Mathematics 401: Mathematical Analysis for Undergraduates

We will explore the wonderful world of sets (usually in $R^n$) and functions and measures in finite dimensions, seeing how far we can push the basic ideas you have already met in calculus. It turns out that the idea go a very long ways into a very wild universe with lots of surprises and delights and in this course you do not get a front row seat — you actually are immersed in the action, doing bare-handed combat with big ideas, learning to tame them and use them to see much more deeply in the mathematical world that is beautiful, surprising and extremely useful. I will also expose you to ideas that are deeply important and usually ignored, at least in undergraduate mathematics courses. Often these ideas and perspectives will be discussed in class and then mastered through extra credit classes.

Mastering the mathematics we study will require a lot of deep, focused thought and exercise — this is not something that you can expect to do in groups or teams. While groups and teams are a very important part of a complete program aimed at mastery, a truly deep grasp is obtained only by taking plenty of time to think, in quiet, away from texting and email and twitter and instagram and other forms of modern distraction! Because this is so important, one extra credit assignment will be to read and write a critique of Deep Work by Cal Newport, a book aimed precisely at the digital distraction that is robbing so many of the opportunity to develop deep expertise in anything. Another early extra credit assignment will be to listen to and then comment on David Levy’s 2008 Google Tech Talk, No Time To Think.

For the more ambitious who want to get a head start on the extra credit and think that a book and a lecture on youtube are just not enough, another article that will make its appearance in this extra-credit list is Bill Thurston’s paper, On Proof and Progress in Mathematics.

Who and Where and When

My teaching assistant is Katrina Sabochick who is my PhD student. If you have already made it to this page, you have found my research group website and have had a chance to learn a bit about myself as well as Katrina. Katrina’s email address is katrina.sabochick@wsu.edu.

Lecture Experiences: MWF 10:10 - 11:00 am, Neill Hall, Room 5W

Office Hours and Collaborative Learning sessions: W 1:00-5:00 pm, Neill Hall, Room 416

Student Run Collaborative Study Session: TBD

What we will cover

This class, together with the continuation in the spring, Mathematics 402, will cover all but the last chapter of Tom Lindstrom’s Book, Spaces: An Introduction to Real Analysis. I will also recommend consulting Wendell Fleming’s Functions of Several Variables, the book I previously used for the class, as well as George F. Simmons’ Introduction to Topology and Modern Analysis . We will exploit the rich connections between geometry and analysis that make analysis an area that can be navigated more easily than most students usually think.

Expectations

You are expected to do the reading before you come to class and try your hand at the exercises that have been assigned. You should expect to spend a minimum of 15 hours a week on the class, though for full mastery, it make take more time. Coming to every class and every collaborative help session will give you a much better shot at mastery!

There will also be a collaborative working session every Wednesday evening that my graduate students will run. I will sometimes also attend the session.

Grades

Grading will be determined through a combination of attendance, quizzes and a take home final test. There will also be extra credit assignments as well. The percentages of the grades will be 25% on the quizzes, 45% on the exercises, and 30% on the take home final. I will make several important comments on grades and how to flourish in the class on the first day. I will permit excellent attendance to make up for as much as 15% of the grade because I believe that committed engagement in this immersive experience is extremely valuable.

Texts, Notes and References

Text for the course: Tom Lindstrom’s Spaces: An Introduction to Real Analysis

Suggested additional texts: These are not required

Here are some Notes and References:

Notes I write for this class:

Assignments and Tests

Reading and exercise assignments will be listed here. You are encouraged to read and study ahead, starting as soon as you read this syllabus for the first time. I will cover a section per lecture, but there will be additional lectures that are not in the book added fairly frequently. These will focus on some aspect of geometric analysis and its applications to data science.

I will put post the assignments here as well as the final take-home test when that is ready.

Reading assignments are the section numbers in the “Spaces”. Exercises are given in the form of <chapter.section.exercise-number>. You should also be playing with exercises that are not assigned.

