Math 3040                  Course Syllabus and Homework Assignments                           Spring 2014

Last updated:  April 27, 2014.

Welcome to the Math 3040 web page (www.math.cornell.edu/~web3040/).  This page contains information about: course organization, homework, and contacting the instructor or TA, as well as important announcements. The information will be updated throughout the term, so please consult this page and the homework assignments regularly to make sure you have the most recent version.

R. Connelly's office hours:  1:30-2:15 Wednesdays in Malott 433.

Z. Tu's office office hours:  2:00-4:00 Fridays (Feb. 6- March 20).

Announcements: The midterm Prelim will be held on February 27, 2014 during class time. Note the change in day.

The final examination for the course will be held on TBA.

What follows is the intended course syllabus for the first few weeks. A syllabus for the remaining weeks will be adjoined to this later in the term.

Text: Martin V. Day: An Introduction to Proofs and the Mathematical Vernacular  

This course will as much about writing clearly and coherently as it is about deciphering and creating proofs.  You will be graded on the overall clarity as well as the correctness of your writing.  You may also want to consult  "The elements of style" by William Strunk Jr. and  E. B. White as well as "Eats, Shoots & Leaves" by Lynne Truss.

Please turn in your homework in a printed legible format.   The following are the reading assignements for the week indicated.  Please read the indicated sections before you come to class so we can have an informed discussion in class.

Some people who are asked and who have volunteered will present a "proof" verbally in class, and it will be subject to the scrutiny of their classmates.  For those people, this can be in place of a homework assignment or take-home prelim.

Please hand in your homework as a typeset document in LaTeX.

 Quizzes, Prelims and Final: There will be:

1) a few unannounced quizzes throughout the term, usually of 10-15 minutes duration;
2) two  in-class prelims, the first  on Thursday,  February 20, 2014 in class,
3) a  final exam on TBA

Grading policy: The following is roughly how the different components of your work will be weighted to determine your final grade: Homework, 15-20%; Classwork (including class participation and quizzes), 20-25%; Prelims, 15% each; Final exam, 30%. The Mathematics Department's (voluntary) guidelines for the mean grade in courses at this level is B+. We expect that this will be close to the mean for this course.

Academic Integrity: The usual rules for academic integrity will apply to all work submitted for credit in this course. Of course, this includes prelims and the final exam. You may consult with other students about the homework exercises---in fact, this can be a helpful way of learning the material---but the homework you submit should show all your work and be in your own words.  Also be very careful about quoting sources that you use, whether it is from your roommate, the web, the bible, or your mother.  Failure to do so could be a basis for plagiarism.

Writing homework in TeX:  I strongly suggest that you write your homework in the language of LaTeX.  You can download the software TeXShop or MiKTeX for Windows for free.  You should follow the instructions for downloading, installing and running the software.   Good luck.  Here is a nice simple template, from Professor Tara Holm, that Louisa Schwartz suggested that could be used to get started.

Possible project subjects:
  1. Euclidean geometry:  Dissections, Scissors congruence, tensegrities, convexity, polyhedra, Platonic solids, Archemedean semiregular solids, what is a fair die?, tilings.
  2. Set theory and logic:  The lambda calculus, Dedekind cuts, Godel's Theorem, etc.
  3. Analysis:  Proof of the intermediate Value Theorem, Proof of the mean value theorem, etc.
  4. Linear Algebra:  one-to-one linear functions from a finite-dimensional space to itself are onto, the spectral theorem, etc.
  5. Topology: Knot theory, classification of 2-manifolds, Hairy ball theorem, Brouwer's fixed point theorem.
  6. Number theory:  Fermat's Last Theorem, Fermat's Little Theorem and cryptography.
  7. Probability: Bayes' theorem, Monty Hall problem.
  8. Collatz conjecture.
  9. A flexible polyhedral surface.
You should use sources beyond the Wikipedia articles above, and a way to find articles, papers, surveys related to a given subject are to use MathSciNet (available from Cornell servers), google scholar, Math arXiv, or the author's web page.  Please discuss with me beforehand the subject you would like to talk about. You should prepare about a 20 talk that involves some proof, as well as a written text in LaTeX (a page or so) with the main statements written carefully.  Be clear to the point, and make your presentation as interesting as you can.  The audience is to participate with questions and be prepared to defend your arguments.

