Scientific

Homological theory of the “algebras in analysis” exists in at least three different versions. First of all, there is the homological theory of Banach and more general locally convex algebras. This is about 40 years old. However, in the last decade of the previous century, a “homological section” appeared in a new branch of analysis, the so-called quantized functional analysis or, more prosaically, the theory of operator spaces. One of principal features of this theory, as is now widely realized, is the existence of different approaches to the proper quantum version of a bounded bilinear operator. In fact, two such versions are now thought to be most important; each of them has its own relevant tensor product with an appropriate universal property. Accordingly, there are two principal versions of quantized algebras and quantized modules, and this leads to two principal versions of quantized homology.

Thus we have now, in the first decade of the 21st century, three species of topological homology: the traditional (or “classical”) one, and two “quantized” ones.

In these lectures, we shall restrict ourselves by studying, in the framework of these three theories, the fundamental concept of a projective module. This concept is “primus inter pares” among the three recognized pillars of the science of homology: projectivity, injectivity, and flatness. It is this notion that is the cornerstone for every sufficiently developed homological theory, let it be in algebra, topology, or, as in these notes, in functional analysis.

Our initial definitions of projectivity do not go far away from their prototypes in abstract algebra. However, the principal results concern essentially functional-analytic objects. As we shall see, they have, as a rule, no purely algebraic analogues. Moreover,
some phenomena are strikingly different from what algebraists could expect, based on their experience.

These lectures are directed at analysts who are interested in learning some of the standard tools of theoretical physics, including functional integrals, the Feynman expansion, supersymmetry and the Renormalization Group. These lectures are centered on the problem of determining the asymptotics of the end-to-end distance of a self-avoiding walk on a $D$-dimensional simple cubic lattice as the number of steps grows. When $D=4$ the end-to-end distance has been conjectured to grow as Const. $n^{1/2}\log^{1/8}n,$ where $n$ is the number of steps. We include a theorem, obtained in joint work with John Imbrie, that validates the $D=4$ conjecture in the simplified setting known as the ``Hierarchial Lattice.''

We construct an action of $\mathbb Z^k$ on a compact zero-dimensional space obtained from a higher graph $\Lambda$ satisfying a mild assumption generalizing the construction of the Markov shift associated to a nonnegative integer matrix. The stable Ruelle algebra $R_s(\Lambda)$ is shown to be strongly Morita equivalent to $C^*(\Lambda)$. Hence $R_s(\Lambda)$ is simple, stable and purely infinite, if $\Lambda$ satisfies the aperiodicity condition.

These notes are for a course I gave while on sabbatical at UBC. The topics covered are: stochastic differential equations, solving PDEs using probability, Harnack inequalities for nondivergence form elliptic operators, martingale problems, and divergence form elliptic operators.

This course presupposes the reader is familiar with stochastic calculus; see the notes on my web page for Stochastic Calculus, for example. These notes for the most part are based on my book Diffusions and Elliptic Operators, Springer-Verlag, 1997.