# The language of mathematics, 2022 Semester 1

The aim of this course is to develop written and oral communication skills in mathematics. Through weekly readings, discussions, and presentations we will learn

- how to write mathematics clearly and coherently,
- how to use symbols and equations within mathematical text,
- how to structure and present arguments,
- how to organise long mathematical documents,
- how to present ideas clearly and effectively.

## Structure of the course

We will meet twice a week, on Tuesday from 12 to 1:30 and Thursday from 2 to 3:30. One meeting will be devoted to writing and one to oral presentation. In the meeting about writing, we will review the week’s reading, do in-class exercises, and discuss the writing assignment. In the meeting about presentation, one or two students will give short talks, and we will discuss various aspects of presentation raised by the talks.

- Tuesdays 12 to 1:30, MSI Board Room (HN 2.48) - presentation
- Thursdays 2 to 3:30, MSI Board Room (HN 2.48) - writing

## Assessment

The mark will be based on class participation (10%), writing assignments (60%), and in-class presentations (30%).

## Resources

- Francesco Vivaldi,
*Mathematical Writing for Undergraduate Students*(Required textbook) - Primer, University of Michigan,
*Demonstration, Proof beyond the possibility of doubt*(Online textbook)

## Rough plan for the writing component

- Weeks 1-3 : Fundamentals
- Weeks 4-6 : Writing proofs
- Weeks 7-9 : Micro writing (small-scale issues)
- Weeks 10-12 : Macro writing (large-scale issues)

## Collaboration and late policy

You may collaborate with others, but **everything you submit must be your own work**. That is, you must write your own submissions, and not copy it from anywhere else. If you get any outside help, you **must acknowledge it** by writing (for example) “I worked with … and … for this homework.” Or “I took help from … website for this homework”. Failing to do this is a form of academic dishonesty.

Late homework will be penalised at 10\% of the grade per day, for a maximum of 3 days, after which it will receive a score of 0. The exception is if you ask me for an extension **in writing** (email is fine) at least **24 hours before** the deadline.

## Detailed weekly plan

The writing assignments are due on Mondays by 5pm. I expect you to be familiar with the reading by the class on Thursday.

### Week 12

#### Reading and discussion

- Igor Pak, How to write a clear math paper

#### Writing

- Submit a revised draft of your final paper.
- Submit the final draft of your final paper (at the end of the exam period).

### Week 11

#### Reading and discussion

- Paul Halmos, How to write mathematics

#### Writing

- Submit a first draft of your final paper.

### Week 10

#### Reading and discussion

- Bruce Berndt, How to write mathematical papers

#### Writing

- Sumit a tentative title and outline of your final paper.

### Week 9

#### Reading and discussion

- Kleiman, Writing a phase two math paper

#### Writing

- Submit the final draft of your first paper.
The final writing assignment.

Your goal is to write a 5-10 page paper about the topic of your second talk. It should have a title, an abstract, a section called “Introduction”, other sections, and a bibliography. There must be more than one mathematical results with proofs. The mark will only be based on the final submission, but there are the intermediate submissions before the final one.

- In week 10, a tentative title and an outline of the paper.
- In week 11, a first draft
- In week 12, a revised draft
- In the exam period, the final draft.

### Week 8

#### Reading and discussion

- Watch Jean-Pierre Serre’s How to write mathematics badly.

#### Writing

- A first draft of your first paper.

#### Presentation

- No presentations this week

### Week 7

#### Reading and discussion

- Keith Conrad, Advice on mathematical writing

#### Writing

This is a longer writing assignment. The first draft is due in Week 8, and a revised draft is due in Week 9. Your mark will be based only on the revised draft.

Write a 1 to 2 page paper about the content of your first talk. Include a title and one or two short introductory paragraphs. Your write-up must contain at least one definition, one example, and one result with proof. See “6. Types of Mathematical Results” and “7. Definitions” in Keith Conrad’s document. For this assignment, you do not have to include references, and it is OK to not divide your document into sections.

### Week 6

#### Reading and discussion

- Primer, Proof techniques: Proof by induction
- Vivaldi, Chapter 8
- In class exercise 6

#### Writing

- Revise and resubmit one proof from assignment 4. Schedule a meeting to talk about the revision.

### Week 5

#### Reading and discussion

- Primer, Proof techniques: Contraposition, Contradiction
- Vivaldi, Chapter 7
- In class exercise 5

#### Writing

- Vivaldi, Chapter 7, Exercise 7.2
- Vivaldi, Chapter 7, Exercise 7.3
- Vivaldi, Chapter 7, Exercise 7.4 (5 more)

### Week 4

#### Reading and discussion

- Primer, Proof Techniques: Uniqueness, Casework, Either/Or Max/Min
- Vivaldi, Chapter 6
- Vivaldi, Sections 7.7, 7.8
- In class exercise 4

#### Writing

- Vivaldi, Chapter 7, Exercise 7.1 (any 2)
- Vivaldi, Chapter 7, Exercise 7.4 (any 5)

### Week 3

#### Reading and discussion

- Primer, Functions (part two), negating universal quantifiers, negating existential quantifiers, negating nested quantifiers
- Vivaldi, Chapter 4
- Vivaldi, Chapter 5
- In class exercise 3

#### Writing

- Vivaldi, Chapter 4, Exercise 4.3 (any 10)
- Vivaldi, Chapter 4, Exercise 4.4 (Only state the negations for any 5)
- Vivaldi, Chapter 5, Exercise 5.1

### Week 2

#### Reading and discussion

- Vivaldi, Chapter 1
- Primer, Fundamentals: Existential quantifiers, universal quantifiers, combining quantifiers
- In class exercise 2

#### Writing

Vivaldi, Chapter 1, Exercise 1.2.

