This syllabus will be filled out more as the semester progresses. Check back often for updates and extras.
I expect the reading to be done prior to coming to class, and the exercises to be started.
Lesson 1
On day 1, we will look at the "Big Picture" of mathematics, and how real analysis fits in. We'll also look at some of the surprising results in store for us this semester, and one of my favorite proofs.
Think About: What are the major themes of mathematics? major techniques?
Lesson | Date | Section | Reading | Assignments | Highlights & Extras |
---|---|---|---|---|---|
1 | 14-Jan | 1.1: Preliminaries | pp. 1-10 | Ch1: 1,3,5 | my favorite proof |
Block I: Real Numbers
In the first block, we will review proof techniques and learn about how to construct the real numbers starting with the natural numbers 0,1,2,… We will also take a look at how the reals are used to construct Euclidean space.
Think About: How many numbers are there? Are there more real numbers than rational numbers? Why do you think so? What does continuity really mean? What are some real-world situations in which the notion of continuity (or discontinuity) appears?
Lesson | Date | Section | Reading | Assignments | Highlights & Extras |
---|---|---|---|---|---|
2 | 16-Jan | 1.1: Preliminaries | pp. 1-10 | Ch1: 6,7 | Hilbert and mathematical rigor (Skill: proof writing) |
3 | 18-Jan | 1.2: Cuts | pp. 10-17 | Ch1: 9,10,11,12,14,15,17 | construction of the Reals |
4 | 23-Jan | 1.2: Cuts | pp. 17-21 | [Turn-in: 1,3,5,6,7,9!,10!] | Dedekind |
5 | 25-Jan | 1.2: Cuts | (Fun Math: geometric spaces) | ||
6 | 29-Jan | 1.3: Euclidean Space | pp. 21-27 | Ch1: 18,26 [Turn-in: 11,14!,15!,18] |
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7 | 31-Jan | 1.6: Continuity | pp. 36-39 | Ch1: 19,38 | |
8 | 4-Feb | 1.6: Continuity | [Turn-in: these problems] | points of discontinuity | |
9 | 6-Feb | 1.4: Cardinality | pp. 28-32 | Ch1: 25,32 | Cantor & the uncountability of the Reals |
10 | 8-Feb | Applications/Review | [Discuss: 19,38] | (Application: cake cutting) The Fair Division Calculator |
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11 | 12-Feb | WPR I | about WPR I | [Turn-in: 19,38] |
Block II: Sets & Sequences (Topology)
In this block, we examine series and sequences of real numbers, and answer the question of when a limit of these numbers exists. We also look at what happens when sets are equipped with a notion of distance. Much of the required terminology comes from the area of Topology, which is essentially a mathematical study of the "closeness" of points.
Think About: What makes a sequence of points converge? Is it possible for a sequence to converge to more than one number? What if you're allowed to remove any terms you want from the sequence? What if you're allowed to rearrange the sequence? What about a series, in which you look at sums of the terms in a sequence?
