Scientific

Entropy and Orbit Equivalence

Author: 
Daniel J. Rudolph
Date: 
Fri, Oct 1, 2004
Location: 
University of Victoria, Victoria, Canada
Conference: 
PIMS Distinguished Chair Lectures
Abstract: 
In these notes we first offer an overview of two core areas in the dynamics of probability measure preserving systems, the Kolmogorov-Sinai theory of entropy and the theory of orbit equivalence. Entropy is a nontrivial invariant that, said simply, measures the exponential growth rate of the number of orbits in a dynamical system, a very rough measure of the complexity of the orbit structure. On the other hand, the core theorem of the orbit theory of these systems, due to Henry Dye, says that any two free and ergodic systems are orbit equivalent, that is to say can be regarded as sitting on the same set of orbits. The goal we set out to reach now is to explain and understand the seeming conflict between these two notions.
Notes: 

On Hilbert's Tenth Problem

Author: 
Yuri Matiyasevich
Date: 
Tue, Feb 1, 2000
Location: 
University of Calgary, Calgary, Canada
Conference: 
PIMS Distinguished Chair Lectures
Abstract: 
Dr. Matiyasevich is a distinguished logician and mathematician based at the Steklov Institute of Mathematics at St. Petersburg. He is known for his outstanding work in logic, number theory and the theory of algorithms. At the International Congress of Mathematicians in Paris in 1900 David Hilbert presented a famous list of 23 unsolved problems. It was 70 years later before a solution was found for Hilbert's tenth problem. Matiyasevich, at the young age of 22, acheived international fame for his solution.
Notes: 

Projective Modules in Classical and Quantum Functional Analysis

Author: 
A. Ya. Helemskii
Date: 
Mon, Aug 11, 2003
Location: 
University of Alberta, Edmonton, Canada
Conference: 
PIMS Distinguished Chair Lectures
Abstract: 
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.
Notes: 

On the Chromatic Number of Graphs and Set Systems

Author: 
András Hajnal
Date: 
Wed, Sep 1, 2004
Location: 
University of Calgary, Calgary, Canada
Conference: 
PIMS Distinguished Chair Lectures
Abstract: 
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.
Notes: 

Self-Interacting Walk and Functional Integration

Author: 
David Brydges
Date: 
Thu, Sep 14, 2000
Location: 
University of British Columbia, Vancouver, Canada
Conference: 
PIMS Distinguished Chair Lectures
Abstract: 

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.''

Notes: 

Convex Optimization

Author: 
Stephen Boyd
Date: 
Mon, Mar 1, 2004
Location: 
University of British Columbia, Vancouver, Canada
Conference: 
IAM-PIMS Distinguished Colloquium Series
Abstract: 
A state-of-the art review of convex optimization with various applications.
Notes: 

Actions of Z^k associated to higher rank graphs

Author: 
U. Kumjian,
D. Pask
Date: 
Thu, Aug 1, 2002
Location: 
University of Victoria, Victoria, Canada
Conference: 
Aperiodic Order, Dynamical Systems, Operator Algebras and Topology
Abstract: 

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.

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

Author: 
Sergei Konyagin
Date: 
Thu, Apr 1, 2004
Location: 
University of British Columbia, Vancouver, Canada
Conference: 
PIMS Distinguished Chair Lectures
Abstract: 

Let $ p $ be a prime. The main subject of my talks is the estimation of exponential sums over an arbitrary subgroup $ G $ of the multiplicative group $ {\mathbb Z}^*_p $:

$$S(a, G) = \sum_{x\in G} \exp(2\pi iax/p),	a \in \mathbb Z_p.$$

These sums have numerous applications in additive problems modulo $ p $, pseudo-random generators, coding theory, theory of algebraic curves and other problems.

PDE from a probability point of view

Author: 
Richard Bass
Date: 
Thu, Jan 1, 2004
Location: 
UBC, Vancouver, Canada
Abstract: 
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.
Notes: 
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