Instructor
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
 There will be group projects, including a final presentation of the projects by each group.
Examples of mathematical writing:
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
List of group projects: TBA
Course Log and assignments:
Week 1: 
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:
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-dimensional 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.

Week 11: Discussed more the 3-sphere, the 3-torus, genus 1 handle body gluing resulting in the 3-sphere and S1 x S2, and Thurston's Geometrization Conjecture (the 8 ``architectures" describing 3-dimensional shapes). We watched part of Curtis McMullen's lecture  titled ``Mathematics as Metaphor", and Jeff Week's lecture ``Shape of Space".

Reading Assignment over Thanksgiving: Finish reading the book "The Poincare Conjecture" by Donal O'Shea.

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


Week 12: Discussed the concept of  curvature of planar curves and surfaces. Showed curvature flow experiments, all with the purpose to get some vague idea what Perleman accomplished.

Writing Assignment due  12/17/2019: Write about your impressions of the class: what you expected from the class, what you liked/disliked about the course, what you thought of the class and of the material covered, whether it helped you see a side of math you had not seen before, whether some of it caught your interest, suggestions of what to do differently etc.