Math 444 Section 202
Mathematical Research and Writing
Online Course Material 

NEWS January 2022

This is my fourth time to teach this course. This is a project course with no final exam. It was designed as a Pure Mathematics alternative to the project course MATH 441 and to meet the Arts degree requirement of a research and writing course in the discipline of Mathematics. When I taught MATH 444 in January 2020 and January 2021, the course became online by necessity and used Canvas for assignments.

Kareem, Taeyeon and Daniel topic: Unlocking Locks.
Victor, Yiqing and Zecheng topic: Divisibility
Hang, Ke and Rui(Ray) topic: Taylor`s Theorem
Michel, Matthew and Jingwen topic: Ham Sandwich Theorem
Keying, Katarina and Yulun topic: applications of linear algebra including Page Rank.
Xingcan, Mandy, Alyssa topic: Congruences including Chinese Remainder Theorem.

The course will be online for a portion of the term. We will start in-person lectures Tuesday February 8. Our classroom has been moved to MATH 203, a more spacious room allowing some distancing. We will be using Zoom for lectures for the online portion of the term. I will email you the Zoom invite/password. It is expected that this will be mostly an in-person course and attendance (online or in-person) will be an important part of the course. One challenge for an online version is the formation of student groups (of 3) for the project. It should be much easier this year with in-person lectures.

By itself, this course is certainly not a doorway to Mathematical Research at the graduate level. For that, students should explore our honours courses with MATH 320 being centrally important. One can certainly go on to graduate school in Mathematics without being an honours student, but success in our honours courses is essential training. We think that this course is a valuable experience that provides an opportunity to all undergrads, Majors or Honours, to explore a Mathematics topic to depth. Moreover, students going on to further degree programs should develop useful skills here.

What is good Mathematical exposition. We had a lively discussion with 13 students offering their thoughts. I have reordered some what the issues.
  • Good English (grammar, organization into paragraphs, coherence etc)
  • Concise, no extra verbiage.
  • Introduction to provide overview of area
  • For example, remarks about result to put it into perspective compared with other results
  • Definition of terms with a clear notation
  • State the theorem. We need clear hypotheses and clear conclusions.
  • Summary of a proof at beginning of a proof
  • Proof should have a clear structure
  • Indicate the nature of the proof such as induction or proof by contradiction etc.
  • The somewhat difficult results has a proof of simple steps. Good logic. One student mentioned the need to not skip steps. Think of the audience as 4th year major students and write accordingly. Make it easy for your readers.
  • The Theorem/Proof model is certainly a Mathematical professional style and differs from other areas.
  • some pictures of graphs are appreciated by the reader. In some cases they may be essential.
  • Give some (perhaps simple) examples and discuss applications
  • Discuss components of your theorem say from the hypotheses of the conclusions.
  • Provide Motivation.

  • Project overview and how I propose to grade the project.  

    Section 202 is taught 12:30-2 TTh. We will using Canvas for assignment submission but the course website will used to present materials. Assignments will appear on the course website as well Canvas. Enrollment is capped at 18. The course prerequisite concerns proofs (MATH 220 or MATH 223 or MATH 226). Discussions and writing up clear proofs will be central to the course. The content may draw on material you have seen in other courses. We hope the Mathematics will be fun for you.

    My office is Mathematics Annex 1114 which I will occupy from time to time. I will try some online office hours during the first two weeks when we are in online mode. Perhaps scheduling an online appointment will work best. I am hoping to confine office hours to TWTh but will be generous with my time on those days.

    Early in the course we will have students presenting proofs in class. As one student put it: I didn`t know that you needed to be so careful. We are less concerned with the difficulty of the proof than clear exposition. Don`t be surprised that your first attempts will need editing. If we have presentations during the online portion of the course, then we will use Zoom. Assignment 2 can be done online in Zoom using written notes in a pdf format. Later presentations will be done using Beamer (a pdf format in TeX that is somewhat like powerpoint).

    Assignment 3 may assist in beginning to choosing a topic. I will encourage the use of Mathematics Magazine or the American Mathematics Monthly or the College Mathematics Journal as a source of inspiration. They are readily available through the UBC library. These publications contain gentler Mathematics Research than a typical journal. They are often problem focused.
  • Journals. 
  • Students will present proofs from their chosen project area as progress reports. Groups of 3 or perhaps 2 students will be formed around a chosen topic. We will discuss the group size when enrollments have been determined.

    The final outcome will be a carefully written project on the topic. At the basic level, we want clearly written theorems/proofs easily understood by the other students. At a higher level, we`d like to see the students to explore some new ideas in the topic. The course has no exam.

    You will have to use Latex and Beamer in the course. The use of beamer software for the presentations will be required. The use of Latex will be required for the final project. You may find Overleaf to be excellent free software that allows two users to edit a file. I use TEXWorks on my Mac and then there is no need for an internet connection while editing.

