Instructor

Franz Pedit, LGRT 1542, pedit@math.umass.edu
Office hours Wed 11-12:30

TA: Sylee Dandekar, email: sdandekar@umass.edu

Course Objectives

The main objective of this class is to practice writing about mathematics.  Additionally, we will write a cover letter and resume for possible job/internship/grad school applications. All writing should be done in the word processing system LaTex (see under resources below).
The mathematical writing will be based on
 There will be more expansive group projects on topics outlined in class in the first half of the semester, including a final presentation of the projects by each group.
Examples of mathematical writing:
Resources

LaTex Installation:  TexShop for Mac  •  MikTex for PC • latex source example • latexed pdf of example source file • latex slides example • latexed pdf of slide example • latex typsetting manual • writing check list

Umass Library
UMass Writing Center:  tutoring and advise on your writing.
UMass Career Center:  advise on job applications, internships, grad school applications, cover letters, vitae.
Upcoming event dates
List of group projects (to be continued and detailed):
  1. Elliptic curves,  circumference of  an ellipse, the pendulum equation.
  2. Cryptography using elliptic curves
  3. 2-dimensional shapes
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 intsallations---those who run linux can fare for themsleves, since they know better anyway). Familiarize yourself with its basics by using the templates provided above. Get help from fellow students if you have difficutlies 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  1/31/2019:
Assume you are given a square crate of y2 oranges and you are supposed to stack them in a pyramid with a square base of x2 oranges.
Notice that the second layer of the pyramid has (x-1)2 oranges and so on. Find the equation x and y have to satisfy so that one can build a complete pyramid out of one crate of organges. Find a solution for this problem, that is, integers x and y fulfilling the equation. Draw an accurate graph of the equation and interpret mathematically  the solution to this pyramid stacking problem.
You are supposed to write an essay explainig your understanding of  the problem and its solution.
The paper should be readable by educated citizens. Everything you explain and write about you need to have a solid understanding first. Avoid putting concepts, words etc. into your writing which you do not understand or whose context is unfamiliar to you.

Week 2:  Peer review of the first writing assignment. Due date Tuesday 2/5/2019.
Reading Assignment :  Serge Lang's  "Beauty of doing mathematics: diophantine equations".
Writing Assignment due 2/7/2019: edited version of first assignment after incorporating peer reviews, the reading assignment, and explanations I provide during  this Tuesday's class.

Week 3: 
Writing Assignment due 2/14/2019: Discuss the pyramid stacking equation with the tools of calculus 1: find the local maxima/minima and inflection points on the curve, determine the intervals where the curve is concave up/down, calulcate the slope of the curve where it intersects the x-axis. Taking two arbitrary distinct points P,Q on the curve, provide an argument why the straight line through those two points will intersect the curve in a unique  third point R, and calculate the coordinates of this third point R in terms of the coordinates of P and Q.
Discuss what happens in the case P=Q, that is, think about what happens to the line connecting P and Q when  Q moves closer and closer to P. What are the coordinates of the third point then. Your paper should have correct formulas and equations, a to scale labelled graph indicating the various characteristics (min/max, inflection points etc.). The paper should be written as if it were a small project including the solution in a  Calc 1 book with sufficient text connecting the various mathematical formulas and drawings. The audience for this paper are people who have studied  Caluclus 1  (thus no need to explain  the fundamentals of Calculus, these are assumed).

Week 4: 
Writing Assignment due 2/21/2019:  Write a paper on the Pythagorean triple problem, that is, the natural number solutions to the equation
X^2 + Y^2 = Z^2. Explain why a Pythagorean triple gives rise to a rational point P (both its x and y coordinates are rational numbers)  on the unit circle x^2+y^2=1. Then find ALL rational points on the unit circle and show that these provide all Pythagorean triples. Read again, more carefully now,  Serge Lang's "Beauty of doing mathematics: diophantine equations", in which the necessary math is explained.
Write out a list of at least 10 Pythagorean triples, non of which are scalings of each other. When you write about this problem, take into account all comments/suggestions/edits made in class and on your graded papers. Start out explaining some of the history of the Pythagoreans, when/where Pythagoras lived, what their accomplishments were etc.. Explain the problem. Then start with explaining your solution. Make it readable, explain in words, make sure there is balance between formulas and text. Carefully typeset equations. Use referencing and citation. You should produce a paper which you would want to show your friends/siblings/parents (who are not mathematicians/only know high school math) and they would actually enjoy reading it and get something out of it.  Or you write the article for the student newspaper. Take it seriously.

