Higher Structures in M‐Theory

Structures in Higher - Dimensional

This is an exposition of some of the constructions which have arisen in higher-dimensional category theory. We start with a review of the general theory of operads and multicategories (cf. Bur], Her], Lei1]). Using this we give a new deenition of n-category (a variation on Batanin's); we also give an informal dee-nition in pictures. Next we discuss Gray-categories and their place in coherence p...

Higher Categorical Structures in Geometry

Structures in Higher-Dimensional Category Theory

This is an exposition of some of the constructions which have arisen in higherdimensional category theory. We start with a review of the general theory of operads and multicategories (cf. [Bur], [Her], [Lei1]). Using this we give a new definition of n-category (a variation on Batanin’s); we also give an informal definition in pictures. Next we discuss Gray-categories and their place in coherenc...

Higher rank homogeneous Clifford structures

We give an upper bound for the rank r of homogeneous (even) Clifford structures on compact manifolds of non-vanishing Euler characteristic. More precisely, we show that if r = 2a · b with b odd, then r ≤ 9 for a = 0, r ≤ 10 for a = 1, r ≤ 12 for a = 2 and r ≤ 16 for a ≥ 3. Moreover, we describe the four limiting cases and show that there is exactly one solution in each case. 2010 Mathematics Su...

Learning higher-order structures in natural images.

The theoretical principles that underlie the representation and computation of higher-order structure in natural images are poorly understood. Recently, there has been considerable interest in using information theoretic techniques, such as independent component analysis, to derive representations for natural images that are optimal in the sense of coding efficiency. Although these approaches h...