Just How Final are Today's Quantum Structures?

I present a selection of conceptual and mathematical problems in the foundations of modern physics as they derive from the title question. Contribution to a panel session, "Springer Forum: Quantum Structures -- Physical, Mathematical and Epistemological Problems", held at the Biannual Symposium of the International Quantum Structures Association, Liptovsky Jan, September 1998. To appear in journal: Soft Computing (2001).Comment: 3 pages, tcilate

R.A.Fisher, design theory, and the Indian connection

Design Theory, a branch of mathematics, was born out of the experimental statistics research of the population geneticist R. A. Fisher and of Indian mathematical statisticians in the 1930s. The field combines elements of combinatorics, finite projective geometries, Latin squares, and a variety of further mathematical structures, brought together in surprising ways. This essay will present these structures and ideas as well as how the field came together, in itself an interesting story.Comment: 11 pages, 3 figure

Construction of Negatively Curved Cubic Carbon Crystals via Standard Realizations

We constructed physically stable sp2 negatively curved cubic carbon structures which reticulate a Schwarz P-like surface. The method for constructing such crystal structures is based on the notion of the standard realization of abstract crystal lattices. In this paper, we expound on the mathematical method to construct such crystal structures

Gauge field theories: various mathematical approaches

This paper presents relevant modern mathematical formulations for (classical) gauge field theories, namely, ordinary differential geometry, noncommutative geometry, and transitive Lie algebroids. They provide rigorous frameworks to describe Yang-Mills-Higgs theories or gravitation theories, and each of them improves the paradigm of gauge field theories. A brief comparison between them is carried out, essentially due to the various notions of connection. However they reveal a compelling common mathematical pattern on which the paper concludes.Comment: 33 pages. To be published in the book: Mathematical Structures of the Universe (Copernicus Center Press, Krak\'ow, Poland, 2014