Franz Pedit, LGRT 1542 & 1535

pedit@math.umass.edu

Office hours: Wed 2:00-3:30 and by appointment

TA

Tiffany Quang

tquang@umass.edu

Course Objectives

The main objective of this class is to practice writing about mathematics. All writing has to be done in the word processing system LaTex (see resources below).

The mathematical writing will be based on

- The Poincare Conjecture, by Donal O'Shea.
- Flatland: a romance in many dimensions, by Edwin A. Abbott 1838-1926.
- Videos of general audience lectures given by
mathematicians on related topics.

- Additional assigned reading.

- Discussions in class.

Examples of mathematical writing:

- Steven Strogatz ''Elements of Mathematics'' and some of his writings you find in the NYtimes.
- Eugene Wigner ''The unreasonable effectiveness of mathematics in the natural sciences''
- Paul Lockhart ''A Mathematician's Lament" (specially recommended for anyone who wants to teach mathematics)
- Robert Kanigel "The Man Who Knew Infinity: A Life of the Genius Ramanujan"
- G. H. Hardy "A Mathematician's Apology"
- Simon Singh "Fermat's Last Theorem"
- Timothy Gowers "Mathematics. A Very
Short Introduction"

- Edward Frenkel "Love and Math"
- Roger Penrose "The Road to Reality" (Introduction & Prologue)

Latex installation

TexShop for Mac • MikTex for PC • various commercial online editors (usually free for single user), e.g. Overleave, LaTex Base

Sample files

latex source example • latexed PDF of example source file • latex slides example • latexed PDF of slide example • latex typesetting manual • various commercial online editors (usually free for single user)

Writing check list

UMass resources

Library

Writing Center: tutoring and advise on your writing.

Career Center: advise on job applications, internships, grad school applications, cover letters, vitae.

Upcoming event dates

- September
3, 2019: no class meeting.

- September 5, 2019: class visit by Science & Engineering Librarian Anne Graham, who will introduce us to the libraries at UMass, online search options, citations, issues with plagiarism etc. This information will be useful whatever you will do after graduation, be it grad school, internships and so on.
- September 17, 2019: Nessim Watson, Assistant Director for CNS Career Center: presentation on the know hows of application writing: cover letters, resumes.

List of group projects: TBA

Course Log and assignments:

Download the full latex installation on your laptop from the links in the resources section (mac users and Microsoft users need different installations---those who run Linux can fare for themselves, since they know better anyway). Familiarize yourself with its basics by using the templates provided above. Get help from fellow students if you have difficulties or google your questions. All writing in this class has to be done in LaTex.

Reading Assignment : Paul Lockhart "A Mathematician's Lament".

Writing Assignment due 9/19/2019:

Write an essay (minimum 1 page single spaced in latex) how your experience with math so far compares to Lockhart's view of mathematics and how it should be taught. Consult the writing check list before you hand in your essay.

Week 2: Discussed the exploration of earth and how humans eventually came to understand its shape.

Make sure you have latex up and running seamlessly.

Reading Assignment : "The Beauty of Doing Mathematics" the first hour of the lecture by Serge Lang, titled "The great problems of geometry and space". Then read the first 2 chapters of "The Poincare Conjecture" by Donal O'Shea (the book is a rather large file, so be patient when downloading via slow servers).

Writing Assignment due 9/26/2019: write a 2-3 page singly spaced essay for say the campus news paper (i.e., your essay should be accessible by all students and faculty) on how the Greek geometer Eratosthenes (275-195 BC) concluded that he earth was round and how he

calculated its circumference to great accuracy. Provide some biographical detail of Eratosthenes and enough mathematical detail for readers to follow the calculations. Include some drawings to aid explanations etc.

Week 3: Discussed the notion of a manifold, boundary, compactness, and simply connectivity.

Reading Assignment : "The Beauty of Doing Mathematics", the second hour, pg. 96--106, of the lecture "The great problems of geometry and space" by Serge Lang. Also read chapters 3 and 4 of "The Poincare Conjecture" by Donal O'Shea (the book is a rather large file, so be patient when downloading via slow servers).

Writing Assignment due 10/3/2019: write a 2-3 page singly spaced essay for say the campus news paper (i.e., your essay should be accessible by students, faculty, and the occasional reader from town) what it would be like to live on a 2-dimensional world. For instance, how does a triangle, square etc. look like to a 2-dimensional being. What kinds of 2-dimensional worlds can you imagine other than the plane? Try to express concepts such a finitely extended, or infinitely extended, boundary or no boundary etc. so that readers get some idea what it is about.

Week 4: Discussed the 2-dimensional Poincare conjecture: surgery and boundary gluing. Watched Carl Sagan's Flat Land video. Listened to an interview with Roger Penrose on ``What things exist'' (make sure you select the ``long interview").

Reading Assignment : "The Road to Reality" by Roger Penrose. Read the Introduction, Prologue, and Chapter 1.

Writing Assignment due 10/10/2019: write a 2-3 page singly spaced essay on what you took away from the interview and the subsequent reading. You may want to include your own viewpoint on the topic of mathematical reality versus physical and mental realities.

Week 5: More on surgery and boundary gluing. Listened to an interview with Roger Penrose on ``Is mathematics invented or discovered?"

(make sure you select the ``long interview"). Also watched a lecture by Curtis McMullen on "The Geometry of 3-manifolds".

