Speaker: Slava Futorny (University of Sao Paulo)

Title: Galois algebras

Abstract: The celebrated Gelfand-Tsetlin formulas for finite-dimensional representations of simple complex Lie algebras lead to a concept of Galois algebras in skew group rings. These algebras admit a general theory of Gelfand-Tsetlin modules and can be viewed as noncommutative analogs of orders in skew group rings. The cases of the general linear Lie algebra and the Yangians will be considered. The talk is based on joint results with S.Ovsienko.

OCIMS PhD Prize Lectures

Speakers:1) Ariane Masuda 2) Karen Meagher (1) Carleton University 2) University of Waterloo)

Abstract:

1) Polynomials over Finite Fields: Relationships with Integers, and Permutation Polynomials

In the first part of my talk I will compare and contrast the ring of integers and the ring of polynomials over a finite field. I will discuss several questions that have analogous versions in both settings. Then, motivated by existing results on smooth and powerful numbers, I will present results on sequences of smooth polynomials, and on the number of powerful polynomials with degrees in short intervals. In the second part of my talk I will present several results concerning the problems of characterizing and enumerating permutation binomials over special finite fields.

2) Two Directions in the Study of Covering Arrays

Covering arrays are a generalization of orthogonal arrays with many practical applications. In this talk I will describe two different approaches to covering arrays that I explored in my Ph.D. thesis. The first approach is to to try to find generalizations of Sperner's Theorem and the Erdos-Ko-Rado Theorem. These two theorems determine the exact size of an optimal binary covering array and it is hoped that extensions of these theorems to set-partition systems will determine the size of optimal non-binary covering arrays. The second direction is to add a graph structure to covering arrays. This improves applications of covering arrays and makes it possible to use methods from graph theory to study these designs.

Speaker: Patrick Weidman (University of Colorado at Boulder)

Title: Model Equations for the Eiffel Tower Profile: Historical Perspective and New Results

Abstract: Two mathematical models for the shape of the Eiffel Tower are reviewed and shown to be inconsistent with Eiffel's writings. Reported here is a third model derived from Eiffel's concern about wind loads on the tower, as documented in a communication to the French Civil Engineering Society on March 30, 1885. A translation of this paper reveals the underlying physics behind the construction of the tower and leads to the formulation of a nonlinear, integro-differential equation describing its skyline profile. Although its solution is exponential, the actual tower profile closely resembles two piecewise continuous exponentials with different growth rates. This is explained by specific safety factors for wind loading that Eiffel & Company incorporated in the design of the free-standing 300 meter tower.

Speaker:Dong Eui Chang

Title:Two Examples of Controlling Mechanical Systems

Abstract:In this talk, I will give two examples of controlling mechanical systems. The first example is the orbit transfer of an artificial satellite around the Earth, and the second example is the time-optimal transfer of a finite dimensional quantum system from one energy level to another. These two examples will illustrate the usefulness of the knowledge of mechanics when one designs controllers.

Speaker: Jerry Marsden

Title: Discrete Mechanics, Variational Principles, Time Stepping, and Optimization

Abstract: Over the last 5 years, the theory and computation for mechanical systems has been developed based on discrete mechanics. The discretization method is based on the discrete variational principles of mechanics. This results in algorithms that respect the geometric structure of mechanics and at the same time are very efficient methods without undesirable things such as numerical dissipation that generic algorithms possess. The theory and practice of these techniques will be explained, along with applications to time stepping (to nonlinear elasticity, for example) and to optimal control.

Speaker: Dr. Dmitri Vainchtein (Georgia Tech.)

Title: Resonance phenomena: a tool for mixing, a tool for control

Abstract: In my talk I discuss several aspects of transport phenomena in the near-integrable multiscale dynamical system. In the first part of the talk I consider mixing via resonances-induced chaotic advection in microdroplets. I show that proper characterization of the mixing quality requires introduction of two different metrics. The first metric determines the relative volumes of the domain of chaotic streamlines and the domain of regular streamlines. The second metric describes the time for homogenization inside the chaotic domain. In the second part of the talk I illustrate how the capture into resonance, that by itself is random in nature and, consequently, is rather inefficient as a mechanism of regular transport, can be structured with little additional cost. As a model problem I consider dynamics of a charged particle in an electromagnetic field.

Speaker:Rob Fry

Title: Smooth approximation on Banach spaces

Abstract: We consider the general problem of uniformly approximating continuous functions on Banach spaces by functions with higher degrees of smoothness. We shall give a short survey of some of the key classical results, as we as more recent and less known developments.

Speaker: Niels Schwartz (University of Passau)

Title: Partially ordered rings as a tool in real algebraic geometry.

