Abstract: River ecosystems are characterized by unidirectional flow; individuals are at risk of being transported downstream. This movement bias gives rise to the `drift paradox': How can a population persist if individuals are washed out of the system? More generally, advection introduces an asymmetry into riverine ecosystems that affects not only persistence of a single population but also spatial spread and interactions between two or more species. In this talk, I will present a number of reaction-advection-diffusion models for populations in rivers and other advective environments. I will start with fairly simple equations and move to increasingly complex models of individual behavior and species interactions. I will explain how advection affects population-level patterns, such as persistence, spread or competitive dominance. This talk is aimed at a general audience.
Sept 17 (Carleton University) - Joint Colloquium
Speaker: Daqing Wan (University of California, Irvine)
Complexity in Coding Theory
Abstract: This is an introductory lecture on fundamental complexity results in coding theory. In particular, we shall explain how the NP-hardness of the decoding problem and the minimum distance problem can be easily deduced from a simple study of Reed-Solomon codes.
Sept 21 (UOttawa) - Joint Colloquium
Speaker: Robert Smith? (UOttawa)
Title: A mathematical model of Bieber Fever: The most infectious disease of our time?
Abstract: Recently, an outbreak of Bieber Fever has blossomed into a full pandemic, primarily among our youth. This disease is highly infectious between individuals and is also subject to external media pressure, further strengthening the infection. Symptoms include time-wasting, excessive purchasing of useless merchandise and uncontrollable crying and/or screaming. We develop a mathematical model to describe the spread of Bieber Fever, whereby individuals can be susceptible, Bieber-infected or bored of Bieber. We analyse the model in both the presence and the absence of media, and show that it has a basic reproductive ratio of 24, making it perhaps the most infectious disease of our time. In the absence of media, Bieber Fever can still propagate. However, when media effects are included, Bieber Fever can reach extraordinary heights. Even an outbreak of Bieber Fever that would otherwise burn out (driven by fans becoming bored within two weeks) can still be sustained if media events are staggered. Negative media can rein in oversaturation, but continuous negative media (the Lindsay Lohan effect) is the only way to end Bieber Fever. It follows that tabloid journalism may be our last, best hope against this fast-moving and highly infectious disease. Otherwise, our nation’s children may be in a great deal of trouble. Note: This talk will accessible to a general audience.
February 15 (UOttawa) - Joint Colloquium
Speaker: Alexander Litvak (Alberta)
Title: Norms of sub-matrices of a random matrix and applications
Abstract: We provide tail estimates for operator norms of random matrices and their sub-matrices in the setting of a log-concave ensemble. More precisely, for every $k$ and $m$ we obtained a uniform bound for norms of sub-matrices consisting of $k$ rows and $m$ columns of a random $n\times N$ matrix, whose columns are isotropic log-concave random vectors. In particular, we establish new uniform estimates for the Euclidean norms of projections of an isotropic log-concave random vector. We show applications of our results to Convex Geometry, Computational Geometry and Compressive Sensing theory.
February 22 (UOttawa) - Joint Colloquium
Speaker: Paul Smrz (Newcastle, Australia)
Title: What we should know about Time
Abstract: Time is one of the best known physical quantities. It is also one of the least understood. The evolution of ideas about the nature of time will be briefly reviewed. The main part of the talk will be devoted to the unsolved problem which dates to 1905 when the special theory of relativity was discovered: If space and time are coordinates of the four-dimensional space-time, why time flows and space does not ? A possible answer based on the formalism of the so called modern differential geometry will be described without technical details in a form accessible to any mathematically thinking person.
March 1 (UOttawa) - CRM - University of Ottawa Distinguished Lecture Series
Speaker: Alice Guionnet (MIT)
Title: Free probability and random matrices
Abstract: Free probability is a theory initiated by D. Voiculescu in the eighties that studies non-commutative random variables. It is equipped with a notion of freeness, which is related with free products, and which plays the same role as independence in standard probability. Free probability is also a natural framework to study random matrices at the limit where their size goes to infinity. Conversely, random matrices provide natural tools and concepts in free probability. In this talk, we will introduce basic concepts in free probability and discuss its relation with random matrices. We will finally describe some uses of free probability theory in operator algebra.
March 8 (UOttawa) - Joint Colloquium
Speaker: Detlev Hoffmann (Dortmund, Germany)
Title: Sums of squares
Abstract: Sums of squares have been a research topic for as long as people have studied algebra and number theory. In modern language, some of the central questions are as follows. Let $R$ be a ring with $1$. Which elements in $R$ can be written as sums of squares of elements in $R$? If an element is a sum of squares, how many squares are needed to write it as such? We give a survey of a few (of the many) classical and more recent results and open problems, focusing on fields, simple (or division) algebras and commutative rings.
April 5 (UOttawa) - CRM - University of Ottawa Distinguished Lecture Series
Speaker: Philippe Gille (Ecole normale supérieure, Paris)
Title: Familles d'algèbres de quaternions et d'octonions.
Abstract: L'algèbre des quaternions de Hamilton H est l'unique algèbre associative à division de dimension finie sur son centre R. Elle est liée aux sommes de quatre carrés et permet notamment de définir une structure de groupe de Lie sur la sphère S3. L'algèbre d'octonions de Cayley O construite en "doublant H" fait le même travail en dimension 8 et le processus s'arrête là. Nous tentons d'expliquer pourquoi en étendant la définition des algèbres de quaternions (resp. octonions) non seulement sur un corps arbitraire mais aussi sur un anneau de base. En d'autre mots, ceci conduit aux "familles d'algèbres", ou encore algèbres à paramètres. Les formes quadratiques multiplicatives sur les anneaux jouent alors un grand rôle dans la compréhension de ces objets.
April 12 (Carleton) - Joint Colloquium
Speaker: Don Dawson (Carleton University)
Title: Some probabilistic objects motivated by evolutionary biology
Abstract: Beginning with the seminal work of R.A. Fisher and Sewell Wright in the 1930’s, probabilistic models have played an essential role in evolutionary biology. This development has dramatically grown since the discovery of the central role of DNA as the source of biological information. In this lecture I will survey some of the basic probabilistic objects inspired by these developments. I will also give a brief description of a new class of set-valued stochastic processes with Boolean dynamics that Andreas Greven and I recently introduced in our study of the emergence of rare mutants in populations undergoing mutation and selection.
May 10 (UOttawa) - Joint Colloquium
Speaker: Asia Ivic Weiss (York University)
Title: Combinatorial Structure of Chiral Polyhedra
Abstract: Classification of regular polyhedra in euclidean 3-space was initiated by Grünbaum in 1977 and completed by Dress by addition of a single polyhedron in 1985. In 2005 Schulte classified the discrete chiral polyhedra in euclidean 3-space and showed that they belong to six families. The polyhedra in three of the families have finite faces and the other three families consist of polyhedra with (infinite) helical faces. We show that all the chiral polyhedra with finite faces are combinatorially chiral. However, the chiral polyhedra with helical faces are combinatorially regular. Moreover, any two such polyhedra with helical faces in the same family are isomorphic. This is a joint work with Daniel Pellicer.
Abstract: In the present talk, we discuss some recent versions of localization methods for calculations in the groups of points of algebraic-like and classical-like groups. Namely, we describe relative localization, universal localization, and enhanced versions of localization-completion. We state several classical results of conjugation calculus and commutator calculus. We also discuss several recent results such as relative standard commutator formulas, bounded width of commutators with respect to the elementary generators, and nilpotent filtrations of congruence subgroups. Overall, this shows that localization methods can be much more efficient than expected. The talk will be accessible to a general audience and graduate students.