Math 499. Senior Seminar: Great Theorems of Mathematics

Spring 2023

Instructor: Enrique Treviño
Lectures: MWF 9:00 - 9:50 am in Brown Hall 116
Office Hours: MWF from 10am and 11:45am. You can also arrange a meeting by appointment.
Office: Brown Hall 123
Email:
Phone Ext.: #6187

Announcements

No class on Monday March 6, Wednesday March 8 and Friday March 10.

Regarding chapter 6, Ebra and KJ talked about the mathematical duels in Italy concerning the solution of the cubic. To learn more about it, there's an Italian book (translated to English) titled "The Secret Formula: How a Mathematical Duel Inflamed Renaissance Italy and Uncovered the Cubic Equation". It's a very interesting book.

Regarding chapter 4, Mike and Trevor talked about Ramanujan and Hardy. To learn more about them, I recommend the book "The man who knew infinity". There is also a movie, from 2015, based on their story.

Eratosthenes estimation of circumference of the Earth segment from the TV show Cosmos narrated by Carl Sagan.

Introduction to Probability by Charles Grinstead and J. Laurie Snell. Free book online that has a very nice reference for Markov chains in chapter 11.

Homework

Schedule for the Presentations.

Course Description

Seminar course to introduce students to various masterpieces in the development of mathematics. Some of the most historically important proofs and ingenious logical arguments from mathematics will be presented and discussed. An emphasis will be placed on the interconnectedness among various subject areas within mathematics. Prerequisite: permission of the instructor.

Textbook

The main book we'll use is:

Journey through Genius by William Dunham.

Goals of the Class

The goals of this class are the following:

Showcase the connections between different areas of mathematics.
Prepare you to know how to give a class-style lecture.
Prepare you to write mathematics professionally (using LaTeX).
Prepare you to give a presentation in Beamer (a package in LaTeX).
Connect with the recreational side of mathematics.
Be able to read mathematical papers of a level just below research level.

Grading

The course grade will be based on:

Class Presentations	40%
Beamer Presentation	20%
Paper	20%
Pi-Day Colloquium	10%
Homework	10%

Class presentation

The class will form into four groups of 2 (or 3 if necessary). The groups will be assigned three chapters from the book Journey through Genius and they will present each chapter to the class over two class periods (two class periods per chapter). The students not presenting will have to attend and they will provide feedback to the presenters by the next class session. I will also get a copy of this feedback.

For the second and third presentations, students should go beyond the text and be ready for questions that can come up for a person reading the text. For example, some questions from Chapter 2 which are relevant are:

The chapter talks about the "collapsible" compass and how Euclid proved that you did not need to assume that you could fix a compass to a length to use it elsewhere. How does Euclid's proof work? (note this proof is not in the book, you'd have to find it elsewhere or figure it out on your own.)
The book mentions proposition I.5 is known as the "bridge of fools" because the diagram vaguely resembles a bridge and many people have trouble understanding the proof. What was the diagram? What is the proof?

The grade for an individual with regard to the class presentation will depend on the presentations (30%) and the feedback they provide to the other teams (10%).
Schedule for the Presentations.

Beamer presentation

Each student will find an article they like from the articles published in 2022 in Mathematics Magazine. The articles will be posted on Moodle.
The student will send their top three picks to me via email by February 22. After I collect all picks, I will assign a student with an article they ranked.
The student will then write a Beamer presentation and give a 15 minute presentation about the article in class. A Beamer presentation is like PowerPoint but written with LaTeX. It has the advantage of making mathematical formulas prettier.
Every student will give feedback on the presentations of the other students.

The presentations will be two-a-day in mid March in the same order as the team presentations. The schedule will be here.
The grade for an individual with regard to the Beamer presentation will depend on the presentation itself (15%) and the feedback they provide to the other students (5%). Note that this presentation is not a team presentation. Using Beamer correctly will be part of the grade.

