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Course Date: 12 September 2014 to 07 November 2014 (8 weeks)
Learn about functional analysis
John Cagnol is currently the Dean of Engineering and Associate
Dean of Studies at Ecole Centrale Paris, an ivy-league engineering
He specializes in Partial Differential Equations
(PDEs) and Modeling and Simulation and has authored over 30 research
papers and edited over five proceedings volumes for various
John has taught and carried out research at the
University of Virginia in the United States and has been the director of
the mathematics department and laboratory of scientific computing at
the ESILV in France.
As for his professional
interests, John is involved with the IFIP (International
Federation for Information Processing) and is the secretary of the
working group on Control of Distributed Systems. He is also on the board
of the ICAST (International Conference on Advancement of Science and
Technology) and is an editor of "Smart Material and Structures", a
publication of the IOP.
Anna Rozanova-Pierrat is assistant professor in Applied Mathematics at École Centrale Paris.
Her research areas include mathematical physics involving the models governed by partial different equations. She specializes in the theory behind inverse problems, equations of non-linear acoustics and diffusion problems on fractals, which
are also subjects of her 10 scientific publications.
Functional analysis is the branch of mathematics dealing with spaces of
functions. It is a valuable tool in theoretical mathematics as well as
engineering. It is at the very core of numerical simulation.
In this class, I will explain the concepts of convergence and talk about
topology. You will understand the difference between strong convergence
and weak convergence. You will also see how these two concepts can be used.
You will learn about different types of spaces including metric spaces,
Banach Spaces, Hilbert Spaces and what property can be expected. You will
see beautiful lemmas and theorems such as Riesz and Lax-Milgram and I will
also describe Lp spaces, Sobolev spaces and provide a few details about
PDEs, or Partial Differential Equations.
Will I get a Statement of Accomplishment after completing this class? Yes. Students who successfully complete the class will receive a Statement
of Accomplishment signed by the instructor.
What resources will I need for this class? For this course, all you need is an Internet connection and the time to
view the videos, understand the material, discuss the material with fellow
classmates, take the quizzes and solve the problems.
What pedagogy will be used? This MOOC is in English but the math will be taught with a "French Touch".
What does "teaching math with a French touch" mean? France has a long-standing tradition where math is addressed from a theoretical
standpoint and studied for its implicit value throughout high school and
preparatory school for the high-level entrance exams. This leads to a mindset
based on proofs and abstraction. This mindset has consequences on problem
solving that is sometimes referred to as the “French Engineer”. In contrast,
other countries have a tradition where math is addressed as a computation
Does it mean it will abstract and complicated? The approach will be rather abstract but I will be sure to emphasize
the concepts over the technicalities. Above all, my aim is to help you understand
the material and the beauty behind it.
Week 1: Topology; continuity and convergence of a sequence in a topological space. Week 2: Metric and normed spaces; completeness Week 3: Banach spaces; linear continuous functions; weak topology Week 4: Hilbert spaces; The Riesz representation theorem Week 5: The Lax-Milgram Lemma Week 6: Lp spaces; Fischer-Riesz Week 7: Sobolev spaces Week 8: Use of functional analysis for Partial Differential Equations
The class will consist of a series of lecture videos, usually between five and twelve minutes in length. There will be approximately one hour worth of video content per week. Some of the videos contain integrated quiz questions. There will also be standalone quizzes that are not part of the video lectures; you will be asked to solve some problems and evaluate the solutions proposed by your fellow classmates. There will be a final exam.
There will be some additionnal contents in the form of PDF files.