Introduction

Mathematics 401 is the 1st of two semesters of undergraduate analysis and this semester I have decided to make sure students gain an instinctive mastery of (1) metric spaces and (2) inequalities so as to facilitate deep understanding and rapd progress next semester when we cover everything from derivatives (of all sorts), measure theory and integration. While I would like all the students to enter the class having mastered the essentials in metric spaces and inequalities, many (most?) have not and so we will focus on that with occasional side trips to gawk at the incredible sights and wild beasts that can be found in the part of the universe explored in analysis and geometric analysis. Ranging from real analysis to geometric measure theory, harmonic analysis and PDE, we will find that analysis is vastly different than simply what happens when you add ϵ’s and δ’s to calculus.

Whiteboard Notes

I will post notes from the lectures — the actual scribbles you see on zoom, essentially a transcript of the whiteboard lecture — as well as various notes I create for you throughout the semester. They will be rather rough and most useful as an aid to remembering what we dicsussed.

Lecture/Discussion Notes

Help session Notes:

My Book

Here is the updated version of the book we used. We covered chapters 7 and 8 — metric spaces and inequalities. There will be extra credit assignments out of other pieces of the book.

Link to the updated book (as of December 2023)

Note that the book is updated from time to time: Last update, Nov 4, 2023

Note: there are typos in the book and the book will be updated frequently. I am mostly focused on finishing a first draft in the next few months, but will also scan from time to time for typos.

Other Excellent References

I am recommending three references. Lindstrom’s Spaces, Fleming’s Functions of Several Variables, and Simmon’s Introduction to Topology and Modern Analysis — All three are excellent references that you should own as a budding mathematician. You will probably find the early chapters in Lindstrom and Fleming’s book helpful.

Reading Assignments

Here are the reading assignments — an important part of the assignment is to take notes and write a question for everything you do not understand and then bring that to class with you!

One other thing: Reading chapters 2 and 3 of Lindstrom’s book, Spaces, will be useful for some of you. Other will find the dissonance between my approach and his difficult. In general, in more advanced mathematics, there are multiple approaches because there are multiple styles of thought and different tastes in how to approach a subject. You can’t really know what fits you until you have sampled different approaches. Chapters 2 and 3 are a more in depth, more details and less concise exploration of the material in Chapter 4 of my book.

  • September 16: Sections 4.1-4.3

  • September 18: Sections 4.1-4.3

  • September 21: Sections 4.1-4.3

  • September 23: Sections 4.4-4.5

  • September 25: Sections 4.4-4.5

  • September 28: Sections 4.4-4.5

  • September 30: Sections 4.4-4.5

  • October 2: Sections 4.4-4.5

  • October 5: Sections 4.6-4.7 (to exercise 4.7.9)

  • October 7: Sections 4.6-4.7 (to exercise 4.7.9)

  • October 9: Sections 4.6-4.7 (to exercise 4.7.9)

  • October 12: Sections 4.6-4.7 (to exercise 4.7.9)

  • October 14: Sections 4.6-4.7 (to exercise 4.7.9)

  • October 16: Sections 4.6-4.7 (to exercise 4.7.9)

  • October 19: Section 4.7 from Exercise 4.7.10 and 4.8

  • October 21: Section 4.7 from Exercise 4.7.10 and 4.8

  • October 23: Section 4.7 from Exercise 4.7.10 and 4.8

  • October 26: Section 4.7 from Exercise 4.7.10 and 4.8

  • October 28: Section 4.7 from Exercise 4.7.10 and 4.8

  • October 30: Section 4.7 from Exercise 4.7.10 and 4.8

  • November 2:Section 5.1

  • November 4: Section 5.1

  • November 6: Section 5.1

  • November 9: Section 5.2

  • November 11: Section 5.2

  • November 13: Section 5.2

  • November 16: Section 5.2

  • November 18: Section 5.2

  • November 20: Section 5.2

  • November 30: Section 5.2

  • December 2: Section 5.2

  • December 4: Section 5.2

  • December 7: Section 5.2

  • December 9: Section 5.2 - 5.3

  • December 11: Section 5.2 - 5.3

Exercises

I expect the exercises to be completed in detail and written up very clearly, so that they are easy to read. If you do not have clear handwriting, you should use LaTeX or some other typed form of communication. I recommend LaTeX since that is a very useful and widely used method. But anything that makes your thoughts clear and easy to read will do. (If you write clearly enough and have an iPad, you could consider using a note taking program to generate the solutions you hand in.)

I expect you to try all the exercises in the sections we cover, and will ask you turn in all of your work, but we will grade a subset of the problems. If you see “Challenge” or “Hard” or “Exploration” at the beginning of the exercise, this is an extra credit exercise.

  • Assignment #1: Exercise 4.4.5. Due Friday, September 18, 2020

  • Assignment #2: Exercises from Sections 4.2, 4.3, 4.4 and 4.5. Due Friday October 2 (We will grade 4.3.1 part 1 — Defn I implies Defn II), 4.4.3, and 4.5.1 carefully)

  • Assignment #3: Exercises from Section 4.6 and Exercises 4.7.1-4.7.9. Due Friday, October 16 (We will grade 4.6.1, 4.7.4, 4.7.7, 4.7.8 carefully)

  • Assignment #4: Exercises 4.7.10-4.7.16 and 4.8.1. Turn in work on 4.7.10-12, 4.7.14, 4.7.15 and choose at least one to do completely and carefully and let us know which one you chose. Due Friday, October 30

  • Assignment #5: Exercises in section 5.1. Turn in work on 5.1.1-5.1.7 — we will grade 5.1.5 and 5.1.6. Due November 6

  • Assignment #6: Exercises from 5.2. Due November 13

  • Assignment #7: Exercises from 5.2. Due November 20

  • Assignment #8: Exercises from 5.2. Due December 4

  • Assignment #9: Exercises from 5.2. Due December 11

Other Homework

Extra credit and other broadening experiences will appear here:

  • Watch David Levy’s Talk, No Time To Think, Youtube Link

  • Read Bill Thurston’s article, On Proof and Progress in Mathematics, Link to arXiv page

  • Read Daniel Coyle’s book, The Culture Code, Link to his website. This book gets a very strong recommendation from me — brilliant selection of stories and choice of central principles.

  • Read David Epstein’s book, Range, Link to his website. I like some chapters a great deal, while others I find very lacking. Overall, every much worth reading. The central message of the power of a richly diverse path to where you are going is very important and often unknown even though it seems like it should not be.

Note: We will figure out, in a collaborative way, how to decide when the exercises will be due. Assignment 1 above is challenging and is weeks of work probably, at least if you have many other classes.

Tests

There are no tests — the problems are enough work already!

Lecture/Discussion Sessions

This class meets every Monday, Wednesday, and Friday from 2:10 - 3:00 on Zoom. I have sent all the registered students the zoom coordinates, I you don’t have them, email me.

Help Sessions

Here are the current help sessions:

  • Enrique Alvarado (with assistance from Yunfeng Hu) will run a help session from 1:00 - 2:00 every Tuesday. The zoom coordinates for the meeting are: 219 936 2865 and the passcode will be emailed to the class.

  • Katrina Sabochick will run a help session as well. She will run her’s Thursdays at 10:00 - 11:00. The Zoom meeting coordinates are: 951 2747 3616. I sent out an email with the passcode. Her fist help session is this week, on the 17th.

  • I will run a couple of help sessions. I will use the same zoom channel as the class. I am starting with Monday and Wednesday 4:30-6:00.

I expect to fine tune these times a bit the week of September 14.