Reading Assignment : Chapters 5 and 6 of "The Poincare Conjecture" by Donal O'Shea.

Writing Assignment due 10/17/2019: We now had many occasions where we described the 2-sphere as two disks glued along their circle boundary. Use stereo graphic projection from the north resp. south poles to explain this model of the 2-sphere in terms of two charts (the images of the stereo graphic projections of the two hemispheres). Calculate explicitly the stereo graphic projection map and its inverse. Also calculate the transition function between the two pages of your atlas of the earth: the north pole projection and the south pole projection. You will need to do some research on your own if you haven't seen the stereo graphic projection before.

Week 6: Watched a lecture by Jeff Weeks on "Shape of Space".

Reading Assignment : Chapters 7 and 8 of "The Poincare Conjecture" by Donal O'Shea.

Writing Assignment due 10/24/2019: Write an essay about Euclid and his importance in geometry. Explain the parallel axiom and why this was controversial for a very long time and how it got resolved.

Week 7: Tiling by regular n-gons of the Euclidean plane: the flat architecture of the doughnut.

Reading Assignment : Chapters 9 and 10 of "The Poincare Conjecture" by Donal O'Shea.

Writing Assignment due 10/31/2019: Go over your stereo graphic projection paper and re-edit it with all the comments I made in class:

- DO
NOT
MESS WITH LATEX!!! Do not underline, center etc. at your own whim. You
cannot improve (you can only destroy) the latex layout of a chosen
document style (maybe experiment with different document styles if you
are unhappy with the current one). Do not make your own section
headers, titles, or captions of images etc. Use the latex
environments provided for that. There also is an extensive
theorem/lemma/corollary/definition/conjecture etc environment if you
need to use that.

- Adjust spacing, image sizes etc so that you get a good layout (no half empty pages, no large empty spaces between paragraphs, or below equations, or between equations etc).
- Think about your chosen notations
and keep them consistent throughout. Do this wisely. If you choose
italics P make sure it is italicized throughout.

- Text in math mode has to be roman
(use \text{...} to achieve that). Use the \sin, \cos etc for standard
functions. This will assure they get printed correctly in equation
mode.Most functions have latex symbols, look up latex math
function/symbols tables.

- NEVER copy and paste formulas from
the internet and include them as a picture file into your essay.
Typeset all formulas by yourself.Balance inline and display formulas
according to importance. Make sure you provide text between displayed
formulas/equations.

- If you include images have a capture and refer to the images in the text. Pay attention to image size and how this compares to your page layout. Try to get images as sharp and readable as possible (usually achieved by high resolution images). Avoid low resolution images from the web, search for high quality images, or make them yourself, by using a drawing program (open source like Gimp etc) or hand draw if you are artistically inclined.
- Use Google queries to find out how things are done if you get stuck.
- If you are not sure about things and have questions ask me during class, we can always make time to go over it, some fellow students might have similar problems.

Week 8: Worked on the connections of Euclid's 5th axiom, the parallel postulate, with the three architectures of compact, connected, orientable 2-dim shapes: the round sphere (positively curved architecture), the torus (flat Euclidean architecture), and the higher genus surfaces (hyperbolic architecture). We discussed tilings in those architectures and noticed that there are interesting tilings of the sphere arising from the Platonic solids.

Writing
Assignment due 11/7/2019:
Explain the Platonic solids, their
historical significance, their connections to the elements (water, fire,
etc.) in Greek philosophy. Then connect the Platonic solids to regular
n-gon tilings of the round sphere: you could try to proof (similar to
what we did for the Euclidean case), that the "puffed out" Platonic
solids are the only possible such tilings. For this, you would have to
explain the condition of a regular tiling on the sphere and etc. Try to
write a cohesive, informative, structured, and mathematically accurate
piece. The audience you should have in mind is the generic reader of,
say, a College news paper. Always check your latex and writing against
the week 6 rubric and the writing
check list.

Week 9: Discussed the hyperbolic plane and its tilings by regular n-gons. Watched a documentary about M. C. Escher narrated by Sir Roger Penrose: The Art of the Impossible: MC Escher and Me (2 parts on YouTube).

Writing
Assignment due 11/14/2019:
Write an essay about M. C. Escher
focusing on his usage of spherical, Euclidean, and hyperbolic geometries
in his art. Include the relevant Escher graphics and elaborate how they
connect to the relevant geometries. Always check your latex and writing
against the week 6 rubric and the writing
check list.

Reading Assignment : Chapters 10 (re-read in the light of the discussions in class), 11, and 12 of "The Poincare Conjecture" by Donal O'Shea.

Week 10: started discussion of 3-diomensional shapes: Euclidean 3-space and various way to think of the 3-sphere.

Writing
Assignment due 11/21/2019:
Define, explain and calculate the
relevant formulas for the stereographic projection from the 3-sphere to
the 3-dimensional equatorial space and its inverse map back to the
3-sphere. Write down your
understanding of it on the level of a math major. Check your latex and
writing against the week 6 rubric and the writing
check list. Make sure
your notation is consistent and formulas a properly typeset.

Handcrafting
Assignment due after
Thanksgiving: choose your
favorite material and your 2 favorite Platonic solids (not both cube and
tetraherdon) and build them as a model. Bring them to class after
Thanksgiving.