Discrete Mathematics

Recent Results on Bootstrap Percolation

Speaker: 
Béla Bollobás
Date: 
Fri, Feb 15, 2013
Location: 
PIMS, University of British Columbia
Conference: 
PIMS/UBC Distinguished Colloquium
Abstract: 
Bootstrap percolation, one of the simplest cellular automata, can be viewed as an oversimplified model of the spread of an infection on a graph. In the past three decades, much work has been done on bootstrap percolation on finite grids of a given dimension in which the initially infected set A is obtained by selecting its vertices at random, with the same probability p, independently of all other choices. The focus has been on the critical probability, the value of p at which the probability of percolation (eventual full infection) is 1/2. The first half of my talk will be a review of some of the fundamental results concerning critical probabilities proved by Aizenman, Lebowitz, Schonman, Cerf, Cirillo, Manzo, Holroyd and others, and by Balogh, Morris, Duminil-Copin and myself. The second half will about about the very recent results I have obtained with Holmgren, Smith, Uzzell and Balister on the time a random initial set takes to percolate.

Alan Turing and Enigma

Speaker: 
John R. Ferris
Date: 
Tue, Mar 27, 2012
Location: 
PIMS, University of Calgary
Conference: 
Alan Turing Year
Abstract: 
Central to Alan Turing's posthumous reputation is his work with British codebreaking during the Second World War. This relationship is not well understood, largely because it stands on the intersection of two technical fields, mathematics and cryptology, the second of which also has been shrouded by secrecy. This lecture will assess this relationship from an historical cryptological perspective. It treats the mathematization and mechanization of cryptology between 1920-50 as international phenomena. It assesses Turing's role in one important phase of this process, British work at Bletchley Park in developing cryptanalytical machines for use against Enigma in 1940-41. It focuses on also his interest in and work with cryptographic machines between 1942-46, and concludes that work with them served as a seed bed for the development of his thinking about computers.

Turing 2012 - Calgary

This talk is part of a series celebrating the Alan Turing Centenary in Calgary. The following mathtube videos are part of this series
  1. Alan Turing and the Decision Problem, Richard Zach.
  2. Turing's Real Machine, Michael R. Williams.
  3. Alan Turing and Enigma, John R. Ferris.

Turing's Real Machines

Speaker: 
Michael R. Williams
Date: 
Tue, Feb 28, 2012
Location: 
PIMS, University of Calgary
Conference: 
Alan Turing Year
Abstract: 
While Turing is best known for his abstract concept of a "Turing Machine," he did design (but not build) several other machines - particularly ones involved with code breaking and early computers. While Turing was a fine mathematician, he could not be trusted to actually try and construct the machines he designed - he would almost always break some delicate piece of equipment if he tried to do anything practical. The early code-breaking machines (known as "bombes" - the Polish word for bomb, because of their loud ticking noise) were not designed by Turing but he had a hand in several later machines known as "Robinsons" and eventually the Colossus machines. After the War he worked on an electronic computer design for the National Physical Laboratory - an innovative design unlike the other computing machines being considered at the time. He left the NPL before the machine was operational but made other contributions to early computers such as those being constructed at Manchester University. This talk will describe some of his ideas behind these machines.

 

Turing 2012 - Calgary

This talk is part of a series celebrating The Alan Turing Centenary in Calgary. The following mathtube videos are also part of this series
  1. Alan Turing and the Decision Problem, Richard Zach.
  2. Turing's Real Machine, Michael R. Williams.
  3. Alan Turing and Enigma, John R. Ferris.

Alan Turing and the Decision Problem

Speaker: 
Richard Zach
Date: 
Tue, Jan 24, 2012
Location: 
PIMS, University of Calgary
Conference: 
Alan Turing Year
Abstract: 
Many scientific questions are considered solved to the best possible degree when we have a method for computing a solution. This is especially true in mathematics and those areas of science in which phenomena can be described mathematically: one only has to think of the methods of symbolic algebra in order to solve equations, or laws of physics which allow one to calculate unknown quantities from known measurements. The crowning achievement of mathematics would thus be a systematic way to compute the solution to any mathematical problem. The hope that this was possible was perhaps first articulated by the 18th century mathematician-philosopher G. W. Leibniz. Advances in the foundations of mathematics in the early 20th century made it possible in the 1920s to first formulate the question of whether there is such a systematic way to find a solution to every mathematical problem. This became known as the decision problem, and it was considered a major open problem in the 1920s and 1930s. Alan Turing solved it in his first, groundbreaking paper "On computable numbers" (1936). In order to show that there cannot be a systematic computational procedure that solves every mathematical question, Turing had to provide a convincing analysis of what a computational procedure is. His abstract, mathematical model of computability is that of a Turing Machine. He showed that no Turing machine, and hence no computational procedure at all, could solve the Entscheidungsproblem.

Turing 2012 - Calgary

This talk is part of a series celebrating the Alan Turing Centenary in Calgary. The following mathtube videos are also part of this series
  1. Alan Turing and the Decision Problem, Richard Zach.
  2. Turing's Real Machine, Michael R. Williams.
  3. Alan Turing and Enigma, John R. Ferris.

Cohomology of Quasiperiodic Tilings

Author: 
Franz Gaehler
Date: 
Thu, Aug 1, 2002
Location: 
University of Victoria, Victoria, Canada
Conference: 
Aperiodic Order, Dynamical Systems, Operator Algebras and Topology
Abstract: 
Topics: • Quasiperiodic tilings • The hull of a tiling • Approximation the hull by CW-spaces • Application to canonical projection tilings • Relation to matching rules • Towards an interpretation

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: 

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 TeX Embedding failed! be a prime. The main subject of my talks is the estimation of exponential sums over an arbitrary subgroup TeX Embedding failed! of the multiplicative group TeX Embedding failed!:

TeX Embedding failed!

These sums have numerous applications in additive problems modulo TeX Embedding failed!, pseudo-random generators, coding theory, theory of algebraic curves and other problems.

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