Mathematics 589: Spring 2022 — Geometric Analysis with a View to Use
In this course, I will present advanced insights and tools in a way that requires very little background to understand. More specifically, if you have a strong grasp of calculus, linear algebra and perhaps differential equations, and you are motivated to work a bit to understand some fascinating and useful tools from geometric analysis, you are well prepared for this class.
This is accomplished by exploiting every intuitive handhold to help those without the usual background to scale impressive heights safely. First I will focus on explaining the meaning of theorems and definitions. Next, we will have a look at how they are used. This will be supported by an explanation of the key ideas that make those theorems true or the definitions useful. Finally, if and only if it is clearly useful for understanding and it does not distract from the narrative we are pursuing, a proof will be outlined, using the intuitions we have previously built up.
There will not be a text for the course, though various notes will be given out from time to time. I will organize a note taking effort and these notes, plus figures should be available soon after each class. We (my group and I) plan on turning the notes into a book that will be published in paperback form, but that will also be given out freely as a pdf. In the notes/book, there will be very carefully chosen references, chosen for the quality of exposition, as well as lists of applications for the ideas and even historical notes. As already mentioned there will be (many) hand drawn figures as well.
The content will focus on (often extremely) economical paths into fairly advanced tools and insghts at the intersection of geometry and analysis. Minimalism not only aids in this ambitious journey, it also forces us to focus on what is important for understanding, for mastery.
Content of the Course (the order may change)
Geometric Linear Algebra (Linear Algebra Done Right According to a Geometric Analyst)
A Concise Tour of Metric Spaces and other Useful Technical Details
Spaces: the Zoo of Countries where Everything Happens
Derivatives = Linear Approximations, and the Huge Zoo of Generalizations
Outer Measures, Hausdorff Measures, Covering Theorems, and Fractals
Inverse Function Theorem, Implicit Function Theorem, and Sards Theorem: Just Three (Powerful) Examples of the Use of Derivatives
Manifolds, Vector Fields and Flows on Manifolds: An Invitation to the World of Dynamical Systems
Degree Theory and Fixed Point Theorems
Inequalities, Geometrically: Concentration of Measure in High Dimensions, Isoperimetric, Jensens, Cauchy-Schwarz, and AM-GM inequalities
Area, Coarea, Crofton and Gauss-Bonnet
Convexity, Legendre-Fenchel Transform, Duality
Forms, Currents, and Minimal Surfaces: A Bird’s Eye View
The Flat Norm Saga: from Image Denoising and Shape Recognition through Properties of Minimizers to Median Shapes and Chemical Interfaces
Smooth, Compact Sets: Distance Functions, Sets With (and Without) Positive Reach, Curvature Measures, and Tube Formulas.
Logistics
What: Mathematics 589, Geometric Analysis with a View to Use
When: Spring Semester, Wednesdays 1:10 - 4:00
Where: Terrell Library 106
Who: Anyone with a background in calculus, linear algebra and differential Equations and a motivation for explore new things
Notes
This is where the notes will appear as they are collected, edited and typed up. This is also where various articles and other notes will be posted.
Course Notes
Other Notes
Original Advertisement For The Course
Topics in Analysis: Geometric Analysis with a View to Use
Semester: Spring 2022
Day and Time: Wednesdays 1:10 - ~4:00 (3 credit hour course)
Room: Tentatively Todd 220
Webpage: https://geometricanalysis.org/spring-2022-topics-in-analysis
Details: In this course, I will focus on covering quite a few modern ideas in geometric measure theory and geometric analysis, but with a rather unusual approach (at least for mathematics courses).
Each lecture will focus on one idea or cluster of ideas and tools, covering (1) the meaning of the theorems/ideas/tools, (2) then how they are or can be used, (3) then the core insights supporting, proving the theorems/tools and (4) then finally, when it is useful for understanding and use, an outline of the proofs showing how those key ideas give you the proofs.
The driving goal = deep, useful, instinctive understanding for participants, irrespective of background and so the class will be appropriate for a wide spectrum of motivated participants from many areas including mathematics, statistics, physics, engineering, economics, and chemistry. (This goal and audience actually helps us find highly efficient paths to understanding.)
Exercises will be given to deepen understanding, but grades will be based on participation because our unusual approach demands it -- a short phrase describing our path to mastery might be "an immersive journey to instinctive mastery".
Topics covered will include (but will not be limited to) derivatives (an entire zoo, actually), inverse and implicit functions theorems, area and co-area formulas, manifolds and currents, deformation and compactness, convexity, convexification and the Legendre-Fenchel transform, concentration of measure in high dimensions, random projections and the Johnson-Lindenstrauss lemma, Hausdorff measures, densities and covering theorems, and curvature measures and topological invariants.
You really cannot be "too applied" for this course -- as long as you use any mathematical methods (or would like to) the chances are you will find the course interesting and useful. Another good title for this course would be modern "Geometric Analysis for the Gifted Amateur" (inspired by Lancaster and Blundell's book on Quantum Field Theory) or "A tour of Geometric Analysis for the Working Scientist".
On the other hand, I will point out numerous tangents that ambitious participants can take, that will fully occupy the most industrious students, so, if you are an advanced student, don't misinterpret the above to mean that the class will not contain enough challenging material to keep you busy!
Please do not hesitate to contact me with questions.