Week 1:     Some Proofs:  Chapter 1  Sections A, B, C.  Read about equivalence relations here, and on page 50 and 60 in Day's text.

Problems:   1.3, 1.6, 1.8.  Read Quinn's article, Quinn-Jaffe article, and Bill Thurston's reply.  On Tuesday be prepared to answer the question:  "Is there a place in mathematics for physical intuition?".  Whatever you think, make your arguments clearly and logically.

Due Thursday, January 30, 2014 in class.

Here is a Proof of the Cauchy-Schwarz inequality that we discussed in class.  This is the basic TeX.

Week 2:     Valid arguments, Quantifiers, and Proofs.  Sections D, E, F. 

Problems:    1.19,  1.20,  1.24,  1.29,  1.31.  Equivalence Relations.

Due Thursday, February 6, 2014 in class.  Solutions.  

Week 3:   Read Chapter 2A.

Problems:  2.1, 2.4, 2.5, 2.7, 2.8, 2.10

Due Thursday, February 13, 2014 in class.  Solutions

Week 4:   Read Chapter 2B.  Induction and Proof techniques. 

Problems:  2.12, 2.14, 2.18, 2.20,  Problems on Pythagorean triples.

Due Thursday, February 20, 2014 in class.  Solutions

Week 5:   Read Chapter 3A, 3B, 3C.  Sets and functions.

Midterm Prelim:  In class covering Chapters 1 and 2.  February 27, 2014 during class time. Note the change in day.

Practice problems and reminder of subjects covered on the Prelim. 

A second chance for Prelim 1.  You can hand in solutions to the problems you did not get during the exam period.   Please LateX your solutions and have them in by Tuesday, March 11.  Here is the exam.

Week 6:   Continue to read Chapter 3A, 3B, 3C, 3D, 3F.  Sets and functions.

Problems:  3.1, 3.5, 3.12, 3.15, 3.16.  Due Thursday, March 6, 2014, in class.  Solutions

Week 7:   Continue to read Chapter 3D, 3F, 3G.  Sets and functions.

Problems:  Set theory problems. (Hint for problems 3 and 4.  A countable union of countable sets is countable, and a finite product of countable sets is countable.)  Due Thursday, March 13, 2014, in class.  Solutions

Week 8:   Read Chapter 1 of Introduction to Probability, by Grinstead and Snell. 

Problems:  Hand in Exercises 8, page 36; 17, page 37; 31 page 40, and the following two problems, one of which is from the book "Pillow Problems and a Tangled Tale" by Lewis Carroll.  Also here are a couple more problems on the Schroeder-Bernstein Theorem.  Due Thursday, March 20, 2014, in class.
Solutions

Week 9:   Read Chapter 4 of Day's Text:

Problems:  Problems 15 and 16 page 73 of Grinstead and Snell's text.  Problems 4.16, 4.17 and 4.24 in Day's text. Due Thursday, March 27, 2014, in class.  Solutions

Week 10:   Continue to read Chapter 4 of Day's Text:

Problems:   4.15, 4.20, 4.25, page 88,  in Day's Text.   Number Theory problems.      Due Thursday, April 17, 2014, in class. (We will have presentations on Thursday.)  Note the later due date for these problems and corrections to the Number theory problems, especially Problems 1 and 2.

Week 11:   Projective Geometry.  Please read the following and hand in the problems on Thursday, April 24, 2014.  Solutions.

  Take  Home Prelim,  due  April 29 in  class.   The rest  of the time will be  taken up with presentations.

Week 12:   Presentations on any subject.  (Suggestions above.)