### Week 1

#### Reading and discussion

- Primer, Fundamentals: Set Theory, Functions (part one)
- Vivaldi, Chapter 2
- Tips on giving talks, Jordan Ellenberg
- In class exercise 1

#### Writing

- Convert a page of handwritten mathematics into LaTeX.

#### Resources

- Learn LaTeX in 30 minutes
- https://www.learnlatex.org/ (especially, lesson 10)

## Comments on presentations (updating list)

Many of these comments also apply to clear writing.

### General

- Make eye contact with the audience as much as possible. Avoid speaking to the board.
- Speak so that everyone can hear clearly (not too softly or too fast).
- Begin informally, before going into technical details.
- Motivate your topic. Get the readers interested in what you are saying.
- Keep the talk focused. Do not try to do everything. Some strategies to keep focus: making simplifying assumptions, doing special cases, explaining by an example.

### Board/slides

- Write the title.
- Write legibly, clearly, in sufficiently big font.
- Develop the habit of speaking when you writing, avoiding long silences.
- Use the board to write the important points, not as a place for scratch work.

### Content

- Illustrate definitions with examples and non-examples. Comment on edge cases.
- Give precise definitions. If that is not possible, explicitly indicate what is left vague or undefined.
- Mention each non-trivial mathematical result you use. Indicate if it is easy or difficult. Give proper attribution.
- Take every opportunity to draw a picture.
- Take every (reasonable) opportunity to tell a story.

### Notation

- Use consistent notation, and alert the audience if notation changes.

### Handouts

- Keep consistent with the talk.
- Comment on everything that is in the handout.

## General comments on the first paper

### Introduction

The introduction to your paper must answer the following two questions

- What is the paper about?
- Why should the readers care?

Number (2) is harder to achieve and takes practice. Number (1) is easier. To achieve (1), your introduction should explain the main topic and the main results. After reading the introduction, the reader should not be surprised by anything that comes later.

### Body

Use the appropriate LaTeX environments for definitions, theorems etc. For example, put the theorems in \begin{theorem} … \end{theorem} and the proofs in \begin{proof}…\end{proof}.

The body must be more than just a sequence of definitions and theorems. You must provide guides to the reader, by periodically reminding them what has been done and what comes ahead. You can do this by including sentences like:

- Having defined eigenvectors and eigenvalues, we now see how to compute them.
- We are now ready to state the main theorem.
- Before we state the main theorem, we need some notation.
- We have finished proving the lemma; now we turn to the proof of the theorem.

All major definitions and theorems should come with an introductory sentence.

For example, before you state a theorem in \begin{theorem}…\end{theorem}, include a sentence like:

- The following is the main theorem of the paper.
- Having established the notation, we state the main theorem.
- The following theorem states that solutions exist and are unique.

Before you give a major definition in \begin{definition}…\end{definition}, include a sentence like:

- We now give the precise definition.
- The following is the key definition of the paper.

Before an example, include a sentence that explains why you are including the example. For example:

- The following example shows that the solution is not always unique.
- We now give an example of a continuous function that is not differentiable.
- Not all matrices admit an eigenvector, as the following example shows.

An introductory sentence is also helpful for minor definitions and theorems, but in those cases, it is more acceptable to skip.

## Topics for second talk and paper

- Fermat’s little theorem
- The chromatic polynomial of a graph
- The four colour theorem
- Modelling radioactive decay
- The mathematics of contagious diseases
- Linear differential equations
- Euler’s method for numerically solving an ODE
- The mathematics of carbon dating
- The heat equation
- The wave equation
- Harmonic functions and the maximum principle
- Primality testing
- The RSA cryptosystem
- The conjugate gradient method
- The Poisson distribution
- The binomial distribution
- The central limit theorem
- The maximum likelihood method
- Primitive roots in modular arithmetic
- Planar graphs
- Recurrence relations
- Exact differential equations
- The prime number theorem
- Taylor Series
- Continued fractions
- Any other topic (with my permission)

## Judgement criteria for the papers

**The title***is descriptive.**is concise.*

**The abstract***adequately conveys the main topic.**is concise.*

**The introduction***motivates the topic.**summarises all the key results.**gives an outline of the rest of the paper.**is not overly technical or overly vague.*

**The sections***are titled informatively.**are arranged in a logical order.**are of appropriate length.*

**Theorems and definitions***are in the correct environment.**are stated precisely.**are stated concisely.**are motivated.*

**Proofs***are in the correct environment.**are correct.**are readable (not overly long or short).*

**Surrounding discussion***guides the reader by periodic recaping and fore-shadowing.*

**Language and grammar***Everything is a part of a complete sentence.**Spelling, grammar, and punctuation are correct.**Sentences are not overly long.**Writing is organised in logical paragraphs.*

**Mathematical matters***Sentences do not start with a mathematical symbol.**Mathematical expressions are adequately separated.**Notation is not too cumbersome.**Notation is consistent.**Mathematical terms are used precisely.*

**References***Proper citations are given.**A list of references is included in a consistent format.*