Abstract: Real algebraic geometry studies zero sets of polynomials over the real numbers (real varieties). Rings of polynomial functions on real varieties carry many partial orders, which have concrete geometric meaning. General partially ordered rings form a category that contains the partially ordered rings of polynomial functions. The category is closed under very many ring-theoretic constructions. Reflectors of the category of partially ordered rings provide tools that can be adapted in an optimal way to the study of specific aspects of real varieties.

Speaker:   Yuliya Martsynyuk (Carleton University & University of Ottawa)

Title:  Invariance Principles via Studentization in Linear Structural and Functional Error-in-Variables Models  

Abstract:  My research brings together two diverse mathematical fields, namely Error-in-Variables Models (EIVM’s) of Mathematical Statistics and current ideas and advances of Probability Theory on self-normalized and Studentized sums of random variables. It develops asymptotic methods in Statistics and Probability along these lines, and also deals with some of their potential applications. EIVM’s, which constitute an over 125-year-old area of Statistics, are models where two or more variables of interest are assumed to have a certain form of a relationship, but are observed with measurement errors, which complicate estimation of unknown parameters of this relationship. EIVM’s have been applied in virtually all areas of science and technology and, in turn, have been stimulated by demand of data analysis in, for example, medicine, agriculture and econometrics. In linear EIVM’s, where variables of interest are assumed to be linearly related, to make an inference about unknown parameters of this relationship, data are frequently summarized and used in the form of partial sums statistics. Hence the necessity of studying the stochastic behaviour of sums of random variables for EIVM’s. Originally, the research was undertaken to improve on Central Limit Theorems (CLT’s) in linear EIVM’s so that they would no longer be of limited use for possible applications. In fact, the main results of my Thesis have turned out to be new, completely data-based and hence readily applicable CLT’s, as well as other similarly featured limit theorems that are also new for EIVM’s. Moreover, all the main results are proved under the most general assumptions used in EIVM’s so far. Mathematical frameworks of some potential applications of the main results to data analysis in linear EIVM’s, such as constructing confidence intervals, are also detailed in my Thesis. My Thesis was initially influenced and inspired by some recent new trends of research in Probability Theory and Mathematical Statistics at the Laboratory for Research in Statistics and Probability (LRSP) at Carleton University. Namely, some most recent papers of Miklos Cso"rgo", Barbara Szyszkowicz and Qiying Wang, and the related theme term seminar series on recent advances in invariance principles and their applications in Probability and Statistics, given by my supervisor Professor Cso"rgo" in Fall 2003/Winter 2004 under the auspices of LRSP, prompted me to get involved in studies of the current state of the art ideas of Probability Theory on self-normalized and Studentized sums of random variables. Such research experience enabled me to introduce a new self-normalization and Studentization approach to limit theorems in EIVM’s. Moreover, augmenting these probabilistic ideas with a few new results that may be of independent interest, led to a unique chain of auxiliary results that are used to prove the latter limit theorems, i.e., the main results of my Thesis. Indeed, beyond their application to linear EIVM’s, they may also prove to be of use in solving other problems as well. The talk is intended for a general mathematical audience.

Speaker: Alastair Scott (University of Auckland)

Title:  Statistical methods for case-control studies and related sampling designs

Speaker: Vladimir Ajaev (SMU)

Title: Nanobubbles, nanoripples, and other wonders of the nanoworld.

Abstract: Experimental techniques such as atomic force microscopy and optical interferometry provide essential tools for accumulating data on many fascinating phenomena in multiphase systems where at least one of the length scales is below 100 nm. However, the physical mechanisms behind some of such recently discovered phenomena are still poorly understood. In this talk we show that the methods of continuum mechanics combined with well-established concepts from physical chemistry can be used to explain some of the recent experimental observations of nanoscale phenomena. Development of mathematical models of fluid flow and fluid-fluid interface deformations and their connection to experimental studies will be discussed. Applications of interest include arrays of nanobubbles on hydrophobic surfaces and their effect on flow and rupture in liquid films, nanoscale ripples in wetting films of typical thickness below 50 nm, as well as jets of molten metal emitted by locally heated metal films on a glass substrate.

Speaker: Partha Lahiri (University of Maryland, College Park)

Title:  Small Area Prediction Interval Problems

Abstract:  For effective planning of health, social and other services, and for apportioning government funds, there is a growing demand to produce reliable estimates for small geographic areas and sub-populations, called small-areas. Empirical best linear unbiased prediction (EBLUP) method has been widely used in small area estimation. The EBLUP method uses a linear mixed model in combining information from different sources. The variability of an EBLUP is measured by the mean squared prediction error (MSPE), and prediction intervals are generally constructed using estimates of the MSPE. Such methods have shortcomings like undercoverage, excessive length and lack of interpretability. In this talk, we will first review different methods available to produce small area prediction intervals. We will then introduce the recently proposed parametric bootstrap method in constructing small area prediction intervals and discuss its advantages over the existing methods.