Paper

For the paper, you have to find a subject on your own and write about it. The assignment has the following rules:

First, pick a mathematical subject. It can be a cool puzzle, an interesting game (that involves thinking), an application of mathematics to some field (maybe an application of math to sports) or an exciting theorem.
Once you pick your subject, find three references to give you background on that subject.
Send the subject and the three references to my email by April 7 . I will then approve the topic, if it is suitable. If not, you have to find one right away. I suggest you have three subjects n mind in case I don't approve one of them.
Once you have my approval, write a paper talking about the subject. Your paper should be between 3 and 5 pages long (double-spaced with 12 point font). You can use the Mathematics Magazine articles as examples of how to write good math articles. You are welcome to include pictures and you should include a bibliography. The first draft is due on April 24.
The paper has to be written in LaTeX. You will send me the .tex file and you will print 4 copies of your pdf (1 for me and 3 for 3 students).
After getting feedback from your peers (by May 1), you will write a final draft and turn it in on May 10 by 11:30 am (that's the date of the final exam)..

Your assignment also includes reading the drafts of three other students and give them feedback so that they can improve their final draft. The comments you write should be written seriously and be helpful.

The grade for an individual with regard to the paper will depend on the first draft (5%), the final draft (10%) and the feedback they provide to the other students (5%). The aesthetic presentation of the paper will be weighed in as part of the grade.

LaTeX and Beamer

As described above, the final paper should be written in LaTeX (and compiled to a pdf) and the individual presentation should be written in a special LaTeX package called Beamer (also compiled to a pdf). Below I include links to useful manuals, websites and also templates for LaTeX and Beamer.

Slides from a talk on LaTeX created by Vadim Ponomarenko of San Diego State University.
The TeX source code for the slides. (Right-click to Save Link As).
LaTeX on Wikibooks. A very useful online manual.
For PC users, you can use TeXnicCenter to compile. You'll need to install MiKTeX. After you install MikTeX, you wil also have TeXWorks in your computer. You can use this compiler instead of TeXnicCenter if you prefer. Or find your own.
For Mac users the things to install are different. You can use MacTeX or find one of your liking online.
Another option is to compile things online in Overleaf. Many of my students have liked how easy it was to use.
Example of a TeX file of mine in TeX and the corresponding pdf.
Example of a Beamer file of mine in TeX and the corresponding pdf.

Pi-Day Colloquium

Working in groups, students will prepare a short sketch of a mathematical paradox or a puzzle to present in mid-March at a Pi-Day Colloquium. Things related to the number pi are preferred, but not necessary. The colloquium will be during class time on Friday March 3. The groups will have to present to me their plans by February 27.

The grade for an individual with regard to the Pi-Day Colloquium will depend on the presentation itself (10%).

Homework

I will post a homework assignment for each presentation from Journey Through Genius. Students need only turn in homework assignments for 5 of the assignments, they get to choose which ones they want to turn in.

Homework

Writing Center

The Writing Center offers free tutorial assistance, information, and resources for every stage of the writing process.

Accommodations Statement

If you are a student who needs an accommodation because of a disability or medical or psychological condition that limits your ability to fully participate in this course, please contact Kara Fifield, Director of Disability Services, to document your disability with the College and with the professor of this course. Academic accommodations should be reasonable and not alter the fundamental nature of this course. Because it can take a week or more to arrange requested accommodations, you are encouraged to establish your semester accommodations as early in the semester as possible. Contact Kara Fifield by email or phone: kfifield@lakeforest.edu or 847-735-5167. For more information about services for students with disabilities at Lake Forest College, see: http://www.lakeforest.edu/academics/resources/disability/

You are also welcome to contact me privately to discuss your academic needs. However, all disability-related accommodations must be arranged through Kara Fifield.

Description of instructional time and expectations:

This course meets 3 times per week for 3 hours per week. The course carries 1.0 course credit (equivalent to four semester credit hours). Students are expected to devote a minimum of 12 hours of total work per week (in-class time plus out-of-class work) to this course.

Academic Honesty

Please read the College's information on Academic Honesty. If a student cheats in an exam, quiz or homework assignment, I will proceed with charging the student with the Academic Honesty Judicial Board. The usual (first) penalty is a 0 in the assignment on which the cheating occured plus some ethics lectures the student would take. The second penalty is usually suspension.

Last modified on February 22, 2023.