Scientific

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: 
PDF icon Hajnal.pdf
Class: 
Scientific
Subject: 
Discrete Mathematics
Logic and Foundations

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: 
PDF icon Brydges.pdf
Class: 
Scientific
Subject: 
Mathematical Physics
Probability

Convex Optimization

Author: 
Stephen Boyd
Date: 
Tue, Mar 2, 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: 
PDF icon Boyd_Stephen.pdf
Class: 
Scientific
Subject: 
Operations Research and Optimization

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.
Notes: 
PDF icon Kumjian_and_Pask.pdf

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

Class: 
Scientific
Subject: 
Operator Algebras

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.
Notes: 
PDF icon Konyagin_Lectures.pdf
Class: 
Scientific
Subject: 
Discrete Mathematics
Group Theory

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: 
PDF icon Bass.pdf
Class: 
Scientific
Subject: 
Partial Differential Equations
Probability

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