    The course has its biggest focus on good Mathematical Exposition.

  • Course Outline: grading scheme etc. 
  • Assignment 1 upload to Canvas by Sunday January 15. A discussion 
  • Assignment 2 upload to Canvas by Thursday January 20.
  • Assignment 3 upload to Canvas by Thursday January 27.
  • Assignment 4 upload to Canvas by Thursday February 3.
  • Assignment 5 upload to Canvas by class time Thursday February 10.
  • Assignment 6 upload to Canvas by Thursday February 17.
  • Assignment 7 upload to Canvas by Thursday March 3.
  • Assignment 8 upload to Canvas by Thursday March 10.
  • Assignment 9 upload to Canvas by Tuesday March 22.

  • Course materials
  • Towers of Hanoi beamer file. Towers of Hanoi tex file. to produce beamer file. (lecture Jan 13) 
  • Three chocolate problems. We discussed the solution to the first problem. (lecture Jan 13) 
  • Runners Problem and Intermediate Value Theorem. (lecture Jan 18) 
  • Sperner`s Theorem and extremal combinatorics. (lecture Jan 20) 
  • Stirling`s Approximation for n! and binomial coefficients. (lecture Jan 25) 
  • Latin Squares Including ideas from Sudoku and transversals. (lecture Jan 27) 
  • Some Theorems you could consider for Assignment 4. you could consider for Assignment 4.  
  • Recurrences and beamer tex file (lecture Feb 1) 
  • A result about tiling a rectangle by rectangles from a paper by Stan Wagon containing 14 different proofs of the result.  
  • The quest for the perfect square including many results by W.T. Tutte.  
  • Pick`s Theorem page 1 page 2 page 3 page 4 page 5 page 6 page 7 page 8 (Lecture March 3) 
  • Matching problems and checkerboard problems. (Lecture March 8) 
  • Catalan Numbers (Lecture March 15) 
  • Your group paper/project is due Thursday March 31.
    This project/paper is 50% of the grade in the course. It will be done in Latex (you have probably seen the tex outline below) and a pdf file will be handed into me on Thursday March 31. You might show it to me earlier to get comments on writing and scope. The project would be something like 10 pages of 12pt Tex. More is fine. The final Assignment 10 is a group assignment of a presentation of say 20 minutes on your project.
    The grading of projects will consider a number of factors. I am looking for an interesting read.

    Factor 1: I will look at the content. More elementary Mathematics will not get as much credit as harder Mathematics. I want to be amazed. At the very least, you want to amaze the first year version of yourself.

    Factor 2: The clarity of the Mathematics will matter greatly. Some of you have rightly pointed out that there is a trade off between Mathematical sophistication and your abilities to present it clearly. My advice is to do both: interesting Mathematics and clear/logical exposition. But it is probably sensible to only tackle Mathematics that you can explain clearly. Note the need for definitions/notation.

    Factor 3: In addition there will be some marks associated with the style of the report. Interesting motivations improve the style. Poor grammar or spelling detract from the style. Sloppy presentation is not welcomed.

    You should avoid words such as clearly or obvious. I will deduct 5% for each usage of such words. You should not copy material from sources, whether cited or not, and instead digest the material and rewrite it yourself. Examples are a helpful addition. And maybe a few pictures.

  • Additional resources
  • Writing Mathematics.  some notes by me.
  • Some Mathematics Journals.  most appropriate for this course. Of course other journals exist but often they are written for specialists and are not as accessible. Taylor Francis will give you free access if accessed through UBC Library
  • Sample beamer file Tex output.  that is explaining the Towers of Hanoi file. Tex File  which includes a few comments to help you read it. It should provide a useful template to which you add your (beautifully written) Mathematical content.
  • Sample standard Tex output for progress report and project.  that uses sections and references of various sorts. Tex File  which includes a few comments to help you read it and also a picture Tutte pic . It should provide a useful template to which you add your (beautifully written) Mathematical content. I included the picture so you could see how to include a picture in your project: usepackage{graphicx}
  • An inequality proved by induction. 
  • Project ideas:

    An outline of a sample project. 

    A sample project from two years ago.  The quality of the writing is excellent and they get far enough into the topic to show a nontrivial application of Gaussian Integers. In the past I have had projects of greater Mathematical sophistication and less Mathematical sophistication. Hopefully you will find a topic to interest you.

  • Strange Facts for your amusement.  
  • Ten Rules for good Mathematical writing by Dimitri Bertsekas  
  • Some infinite jokes   from a professor Chris Ryan in Sauder.
  • You may find amusing parts of a youtube video by Jean-Pierre Serre on How to Write Mathematics Badly