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Data-Driven Methods for Dynamic Systems

Vendredi 1er décembre | 13h - 15h
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(Concordia University)

With the growing size of experimental data and the increase of computational power, there have been revolutionary new tools developed to model these datasets. Broadly falling under machine learning, these techniques have the power to not only model new systems, but fundamentally alter how we analyze the systems we already have. The goal of this mini-tutorial is to provide an example-driven understanding of how such modern computational tools can be applied to interpret dynamic data.

 

Throughout these demonstrations we will bring together state-of-the-art methods involving optimization, sparse regression, and neural networks to provide accurate descriptions of dynamic data. The methods presented in this tutorial draw inspiration from the theory of dynamical systems analysis. In particular, we will illustrate how classical concepts from dynamical systems such as variable transformation, linearization, and the identification of coherent sets can be implemented with a suite of numerical methods, often with performance guarantees. Major components of this mini-tutorial will focus on:

 

  • The Koopman operator and its approximation from data.

  • Model identification using sparse regression.

  • System simulation using physics-informed neural networks.

  • Using autoencoder neural networks to approximate conjugacies.

 

As will be demonstrated, these methods seek to compliment, not replace, the analysis of time-dependent systems in much the same way that numerical time-stepping did in the twentieth century.

Data-driven dynamic systems

The objective of this mini-course is to introduce the audience to some recent developments in the area of arithmetic statistics. Particular emphasis will be given to the problem of counting number fields. We will pay special attention to Malle’s conjecture, a conjectural asymptotic formula for the number of extension of the rational numbers with given Galois group and bounded ramification invariants (discriminant, product of ram- ified primes, Artin conductor, etc). This is a quantitative strengthening of the inverse Galois problem and, in discussing known results and current areas of progress, we will restrict our attention to groups where this difficult conjecture is well-understood. We will also pay attention to counterexamples, classical and more recent, to this conjecture and the consequent modification that is currently believed.

 

A basic background in algebra (basics of Galois theory) will be essential, but not much more. Some earlier exposure to number theory will be helpful. I will aim to keep the presentation as self-contained as possible.

 

This course will be an opportunity for introducing attendees to a cutting-edge area of pure mathematics

Introduction to Arithmetic Statistics

Vendredi 1er décembre | 13h - 15h
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(Concordia University)

Arithmetic Sysems
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Les mathématiques comme activité d’enseignement

Vendredi 1er décembre | 13h - 16h
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Eva Knoll

(Université du Québec à Montréal)

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Valériane Passaro

(Université du Québec à Montréal)

La séance sera donnée en français et en anglais

Nous faisons tous des mathématiques parce que cela nous donne du plaisir, que ce soit l’activité elle-même ou ses résultats, elles donnent satisfaction. 

Ce n’est pourtant pas la même chose que de les enseigner ou de les apprendre. Ce mini-cours est destiné aux nouveaux profs et aux étudiant.e.s des cycles supérieurs qui se préparent à enseigner pour la première fois.

Pour vous aider à créer un apprentissage significatif, nous vous proposons de commencer par examiner ce que peuvent vouloir dire savoir, ou comprendre des mathématiques, avant de faire l’expérience du passage du « ne pas savoir » au « savoir ». Cette expérience servira de canevas pour explorer les activités, approches et tactiques qui favorisent un bon développement mathématique chez vos étudiant.e.s. Suite à ce travail, nous allons collectivement examiner des concepts mathématiques pour explorer les approches possibles et leurs retombées pour la compréhension.

Maple

Maple for Mathematics Research and Teaching

Vendredi 1er décembre | 10h - 12h
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Paulina Chin

(Senior Architect, Mathematical Software, MapleSoft)

The Maplesoft Mathematics Suite is a family of mathematical software products that have been used by educators, scientists and researchers for over 40 years. It includes:

 

  1. Maple: a hybrid symbolic-numeric math engine and programming environment   

  2. Maple Learn: an interactive online environment for exploring mathematical concepts, solving problems, and sharing interactive content

  3. Maple Calculator: a free mobile app for math calculations, step-by-step solutions, and visualization

  4. Maple Flow: an easy-to-use math tool for engineering design calculations

  5.  MapleSim: a modern modeling and simulation tool for engineering students and professionals

 

In this workshop, we will concentrate on Maple and show you the aspects of this product that will most benefit your research work. These include specialized packages for computations in areas such as graph theory, number theory, differential equations and mathematical physics. Maple allows exact arithmetic or arbitrary-precision computations using software floats, but it also features fast numeric libraries for linear algebra, optimization, data analysis and more. In addition, you can take advantage of our publishing, connectivity and programming tools.

 

If you teach mathematics, Maple has tools to make your life easier. These include the popular Student packages that feature tutors and step-by-step walkthroughs of computations, and an extensive library of interactive educational applications. In addition, we will look at Maple Learn, our online product, which was designed with education in mind and has a large gallery of ready-made applications that instructors can use. While Maple is primarily used at the university level, Maple Learn is also effective for high-school and 2-year-college classrooms.

 

In this workshop, you will be able to get hands-on experience with both Maple and Maple Learn and find out which features will best serve your teaching and research needs. All registrants will receive limited-term licenses for these products so that you can continue exploring them after the workshop ends.

 

Please note: this course is provided free of charge and snacks will be available.

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Karishma Punwani

(Director of Product Management, Maplesoft)

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