# Scientific

## Projective Modules in Classical and Quantum Functional Analysis

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.

## On the Chromatic Number of Graphs and Set Systems

During this series of lectures, we are talking about infinite graphs and set systems, so this will be infinite combinatorics. This subject was initiated by Paul Erdös in the late 1940’s.

I will try to show in these lectures how it becomes an important part of modern set theory, first serving as a test case for modern tools, but also influencing their developments.

In the first few of the lectures, I will pretend that I am talking about a joint work of István Juhász, Saharon Shelah and myself [23].

The actual highly technical result of this paper that appeared in the **Fundamenta** in 2000 will only be stated in the second or the third part of these lectures. Meanwhile I will introduce the main concepts and state—--and sometimes prove—--simple results about them.

## Self-Interacting Walk and Functional Integration

## Convex Optimization

A state-of-the art review of convex optimization with various applications.

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## Actions of Z^k associated to higher rank graphs

Published in: *Ergodic Theory Dynam. Systems* **23** (2003), no. 4, 1153-1172.

## Exponential Sums Over Multiplicative Groups in Fields of Prime Order and Related Combinatorial Problems

## PDE from a probability point of view

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.