sketch the development of topological proofs in discrete mathematics beginning with the proof of the Kneser conjecture about 25 years ago, which eventually led to a new discipline: topological combinatorics. In the beginning of the twentieth century the discipline of combinatorial topology already made use of combinatorial concepts in topology...

In this paper, a method for image deformation is presented. It is based upon decomposition of the deformation problem into basic physical laws. Unlike other methods that solve a differential or an energetic formulation of the physical laws involved, we encode the basic laws using computational algebraic topology. Conservative laws are translated into exact global values and constitutive laws ar...

Duality in the general course of human affairs seems to be a juxtaposition of complementary or opposite concepts. This frequently leads to poetical sounding uses of language, both in the common language and in the precision of mathematical theorems. Thus the duality of Projective Geometry: Two points determine a line; two lines determine a point. Gergonne first introduced the word duality in ma...

These are some notes for a second-semester algebraic topology course at UC Berkeley which I taught in 2004 and 2005. The topics of the course are: higher homotopy groups and obstruction theory, bundles and characteristic classes, spectral sequences, and Morse theory and applications to differential topology. Disclaimers: These notes are intended as an elementary introduction to selected core id...

The problem of second-order consensus is investigated in this paper for a class of multi-agent systems with a fixed directed topology and communication constraints where each agent is assumed to share information only with its neighbors on some disconnected time intervals. A novel consensus protocol designed based on synchronous intermittent local information feedback is proposed to coordinate ...