# Distinguished Colloquia

The Department of Mathematics and Statistics has been extremely fortunate to have benefitted from visits by several highly prominent researchers in mathematics and statistics over the years. The seminars given by these individuals have been united under the prestigious banner of Distinguished Colloquia. Those who have participated in this series include:

Dr. Brian Alspach
University of Regina
Thursday, May 11th, 2006
2:00pm, AA-1046

Groups acting on graphs

A permutation group G is s-transitive if for any two s-tuples (a1,a2,...,as) and (b1,b2,...,bs), where there are no repeated elements in either s-tuple, there is a permutation g G satisfying g(ai)=bi for i=1,2,...,s. When s=1, we simply say that G is transitive, and when s=2, we say that G is doubly transitive. A graph X is vertex-transitive if Aut(X) is transitive on the vertex set of X. There are many instances of interesting permutation groups that act multiply transitively on a variety of discrete structures such as codes, designs, configurations, etc., but in the case of graphs, if Aut(X) is doubly transitive on the vertex set of X, then X is forced to be either the complete graph or its complement. In other words, the graphs are forced to be trivial. However, this is not the end of story as we can look at highly transitive action on substructures. My talk deals with this approach to symmetry in graphs.

Prof. Vasily M. Babich
St Petersburg State University and the Steklov Mathematical Institute of the Russian
Tuesday, April 11th, 2006
1:00pm, AA-1046

On asymptotic methods in mathematical physics

If a physical process is described by a solution of some mathematical problem and this solution does not contain any large or small parameter, then to derive certain physical properties of the process analytically is only possible in very rare cases. Consideration of mathematical problems containing small or large parameters is the domain of asymptotical methods of mathematical physics.

A crucial point in obtaining asymptotics is to suggest a correct ansatz -- i.e. an analytical pattern of the asymptotic expansion. Constructing it is a sort of art. Successful ansatzes are the WKB expansion, Olver's ansatz, etc.

In the talk I will explain some ideas, which are able to help in this difficult problem: to find the appropriate expansion.

1. The first idea is connected with “adiabatic” character of some physical processes: a quickly developing process may be governed by slowly varying parameters. An example is the Wentzel-Kramers-Brillouin (WKB) expansion.
2. Some solutions have boundary layer properties. A well known example is the famous boundary layer theory in hydrodynamics. Another example is boundary layers in high frequency diffraction theory.

Raymond J. Carroll
Distinguished Professor, Department of Statistics, Texas A&M University Monday, September 11th, 2006
3:00pm, IIC-2001

Measuring dietary intake

Newspaper articles routinely report the results of epidemiological studies of the relationship between what we eat and disease outcomes such as heart disease and various forms of cancer. One of the most-quoted studies is the Nurses Health Study, which follows the health outcomes 100,000 nurses and asks them questions about their dietary intakes. While there are exceptions, for the most part one can find a relationship between heart disease and diet, e.g., less fat, more fruits, etc. On the other hand, it is rare that prospective epidemiological studies of human populations find links between cancer and dietary intakes. Perhaps the most controversial of all is the question of the relationship between dietary fat intake and breast cancer. Countries with higher fat intakes tend to have higher rates of breast cancer, and yet no epidemiological study has shown such a link. The puzzle of course is to understand the discrepancy.

Obviously, the etiology of disease may explain why heart disease, with its intermediate endpoints such as serum cholesterol, has confirmed links to nutrition while the evidence is mixed with cancer. I will focus instead on a basic question of study design: how do we measure what we eat? Try this out: how many days per year do you eat apples? I am going to review the accumulating evidence that suggests that with complex, subtle disease such as cancer, with no good intermediate endpoints such as serum cholesterol for heart disease, finding links between disease and nutrient intakes will be the exception rather than the rule, simply because of the way diet is measured. I will close with remarks about the Womens Health Initiative Dietary Intervention Trial and two new cohort studies, along with my own views of the subject.

Dr. Barbara Lee Keyfitz
Director of Fields Institute, Toronto
Wednesday, October 18th, 2006
1:00pm, IIC-2001

Multidimensional conservation laws -- more questions than answers

The analysis of quasilinear hyperbolic partial differential equations presents a number of challenges. Although equations of this type are important in a number of applications, ranging from high-speed aerodynamics, through magnetohydrodynamuics, to multiphase flows important in industrial technology, there is little theory against which even to check the reliability of numerical simulations.

Development of a theory for conservation laws in a single space variable has led to remarkable advances in analysis, including the theory of compensated compactness and the study of novel function spaces. Recently, a number of groups have begun to approach multidimensional systems via self-similar solutions.

In this talk, I will give some history of the development of conservation law theory, including an indication of why the applications are important. I will describe some of the recent results on self-similar solutions, and the interesting results in analysis that they involve. Finally, I will outline some of the paradoxical questions that remain.

