MA387 Course Syllabus

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]
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
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
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!