  • Reading #1 — Due August 23, 2019: 1.1-1.3

  • Exercises #1 — Due August 23, 2019: 1.3.7(c)

  • Reading #2 — Due August 30, 2019: 1.4-1.6

  • Exercises #2 — Due August 30, 2019: 1.4.4 and 1.5.3

  • Reading #3 — Due September 6, 2019 Chapter 2 (read over, begin to master it)

  • Exercises #3 — Due September 6, 2019 2.1.6 (typo: a > 0 —> x > 0)

  • Reading #4 — Due September 13, 2019 Chapter 2 (master the whole chapter)

  • Exercises #4 — Due September 13, 2019 Chapter 2.3.10

  • Reading #5 — Due September 20, 2019 Chapter 3.1-3.3

  • Exercises #5 — Due September 20, 2019 3.1.9 and 3.2.4

  • Reading #6 — Due September 27, 2019 Chapter 3.3-3.5

  • Exercises #6 — Due September 27, 2019 3.3.13

  • Reading #7 — Due October 4, 2019 Chapter 3.5-3.7

  • Exercises #7 — Due October 4, 2019 3.4.8 (extra credit 3.5.11)

  • Reading #8 — Due October 11, 2019 re-read Chapter 3

  • Exercises #8 — Due October 11, 2019 re-do any single exercise (from chapter 3) you would like to redo

  • Reading #9 — Due October 18, 2019 4.1-4.2

  • Exercise #9 — Due October 18, 2019 4.2.6 (extra credit 4.2.11)

  • Reading #10 — Due October 25, 2019 4.3 and 4.4

  • Exercises #10 — Due October 28 2019 4.3.5

  • Reading #11 — Due November 1, 2019 4.5 and 4.6

  • Exercise #11 — Due November 1, 2019 4.6.4 a,b,c (Extra Credit 4.4.1)

  • Reading #12 — Due November 8, 2019 4.7

  • Exercise #12 — Due November 8, 2019 4.7.1

  • Reading #13 — Due November 15 2019 4.8 and 4.9

  • Exercise #13 — Due November 15, 2019 4.8.5

  • Reading #14 — Due November 22, 2019 4.9 and 4.10

  • Exercise #14 — Due November 22, 2019 4.10.2 (This is the last Exercise for the Semester)

Other Reading: Mathematical and Otherwise

I will post here the other books and papers I recommend.

  1. Range by David Epstein — Caution: stories are good, but there is a serious deficiency at times regarding the implied wisdom which is not wisdom. for example, the idea of domain experitise being valuable in solving problems is called into question by the InnoCentive and Kaggle stories. This is deeply misleading. The truth is only clear with a deeper reading of those stories and a willingness to distrust the experts that Epstein quotes. In particular, the book does not expose an understanding of the importance of deep work in every area of endeavor. Nevertheless, the book tells some good stories and raises good questions and is very valuable in its pointing out the immense value in being careful where and when you work deeply, the immense value in cultivating a “renaissance man”, “renaissance woman” perspective, and the immense value inbreaking down silos.

  2. The Culture Code by Daniel Coyle. Superb book. While I have minor quibbles, Everyone should read this book. I wrote a review here: https://notesfromkevinrvixie.org/2019/07/09/cultures-of-creativity-and-innovation/.

  3. Deep Work by Cal Newport. Excellent chapters and then chapters that make me cringe. It is rather unique in the extremes I felt when I was reading it. It is worth reading, worth paying attention to the core message of the value of deep, focus, deep flow, as well as an alert to the stuff that distracts and disrupts flow and focus. Most of what I did not like stems from the inexperience of the author. Things like his arrogance, the elitism that infects the TED talk/thought leader/best seller crowd and a lack of deeper wisdom. But it remains that there is a lot to recommend the book, chapters that are excellent, and a deeper conversation that this will spark as you read the book thoughtfully, deeply.

  4. On Proof and Progress in Mathematics by Bill Thurston. This article, written in 1994, is an inspiring look at how we can generate a generous path to progress in mathematics. I found it very inspiring and so have many other mathematicians. Here is a link: https://www.ams.org/journals/bull/1994-30-02/S0273-0979-1994-00502-6/

  5. You should become acquainted with the arXiv: Here you go https://arxiv.org/

  6. David Levy’s 2008 Google Tech Talk, No Time To Think.

  7. The Age of Surveillance Capitalism by Shoshana Zuboff. I have not read a lot of this book yet, but already I believe it is an incredibally important book. Hearing her interviewed was the last nudge it took for me to delete by Facebook account. The thing that makes this book important is not only the scarcity of books that deal with this subject in a deeper way, but the fact that this subject is so critically important to the future. While there are places I would go deeper, it was probably not clear that you could do that and keep a wider audience. I believe that the climate and information/freedom/individuality/humanity disasters that the world is heading towards are fundamentally spiritual in nature, but navigating that, in a book intended for a very diverse audience, is hard -- though I believe not impossible. She may do a bit of it. I have only read a limited pieces of it so far, but there are points I have read at which it would be natural to hint at this where she has not. At any rate, I expect to end up with a very high opinion of the book when I am done with it, even if I find places where she could have delved more deeply.