Lesson | Date | Section | Reading | Assignments | Extras |
---|---|---|---|---|---|
12 | 14-Feb | 2.1: Metric Spaces (sequences and subsequences) |
pp. 51-54 | Ch2: 26* | Exams returned and discussed sequences/series |
13 | 19-Feb | 2.1: Metric Spaces (metrics; continuity) |
pp. 51-56 | Ch2: 15,24,82 | Mini-project assignments metric spaces |
14 | 21-Feb | 2.1: Metric Spaces (open and closed sets) skip Homeomorphism |
pp. 58-64 | Ch2: 3*,5*,13* | Leonhard_Euler and the Konigsberg Bridges |
15 | 25-Feb | 2.1: Metric Spaces (continuity; inheritance) |
pp. 55-56,64-69 | Ch2: 6,17 [Turn-in: these problems] |
(Fun Math: donuts & coffee cups) |
16 | 27-Feb | 2.1: Metric Spaces (product metrics; Cauchy sequences) |
pp. 71-75 | Ch2: 27 | the taxicab & Facebook metrics |
17 | 29-Feb | 2.1: Boundedness | p. 75 | Ch2: 29*,30* Paper selection for projects |
contractions & fixed points (Fun Math: The Brouwer fixed point theorem) (Skill: counterexamples) |
18 | 4-Mar | 2.2: Compactness I | pp. 76 | [Turn-in: these problems] | Frechet and the importance of compactness |
19 | 6-Mar | 2.2: Compactness II | pp. 76-82 | Weierstrass; the Bolzano-Weierstrass & Heine-Borel Theorems | |
20 | 10-Mar | 2.5: The Cantor Set | pp. 95-99 | [Turn-in: these problems] | Cantor sets |
21 | 12-Mar | 2.5: The Cantor Set | pp. 95-99 | Mini-projects due by 13-Mar | fractional dimensions (Fun Math: weird spaces) |
Spring Break |
After Spring break, we will turn to the question: why is compactness SO important? We will also look at the most significant theorems which require the condition of compactness in a very fundamental way. Finally, we'll look at an alternate definition of compactness… which looks very different but turns out to be equivalent.
Lesson | Date | Section | Reading | Assignments | Extras |
---|---|---|---|---|---|
22 | 25-Mar | Review of 2.1-2.2 2.3: Connectedness |
pp. 82-87 | Bernard_Bolzano the topologist's sine curve |
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23 | 27-Mar | Applications of Compactness I | pp. 79-82 | (Application: optimization) | |
24 | 31-Mar | 2.4: Coverings | pp. 88-95 | [Turn-in: these problems] | space-filling curves |
25 | 2-Apr | Applications of Compactness II | (Application: approximations) | ||
26 | 4-Apr | 2.4: Coverings | (Application: sensor coverings) | ||
27 | 8-Apr | Applications | [First three glossary submissions due] | (Application: voting) | |
28 | 10-Apr | Review | [Turn-in: these problems as part of WPR grade] | ||
29 | 14-Apr | WPR II | about WPR II | All glossary entries must be completed by 1600 on 13-Apr |
Block III: Functions, Continuity, and Differentiation
In this final block, we will take a look at functions, continuity, and differentiation. Now that the concept of limit has been made rigorous, the concept of a derivative can be as well!
Think About: What conditions must you add to make a continuous function differentiable? What is "lost" (or "gained") at points where a function is not differentiable?
Lesson | Date | Section | Reading | Assignment | Highlights & Extras |
---|---|---|---|---|---|
30 | 16-Apr | 3.1: Definition of the Derivative Mean Value Theorem |
pp. 139-143 | ||
31 | 18-Apr | Course Drop | |||
32 | 22-Apr | 3.1: L'Hopital's Rule and Discontinuities | pp. 143-146 | [Turn-in: these problems] | |
33 | 24-Apr | 3.1: Power Series and Taylor's Theorem | pp. 149-151 | ||
34 | 28-Apr | 3.1: Power Series and Taylor's Theorem | pp. 149-151 | ||
35 | 30-Apr | Review | |||
36 | 2-May | WPR III | about WPR III | All glossary entries must be completed by evening on 1-May |
Course End
The last few weeks will be spent on review and topics of your choosing. You will get the chance to research, write, and speak about a topic related to analysis.
Lesson | Date | Section | Reading | Assignments | Highlights & Extras |
---|---|---|---|---|---|
37 | 6-May | Radius of Convergence/Power Series | 3.1,3.3 | [Turn-in: these problems as part of WPR grade] | |
Projects Day (8-May) | |||||
38 | 9-May | Project Drop/IPR's | |||
39 | 13-May | Project Presentations I | |||
40 | 15-May | Project Presentations II | |||
TEE | 20 May | TEE | Comprehensive | Th319 0735-1005 | Good luck! |