Week 5: 
Reading Assignment :  Serge Lang's  "Beauty of doing mathematics: Prime Numbers".
Writing Assignment due 2/28/2019: Write about the documentary Fermat's Last Theorem we saw in class. Put yourself in the position of a reviewer who is writing a review about the movie for the general public. Explain the mathematical background, history, events, the overall dramaturgy of the documentary, the people appearing, who made the documentary, whether you would recommend the documentary, for who it is suitable etc.

Week 6: 
Writing Assignment due 3/7/2019:  This will be a mathematical paper written for peers. Define the mathematical concept of a group.
Then discuss in detail the following examples and
explain and prove in each case whether this is a group or not (what is the neutral element, what is the inverse etc):

Example 1: the natural numbers 0,1,2,.... with addition.
Example 2: the integer numbers ...-2,-1,0,1,2,3.....with addition.
Example 3: the integer numbers without zero ...-2,-1,1,2,3.... with multiplication.
Example 4: the numbers  Z/nZ modulo n for n a non-zero natural number with addition modulo n.
Example 5: the numbers  Z/nZ modulo n without zero with multiplication modulo n.
Example 6: the numbers  Z/pZ without zero modulo a prime number p  with multiplication modulo p.
Example 7: provide an example of a non-commutative group.

Use the theorem/definition/example latex environment to
state definitions and theorems/examples etc you need (see the latex template I gave you).
For instance:
Theorem. The natural numbers modulo n are a commutative group under addition.
Proof: .....
Don't forget to give the paper a \title, \author, \date. Use \section, \subsection etc to structure your paper.
Give sections a title which reflects what the section is about.

Week 7: 
Writing Assignment due 3/21/2019:  Wilson's Theorem on primes. Discussions with TA Sylee Dandekar on writing, what to do and what not
to do---see the writing check list above in the resource section. Real time edits/discussions of samples of  the class' work.


Week 8: 
Reading Assignment :  Roger Penrose's "The Road to Reality" (Introduction & Prologue).
Writing Assignment due 3/28/2019: Write about the museum visit. Your impressions, your understanding/comprehension of it,
your opinions about art and how/whether it may relate to math in general, whether this exhibition conveyed this point in the specific, whether you agree/disagree to the artists intentions and comprehension/use of math etc. Read the statements of the artists  on the museums website, the curators' pamphlete, and/or any other information you can get on this specific exhibit.  You need to write at leat two pages single spaced.
Put yourself in the shoes of writing for a newspaper/magazine (e.g. New Yorker Magazine) for readers whom you provide information, background, and  your critique/impressions of the exhibit.

Week 9: 
Reading Assignment :  Roger Penrose's "The Road to Reality" (Introduction & Prologue).
Writing Assignment due 4/4/2019: Write about your understanding/critique/doubts/alternatives  of Penrose's threefold view of reality as explained in his Prologue. At least 2 pages, single spaced.

Week 10: 
Reading Assignment : 

Week 10: 
Writing Assignment due 4/18/2019: In class we discussed how to make new shapes out of existing ones via boundary glueing.
For example, a rectangle in the plane (filled wih bulk/area inside) has its  4 sides  as boundary. Glueing opposite sides
makes a cylinder (no lids! A tube you casn look through)  with two boundary circles. Glueing those boundary circles makes a dougnut surface.
Now to your writing project:
take a rectangle (it is easier to visualize this when the rectangle is a longer strip), and twist the shorter side 180 degress and then glue it to the other shorter side. What you get is called a Moebius strip. What is the boundary of the Moebius strip? Explain why it is only one circle (abstractly a circle, in space it is somewhat twisted). Now you can make two new surfaces: glue a disk (which has a circle boundary) to the boundary of the Moebius strip. Describe this surface, make drawings, try to understand what you get this way. Can all loops be reeled back in---the famous rope reeling in test. You can also glue two Moebius strips together along their respective circle boundaries. What surface do you get this way? Draw pictures,  make a model (if you are so inclinded), describe in words,  and answer the question whether all loops can be reeled in. Write at least 2-3 pages with a general audience in mind. Your mission is to explain the situation with all means you have at your disposal, reflecting your understanding, and your ideas about how to convey what you have in mind  best. Do not just point a reader to existing literature or videos etc. If you want to make a video/series of drawings/story book etc.  on your own, this is fine.

Week 11: 
Writing Assignment due 4/30/2019: write about your class experience, your expectations (if any) before you took the class, your impressions during the course of the semester, what you thought was interesting and what was not, whether you learned something or not, your overall opinion. The reader (who has no idea what Junior Writing entails)  should come away with a better sense of what this course (at least) tried to achieve.