Project Proposal:

$1)$ The Project Proposal identifies the Person, Place and Problem$($s$)$ you have chosen and the primary sources that you will begin your research with. It is important to be as specific as possible and have resources that can support what you would like to do$.$ To help with this, review the MMT Library and Research Resources modules. These modules will help you access high quality research materials, cite sources $($required$),$ and connect with a University of Waterloo Librarian if you need further assistance. You will need to cite all your sources in APA style. Review the APA style citation resource page.

The first part of your submission will identify the Person, Place, Problem$($s$),$ and hook. A hook is a single sentence that grabs the attention of the reader and indicates the content of the project. The second part of your submission will contain a detailed outline. The third part will identify at least five scholarly sources. For each source, identify the chapter, section, or pages that will be relevant.

Past students have generously offered their Project Proposals as exemplars. Please note, these are examples only and do not necessarily represent work that received perfect marks.

Project Proposals as exemplars

Project Summary

Person: Pierre De Fermat

Place: Toulouse, France, $17$th Century $1601-1665.$

Problem: Theorems of Fermat. Note of explanation: My aim is to focus on the mathematics of

Fermat, and why it would trigger the mathematical odyssey" I mention in the hook below. Although,

I may briefly explore $($section $4)$ the historical effect of some of Fermat's theorems, including

the effect on mathematics in general, my intention is not to address the subsequent mathematics.

Hook: I explore why a mathematical theorem scribbled by an amateur mathematician in the margins

of a newly$-$printed version of an ancient text by Diophantus, would lead to a $300-$year mathematical

odyssey.

Project Outline

$1.$ Introduction

$($a$)$ The idea of mathematical beauty. Here I will develop a possible answer to the question

posed in the hook to be tested in the project. Reference $1($see below$)$ will be useful here.

$($b$)$ Ancient Greek Manuscripts: This section will include a brief history of these manuscripts,

including how they survived from antiquity. Reference $2$ contains some information, and

reference $3$ has a great deal of information.

$($c$)$ Arithmetica of Diophantus: The version used by Fermat appears by Claude Gaspard de

Bachet $(1591-1639).$ Here I will include a description of some of the contents of this

work, in particular ideas that interested Fermat such as "perfect and amicable numbers,

figurate numbers, magic squares, Pythagorean triads, divisibility and prime numbers"

$(2).$ Reference $3$ contains much information, and I will obtain much of the mathematical

background from References $4$ and $5.$

$2.$ The Setting $($Reference $6$ contains much of the information required for this section$)$

$($a$)$ The Setting $-$ France in the seventeenth century

$($b$)$ Fermat, Early Life

$($c$)$ Fermat's Career as a Lawyer

$($d$)$ Character of the man

$($e$)$ Mathematical Colleagues and Interactions $($Decartes$,$ Pascal$)$

$3.$ Fermat's Theory of Numbers $($References $2$ and $7)$

$($a$)$ Method of Infinite Descent

$($b$)$ Contributions to Number Theory: Proofs of his theorems using infinite descent.

$($c$)$ Fermat's Lesser Theorem and Primes

$($d$)$ Fermat's Last Theorem: There are no solutions for x$^$n$+$ y$^$n $=$ z$^2$ for n $>2.$

$4.$ After Fermat and Conclusion

$($a$)$ How do Fermat's findings fit into the modern field of Number Theory?

$($b$)$ Brief history of the search for the proof to Fermat's Last Theorem $($Reference $7)$

$($c$)$ Conclusion$-$ My answer to the question in the hook. Some final questions I may address:

How does this story inform us about the progress of mathematics? Does the pursuit

of aesthetics lead to new theories and ideas in mathematics? Why would pursuit of

symmetry and aesthetics lead to profound breakthroughs in mathematics? Reference $8$

has some important thinking on these questions.

Source material

Source Material

$1.$ Sinclair, N$.,$ Pimm, D$.,$ & Higginson, W$.(2007).$ Mathematics and the Aesthetic: New Ap$-$

proaches to an Ancient A$\_$nity. Springer, New York. Relevant sections: Chapter $\_,\backslash$A

Historical Gaze at the Mathermatical Aesthetic" and Chapter $2,\backslash$Beauty and Truth in Mathematics."

$2.$ Merzbach, U$.$ C$.$ and Boyer, C$.$ B$.(2011).$ A History of Mathematics, Third Edition, John Wiley

& Sons, Inc., Hoboken $,$ New Jersey. Relevant sections: Chapter $15$: $\backslash$Analysis$,$ Synthesis,

the in$\_$nite and Numbers".

$3.$ Meskens, A$.(2010).$ Travelling Mathematics $-$ The Fate of Diophantos' Arithmetic. Springer

Basel. Relevant sections: Chapter $3,\backslash$Diophantus and the Arithmetica", and Chapter $7$:

$\backslash$Renaissance or the rebirth of Diophantus" and Chapter $8\backslash$Fair Stood the wind for France,

in particular section $8\mathrm{.}2,\backslash$Emulating the ancients: Claude$-$Gasper Bachet de M$\_$eziriac"

$4.$ VanderBurgh, I. MATH$681(2018).$ Problem Solving. University