Number Theory (Concordia University)
Chantal David got her PhD at McGill University in 1993, and she is currently a professor at Concordia University in Montreal. She was also a member of the Institute for Advanced Studies in the academic year 2009-2010. She is interested in analytic number theory, and distribution questions related to arithmetic objects, such as elliptic curves, or curves over finite fields. She was awarded the Krieger-Nelson Prize of the Canadian Mathematical Society in 2013.
Average distributions for elliptic curves
Let E be an elliptic curve over Q . There are many open conjectures about the distribution of local invariants associated with the reductions of E modulo p as p varies over the primes. Perhaps the most famous examples are the conjectures of Lang and Trotter (1976) and Koblitz (1988), the last one being a generalisation of the twin prime conjecture in the context of elliptic curves. We will explain those conjectures, and how one can gain evidence for them by averaging over some families of elliptic curves. We explain how the average results fit the conjectural asymptotics, in terms of the order of magnitude, but also in terms of the precise constants associated to each given conjecture, giving evidence for the probabilistic models defined in terms of local probabilities. More recently, applying those average techniques to different distribution questions, as counting the frequency of occur- rence of a given abelian group appearing as the group of points of elliptic curves over finite fields, we found that the resulting average distribution is also governed by the Cohen-Lenstra Heuristics, which predict that random abelian groups occur with probability weighted by the number of elements of their automorphism group.
Spectral Geometry (Laval University)
Alexandre Girouard obtained his PhD from Université de Montréal in 2008. After a few years of mathematical adventures in Europe (Wales, Switzerland and France) he is now back home. He is currently a professor at Laval University (Québec). His research interests are in geometric analysis, spectral geometry and isoperimetric problems.
L’inégalité isopérimétrique à travers les âges
Parmi toutes les figures planes de même périmètre, quelle est celle dont l’aire est la plus grande? La légende veut que la princesse Elisha, ayant débarqué sur les côtes de l’actuelle Tunisie autour de 814 av. J.-C., ait obtenu autant de terre qu’elle pourrait en délimiter à l’aide de la peau d’un boeuf. Elisha découpa donc la peau en une fine lanière, la plus longue possible, et forma avec celle-ci un demi cercle s’appuyant sur la côte, rectiligne à cet endroit. Elle fonda ainsi la ville de Carthage, dont elle devint la première reine. La princesse Elisha venait de trouver la solution du problème isopérimétrique classique: c’est le cercle qui a l’aire la plus grande parmi les figures planes de périmètre donné. L’influence du problème isopérimétrique sur le développement des mathématiques est immmense, mais malgré tous les efforts déployés, il a fallut attendre la fin du 19ième siècle pour qu’une preuve satisfaisante émerge. Pourquoi? Après avoir tenté de répondre à cette question, nous survolerons certains des développements récents de l’isopérimétrie en analyse géométrique et en géométrie spectrale.
Operator Theory (Carleton University)
Matt Kennedy has been an assistant professor at Carleton University since 2011. He obtained his PhD in 2011 from the University of Waterloo and was awarded the 2012 Doctoral Prize from the Canadian Mathematical Society. His research interests are in operator algebras and their connections with other areas of mathematics like algebraic geometry and group theory.
The noncommutative world
An important consequence of the discovery of quantum physics and the work of people like von Neumann and Wigner is the fact that we live in a “noncommutative world.” Mathematically, this means that in the passage from the classical to the quantum world, familiar commutative objects like functions should be replaced by noncommutative objects like operators. This process, which we call quantization, has led to the development of some exciting new topics in mathematics, like quantum computing and noncommutative geometry. In this talk, I will give a brief introduction to some of these ideas.
Applied Statistics (Carleton University)
Dr. Shirley Mills is a professor of both Mathematics and Statistics. She has
been a professor for 43 years, working at the University of Winnipeg, the
University of Alberta and, since 1983 at Carleton University. Throughout her
career, Dr. Mills has received several awards for teaching excellence. While
she began her career as a mathematical statistician working in the area of
detection and estimation in the presence of outliers, her career has evolved to include research in hardware and software reliability and to numerous areas of applied statistics, with extensive collaborations with a wide variety of researchers in science, engineering and social science on a wide variety of topics. In particular she has collaborated extensively with researchers in Engineering on risk modelling and analysis of environmental datasets and has worked closely with researchers in Science on geostatistical modelling. She also founded the Statistical Consulting Centre at Carleton University in 1987 and directed it for 7 years. She is recognized as an expert in the analysis of complex, massive datasets and is an expert in statistical computing, particularly in R and in SAS and SPSS. She also developed the first graduate university course in Canada in Statistical Data Mining (i.e. “Big Data”) and has taught it for the past 17 years. She has also given numerous invited lectures on this topic. She is a former President of the Statistical Society of Ottawa, a former member of the Executive of the Statistical Society of Canada and of the Canadian Association of University Teachers and is currently the Executive Director of the Statistical Society of Canada. She leads a large team of graduate students in research involving statistical computing and data mining and has to date graduated over 40 such students who are in great demand in industry and government.
Big Data: Mathematics, Statistics and Data Science
“Big Data/Data Science/Data Analytics” is revolutionizing how we work, live and communicate. The October 2012 Harvard Business Review article declared: “Data Scientist: The Sexiest Job of the 21st Century”. Data scientists have been described as like Renaissance individuals - part mathematicians, part business strategists, part statistical savants, able to apply their background in mathematics to help tame the data dragons. My talk will focus on some of the statistics and mathematics underlying Big Data/Data Analytics, with applications to diverse fields such as security, healthcare, sports.
Number Theory (Queen's University)
M. Ram Murty obtained his B.Sc. (Hons.) from Carleton University and his PhD from MIT in Cambridge, Massachusetts in 1980. Afterwards, he spent a year at the Institute for Advanced Study in Princeton, New Jersey and in 1981, a year at the Tata Institute of Fundamental Research in Mumbai, India. In 1982, he joined the faculty of McGill University in Montreal and in 1996, he moved to Queen's University in Kingston, Ontario, where he holds the Queen's Research Chair in mathematics and philosophy. In 1990, he was elected as a fellow of the Royal Society of Canada. In 1991, he was awarded the E.W.R Steacie Fellowship by NSERC and in 1998, the Killam Fellowship by the Canada Council for the Arts. He has written more than 10 books and 190 research papers. His book, Non-vanishing of L-functions and applications, written jointly with his brother Kumar, won the 1996 Balaguer Prize. Last year, his book on Indian philosophy was released by Broadview Press.
Prime Numbers and Zeta Functions
The study of the distribution of primes has been a source of inspiration and creativity for much of mathematics for many centuries. In this talk, I will outline how zeta functions can be used to study prime numbers and highlight some recent developments in this context.
Numerical Analysis and PDE's (McGill University)
Dr. Nave obtained his Ph.D. from UCSB under the supervision of Prof. Banerjee and Prof. Liu. His thesis work focused on developing numerical methods for the falling liquid film problem. He then joined the math department at MIT where he developed numerical methods for the advection equation and for PDEs with discontinuous solutions. In 2010 he joined the math and stats department at McGill University. His current research covers a wide range of applied math topics from symmetry-preserving discretization to fluid dynamics to computational geometry.
Evolving Curves, Surfaces, and More…
In this talk I will present numerical approaches for the time evolution of curves, surfaces, and general sets. To this end, one must solve the linear advection equation. Although this equation is arguably the most elementary PDE, its numerical solution is one of the most challenging problems in numerical analysis. I will highlight these challenges, present some numerical schemes to address them, and finish with a collection of examples from fluid dynamics to computer graphics.