Operads, multicategories, and their representations (also called operadic/multicategorical algebras) play a key role in organizing hierarchies of higher homotopies in any category with a good notion of homotopy theory. In this talk we show how one can generalize work of Toen and Rezk to provide a description of the derived category of any multicategorical algebra. Time permitting, we will discuss applications of this theory to problems in combinatorial representation theory.

We do not assume prior knowledge of the theory of operads and multicategories.

The Theory of Real Submanifolds in a Complex Space (which is sometimes called, in some more general settings, "CR-geometry") goes back to H.Poincare and was deeply developed in further works of E.Cartan, N.Tanaka, S.Chern and J.Moser. In the present series of lectures we consider the classical aspects of this theory, as well as some recent results, focusing mainly on the holomorphic equivalence problem, groups of holomorphic symmetries and the holomorphic extension problem for real submanifolds in a complex space.

We identify two seemingly disparate structures: supercharacters, a useful way of doing Fourier analysis on the group of unipotent upper triangular matrices with coefficients in a finite field, and NCSym, the symmetric functions in noncommuting variables. Each is a Hopf algebra and the two are Hopf isomorphic. This allows developments in each to be transfered. The identification suggests a rich class of examples for the emerging field of combinatorial Hopf algebras.

The resulting theory is very nicely behaved — there is a rich combinatorics describing induction and restriction along with an elegant formula for the values of superclasses on superclasses. The combinatorics is described in terms of set partitions (the symmetric groups theory involves integer partitions) and the combinatorics seems akin to tableau combinatorics. At the same time, supercharacter theory is rich enough to serve as a substitute for ordinary character theory in some problems.

In the first part of my talk I will give an introduction to Chow motives and explain their most important properties. In the second part I will discuss some applications. In particular I will present some recent results about the relation between motives and canonical dimension.

In the first part of my talk I will give an introduction to Chow motives and explain their most important properties. In the second part I will discuss some applications. In particular I will present some recent results about the relation between motives and canonical dimension. (This Algebra Seminar talk is Stefan's second talk, which follows Stefan's first talk, which is a Colloquium talk.)