Conférences - Algèbre - Canada

Banff International Research Station for Mathematical Innovation and Discovery

An equation is called Diophantine if whole number or fractional solutions are sought. Diophantine equations are named after Diophantus of Alexandria, a 3rd century mathematician and were popularized by Pierre de Fermat, a 17th century lawyer and amateur mathematician. Unlike most branches of mathematics, many Diophantine questions and theorems can be understood and appreciated by the general public, and the subject is a favourite with popularizers of mathematics. The methods for tackling Diophantine equations are however deep and varied, and the subject has recently experienced an explosion of new directions and results. This workshop will bring together experts and young scholars to explore these new directions and stimulate further research into Diophantine problems.

The field of algebraic dynamics has emerged over the past two decades at the confluence of algebraic geometry, discrete dynamical systems, and diophantine geometry. In recent work, striking connections have been observed between algebraic dynamics and much older theories of difference and differential equations. This meeting brings together mathematicians with expertise in such diverse fields as ring theory, complex dynamics, differential and difference algebra, combinatorics and algebraic geometry. New work towards the dynamical Mordell-Lang and dense orbit conjectures as well as theorems on hypertranscendence and functional independence proven by connecting difference Galois theory, algebraic dynamics and other algebraic approaches to the study offunctional equations will be presented at this meeting.

Banff International Research Station for Mathematical Innovation and Discovery (BIRS)

Recently, the study of Hénon maps has undergone new exciting developments, like the recent proof of the existence of wandering domain for complex Hénon maps. This result was achieved by mixing techniques from different parts of dynamics, namely real and complex. In this workshop we intend to pursue this trend by bringing together experts on real, complex and non-archimedean dynamics to get new perspectives in the study of Hénon maps.

Banff International Research Station for Mathematical Innovation and Discovery (BIRS)

One of the most fundamental notions when studying functions of a real variable is the rate of change of a function, as measured by its derivative. Geometrically, the derivative is the slope of the tangent line. Both the notion of the derivative and the tangent line (or more generally, the tangent bundle of a smooth manifold) can be defined purely axiomatically because of the underlying structure of the category of smooth functions. The derivative, for example, is determined by its properties (such as the sum and product formulae for differentiation, the chain rule, etc.). This structural approach to the derivative leads to the notion of a differential category. When working with models of differential categories such as smooth functions, the listed properties seem like natural consequence of the model. On the contrary, the differential category structure shows that these properties determine the structure of the derivative, rather than the other way around.

Banff International Research Station for Mathematical Innovation and Discovery (BIRS)

The aim of this workshop is to bring together experts on many developing perspectives on knot homology (physical, geometric and algebraic) to draw connections between them, and to explore applications.

Cours et évènements pour étudiants en mathématiques,    Géométrie et topologie

Banff International Research Station for Mathematical Innovation and Discovery (BIRS)

Algebraic dynamics is the study of discrete dynamical systems on algebraic varieties. It has its origins in complex dynamics, where one studies self-maps of complex varieties, and now encompasses dynamical systems defined over global fields. In recent years, researchers have fruitfully investigated the latter by applying number-theoretic techniques, particularly those of Diophantine approximation and geometry, subfields which study the metric and geometric behavior of rational or algebraic points of a variety.

Cours et évènements pour étudiants en mathématiques,    Simulation

Banff International Research Station for Mathematical Innovation and Discovery (BIRS)

Irregular growth is a ubiquitous phenomenon in nature, from the growth of tumors, crystals, and bacterial colonies to the propagation of forest fires and the spread of water through a porous medium. Mathematical models of random growth have been a driving force in probability theory over the last sixty years and a rich source of important ideas. The analysis of random growth models began in the early 1960s with the introduction of the Eden model by Eden and first-passage percolation by Hammersley and Welsh. The field witnessed several breakthroughs in the 1980s and 1990s, from the introduction of more random growth models, including the Kardar, Parisi, and Zhang (KPZ) equation and the KPZ universality class, to the ground-breaking works of Tracy and Widom and of Baik, Deift, and Johansson, and the seminal works of Newman and coauthors. These results caused a flurry of activity and more analytical and geometrical tools were developed.

Banff International Research Station for Mathematical Innovation and Discovery (BIRS)

Nonlinear inverse problems are ubiquitous in medical sciences, engineering, physics and other natural sciences. This workshop will advance a statistical paradigm that allows underpinning statistical decision making in such problems in a scientifically rigorous way. This will be achieved by providing `frequentist' large sample guarantees for inferences arising from commonly used (Bayesian or non-Bayesian) algorithms in such problems. Thereby, a large cluster of applied problems, where algorithms are used for day-to-day decision making with statistical data, will be put on a solid and robust foundation, facilitating its confident use in modern society.

Cours et évènements pour étudiants en mathématiques,    Statistique, Théorie des jeux

Banff International Research Station for Mathematical Innovation and Discovery (BIRS)

The Functoriality conjectures, formulated by Langlands in 1967, have been a driving force behind much of the work in both Number theory and Representation theory in the last 50 years. This workshop will explore a relatively new approach to the conjectures stemming from Langlands' recent proposal called ``Beyond Endoscopy''. This proposal has since blossomed in the work of many mathematicians and has produced very interesting recent results in several areas of mathematics from algebraic geometry through representation theory to analytic number theory.

Cours et évènements pour étudiants en mathématiques,    Théorie des nombres, arithmétique