Prof. Alexander Premet
University of Manchester, UK
Wednesday, May 23rd, 2007
1:00pm, EN-2006

Classification of the finite dimensional simple Lie algebras of characteristic p > 3

Recently Helmut Strade and I announced that every finite dimensional simple Lie algebra over an algebraically closed field of characteristic p > 3 is either classical (that is, comes from characteristic 0) or a filtered Lie algebra of Cartan type or one of the Melikian algebras of characteristic 5.

In my talk I will give a brief description of all Lie algebras mentioned above and outline some methods used in the Classification Theory.

Prof. Amitai Regev
Weizmann Institute, Israel
Wednesday, October 4th, 2006
1:00pm, IIC-2001

On asymptotic growth in PI algebras

Let \$lambda\$ denote a partition of a natural number \$n\$ and \$f^lambda\$ the number of standard tableau of shape \$lambda\$. The value of \$f^lambda\$ is given -- for example -- by the so called “Hook Formula”. Various natural questions -- in Combinatorics, Representation Theory, Polynomial Identity Algebras, and even in Physics -- motivate the study of the sums \$\$S_k^{(beta)}(n) = sum_{lambdainGamma_k(n)} (f^lambda)^beta\$\$ where \$Gamma_k(n)\$ are the partitions with \$leq k\$ parts. The asymptotics of \$S_k^{(beta)}(n)\$ involves the Selberg integral, and establishes a connection between the combinatorics and “Random matrices”. Related to \$S_k^{(beta)}(n)\$ are also the codimension of Polynomial Identity algebras, and we discuss some recent results on their asymptotics.

Prof. Roderick Wong
Department of Mathematics
City University of Hong Kong, Hong Kong

Distinguished Colloquium
Thursday, September 20th, 2007
2:00pm, IIC-2001

Boundary layer theory

In 1904, Prandtl introduced the boundary layer method (matched asymptotics) which has become one of the most important tools in applied mathematics. Despite its usefulness, there still does not seem to have a rigorous analysis to validate the accuracy of Prandtl's treatment. In this lecture, we will point out that errors can occur even in very well-known formulas in boundary layer theory, and present some rigorous results for two non-linear two-point boundary value problems, involving Carrier-Pearson equations.

Dr. Anthony Davison
Ecole Poytechnique Federale de Lausanne (EPFL)
Switzerland
Friday, Septmenber 12th, 2008
3:00 - 4:00pm, A-2071

Geostatistics of extremes

Climatic change is forecast to change the frequency and sizes of extreme events such as major storms, heatwaves and the like, and the effects on human mortality, health and infrastructure are starting to become of major concern to public health authorities, engineers, and other planners. Predicting the possible impacts of such events necessarily entails extrapolation outside the range of the available data, and the usual basis for this is the statistics of extremes and its underlying probability models. Analysis of extreme events for single series of data is now well-established and used in a variety of disciplines, from hydrology through metallurgy to finance and insurance, but the corresponding theory for events in space is underdeveloped. After some motivating material, this talk will describe the basic probabilistic theory of extremes, and then will outline how it may be extended to the spatial context, before turning to more statistical matters such as fitting of appropriate models to data and their use for prediction of future events. Please click here to view in pdf format.

Dr. Stephan De Bièvre
Laboratoire Paul Painlevé et UFR de Mathématiques
University of Lille (USTL)
Thursday, October 16, 2008
12:00 - 1:00pm, A-1043

Quantum maps: a case study in quantum chaos

From a physics point of view, quantum chaos is the study of quantum dynamical systems having a chaotic classical limit. Mathematically, it is a branch of semi-classical analysis, with elements of spectral analysis, geometry and number theory. We will in this talk present some of the questions in the field through illustrative examples. We will in particular explain the Shnirelman theorem and some of the more recent developments surrounding it.

Prof. Der-Chen Chang
Georgetown University, Washington
Monday March 9, 2009
2:00p.m., HH-3017

On the d-bar Neumann Problem

In this talk, we give a detailed discussion of the background and some recent progress of one of the most important problems in several complex variables: The d-bar Neumann Problem. In particular, through the discussion we obtain the solving operator for the inhomogeneous Cauchy-Riemann equation in a bounded, pseudo-convex domain of finite type with smooth boundary, under a given (0,1)-form, but also some possible optimal estimates for the associated solution.

Prof. Alberto Elduque
Departamento de Matematicas
Wednesday, May 6, 2009
2:00p.m., HH-3017

Exceptional Numbers

By doubling the complex numbers we obtain the Hamilton quaternions. These form a real four dimensional division algebra which is quite useful in dealing with some geometrical problems. It will be explained how to compose rotations on the three and four dimensional euclidean spaces, and will derive some consequences. Finally, by doubling the algebra of quaternions, we will obtain the algebra of octonions, which appears in many exceptional situations in Algebra and Geometry. A review of some of these exceptional situations will be given.

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