Geometric Measure Theory

De Pontão Nós Digitais

This is the main page of a graduate-level course in geometric measure theory, being taught in 2017/1 at the Polytechnic Institute IPRJ/UERJ.

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It is a follow-up course to Riemannian Geometry, Differential Varieties, Functional Analysis/graduate Real Analysis and basic Measure Theory. It begins with a review of exterior algebra, cohomology and measure theory. The subject, specially the concept of currents, has largely been developed at the instructor's alma mater, Brown University, mainly by Herbert Federer, and largely popularized by fields medalist David Mumford. These subjects have wide application, and came to the instructor's knowledge while being used in Mumford's course on Pattern Theory at Brown, jointly delivered with Peter Michor.

The spaces dealt with in this course extend the theory of finite-dimensional differentiable varieties, covered in a previous Differential Varieties course, to more general sets. Most importantly, this course provides fundamental language for researchers using advanced mathematics in many different fields, such as probability theory and machine learning in high dimensions.

The public is interdisciplinary scientists and mathematicians. Emphasis is given to a conceptual and intuitive understanding of this difficult but powerful subject, regarded as very hard within mathematics, while working out the most fundamental formalisms.

General Info

  • Instructors: prof. prof. Ricardo Fabbri, Ph.D. Brown University
  • Meeting times: Tuesdays 3:10pm-5pm, and Thursdays 10:40am-12:20pm, room

Course Format

  • Evaluation criteria: Final grade = exam (60%), class participation (20%) and exercises/reading summaries (20%)
  • There will be assigned exercises and reading almost every class (papers, book chapters, etc)
    • Readings must be summarized with personal opinions and reflexions and a summary must be typed and handed in

Pre-requisites

  • Strictly, one is expected to be familiar with traditional measure theory as seen in graduate-level real analysis or functional analysis (review will be provided for those coming from an engineering background)
  • Riemannian Geometry and Differential Varieties: we will review tensors at a deep level, from scratch, but your life will be a lot easier if you have already seen these concepts.
  • Linear Algebra: the only real pre-requisite. Students are expected to know linear algebra at the level of Hoffman and Kunze, and also in the context of its application to the theory of differentiable varieties and riemannian geometry.

Approximate Content

We will be reading sections of interest from Federer's book (starting with Ch 1), together with complements from the others. Focus may shift based on research demand and demand from student's individual projects. We plan to focus on the following topics.

Main Resources

Textbooks

We will be following Herbert Federer's famously hard book as a guide, mainly learning his language and style through the first couple of chapters, then selecting topics for advancing right to the theory of currents (ch. 4.1) as fast as possible [1]

Complementary Textbooks on Measure Theory

  • Geometric Measure Theory: A beginners guide, 5th ed, F. Morgan, Academic Press, 2016
  • Real Analysis, G. Folland, 2nd ed
    • This is the graduate course book used at Brown during my time, covering basic measure theory from a modern standpoint. Great for reviewing sigma-algebras etc as treated by modern mathematicians, but needs other less modern books (like Kolmogorov's) to complement insight.
  • An Introduction to Measure Theory, by fields medalist Terence Tao
    • Terence Tao gives great insight to basic constructions that seem arbitrary at first

Complementary Textbooks on Differential Geometry, Manifolds and Tensors

  • Foundations of Differentiable Manifolds and Lie Groupds, Frank W. Warner
    • A beautiful and modern intro to tensors on Chapter 2 provides an initial setting for Federer's also beautiful Chapter 1.
  • See also Geometria Riemaniana.

Lectures

Partial listing & Tentative Outline

Misc. Notes

Homework

Assignment 1

Assignment 2

Links

Keywords

Differential Geometry, Pattern Theory, Machine Learning, Fractals, Functional Analysis, Currents, Tensors, Exterior Calculus, Measure Theory, Measurable Sets, Computer Vision, Artificial Intelligence, High-dimensional pattern analysis.