Course Goals
 To prepare you for Part 1 of the Comprehensive Exam by helping you to review and develop mastery of the five fundamental courses of the math major: Differential and Integral Calculus, Multivariable Calculus, Discrete Mathematics, and Linear Algebra.
 To help you develop your abilities to communicate your ideas, especially your mathematical and technical ideas, through writing and through speech.
 To help you become informed about the range of opportunities available to you as math majors, both over the next year and after you graduate.
You can find a link to the syllabus here.
Assignments
Week 1: Due Thursday, January 31
 Complete a practice Calc 1 section of the comps exam.
 Use $\LaTeX$ to typeset your answers for the Calc 1 exam; you should include a
\maketitle
command that gives a title and your name and the date.  If you haven’t already, fill out this short survey on how comfortable you feel with each topic on the comps study guide. Use 1 for topics you don’t remember at all, and 5 for topics you feel completely comfortable with. Please complete this by Saturday.
Week 2: Due Thursday, February 7
 Study for the Calc 1 comps exam.
 Read the solutions to the practice Calc 1 section.
 Submit an abstract you have written for the Krebs and Wright paper we discussed in class.
 Write it in $\LaTeX$ and use the
\begin{abstract}
\end{abstract}
environment.
 Write it in $\LaTeX$ and use the
 Sign up for comps review talk (link to come soon).
Week 3: Due Thursday, February 14
 Complete a practice Calc 2 section of the comps exam. (You don’t need to type it up this time; feel free to handwrite it, but turn it in).
 Sign up for a slot to talk about one comps review topic at this link. Note: Due to a technical screwup on my part, there is one multi topic and one calc 2 topic listed at the end.
 If you are giving a talk next week: Send me a copy of your notes for the talk by Monday night. That way I can give you feedback and make sure you’re leading everyone else in the right direction.
 Choose a paper to summarize for your first paper of the course (see below). Print out and bring to class a brief abstract for the paper, written in $\LaTeX$.
 Review notes on giving a talk.
Week 4: Due Thursday February 21
 Study for the Calc 2 comps exam.
 Read the solutions to the practice Calc 2 section.
 Bring four printed copies of your resume into class on Thursday.
Week 5: Due Thursday February 28
 Complete a practice multi section of the comps exam. (You don’t need to type it up this time; feel free to handwrite it, but turn it in).
 If you are giving a talk next week: Send me a copy of your notes for the talk by Monday night. That way I can give you feedback and make sure you’re leading everyone else in the right direction.
 Turn in a printed out hardcopy rough draft of your summary paper assignment.
 Review notes on writing a paper.
Week 6: Due Thursday March 7
 Read the solutions to the practice multi section.
 Look over my notes on multivariable calculus from Thursday.
 Study for the multi comps exam.
 Make any revisions you feel like to your essay. You don’t need to turn this in.
 Bring three printed copies of your draft paper in to class.
Week 7: Due Thursday March 21
 Complete a practice linear section of the comps exam. (You don’t need to type it up this time; feel free to handwrite it, but turn it in).
 If you are giving a talk next week: Send me a copy of your notes for the talk by Monday night. That way I can give you feedback and make sure you’re leading everyone else in the right direction.
 Turn in a printed out hardcopy final draft of your summary paper assignment. Some tips:
 Please identify the paper you’re summarizing somewhere in your paper.
 I encourage everyone to use $\BibTeX$ for citations, but it can be a little tricky if you’ve never done it, and it’s not a requirement. This tutorial should explain how; with the note that you can use Google Scholar to get you your .bib file and shouldn’t need to format it yourself.
 $\LaTeX$ doesn’t automatically create smart quotes for you, and thus you shouldn’t type
"words"
if you want quotation marks. Many editors will fix this for you, but if yours doesn’t, it is much better to type``words''
instead. (The backtick mark is probably under the tilde in the upperleft of your keyboard, right below the escape key).  If you’re using a function with a long name in math mode, make sure you mark it as a function.
 If $\LaTeX$ already recognizes your function you can just type it with a backslash: so
\log
instead oflog
or\gcd
instead ofgcd
.  If $\LaTeX$ doesn’t recognize your function, you need to tell it. You can either type something like
\operatorname{area}
every time instead of justarea
, or you can add\DeclareMathOperator{\area}{area}
once to the preamble. (Both of these options will only work if you are using the packageamsmath
).
 If $\LaTeX$ already recognizes your function you can just type it with a backslash: so
 If you’re typesetting an integral, it can be helpful to type
\,dx
instead ofdx
Week 8: Due Thursday March 28
 Read the solutions to the practice linear section.
 Study for the linear comps exam.
 Submit an abstract for the Theorem paper.
Week 9: Due Thursday April 4
 Complete a practice Discrete section of the comps exam. (You don’t need to type it up this time; feel free to handwrite it, but turn it in).
 If you are giving a talk next week: Send me a copy of your notes for the talk by Monday night. That way I can give you feedback and make sure you’re leading everyone else in the right direction.
 Turn in a printed out hardcopy rough draft of your theorem paper assignment.
Week 10: Due Thursday April 11
 Read the solutions to the practice discrete section.
 Study for the discrete comps exam.
 Bring three printed copies of your draft paper in to class.
 Email me an abstract for your final talk.
Talks: April 18
 Patrick Bender: Newton’s Method
 Theo FrareDavis: Bayes’s Theorem
 Adeline Zhang: ZeroSum Games
 Dan Huth: Savitch’s Theorem
 Jane Bellamy: The CauchyGoursat Theorem
 Daniel Hermosillo: The CauchyRiemann Equations
 Andrea Stine: Cauchy’s Integral Formula
Talks: April 25
 Summer Li: Greedy Coloring Algorithms
 Jenny Yu: Four Color Theorem
 Jan Yan: Kruskal’s Algorithm
 Kate Grossman: Pick’s Theorem
 Junepyo Lee: Pairwise Integral Planar Distances
 Yuxin Xu: An Introduction to Groups
 Andrew Porter: Algebra and Convergence
 Alex Hernandez: Integral Tricks from Complex Analysis
Mathematical Communication
You may find the introduction to LaTeX I wrote elsewhere on this site helpful.
Summary Paper

The goal of this assignment is to learn to communicate about mathematical ideas to an audience unfamiliar with the ideas you’re discussing. You should think of the audience as being your fellow math majors: mathematically skilled but unfamiliar with the specific topic and field you are working in.
You should communicate the basic ideas, goals, and methods of the paper, but not get bogged down in the weeds of detailed calculations, technical arguments, or details of proofs.
 Choose a paper to summarize. The paper should probably be in the 410 page range.
 AMS Chauvenet Prize winners
 Fermat’s Library (click the “Mathematics” button)
 College Mathematics Journal
 American Mathematical Monthly (Alternate link here)
 Mathematical Intelligencer
 Mathematics Magazine
 Or find your own!
 Rubric for the summary paper is here.
 You need to tell me the paper you have chosen and write an abstract for your summary paper by Thursday, February 14.
 Rough draft is due Thursday, February 28.
 Final draft is due Thursday, March 21.
Theorem Paper

The goal of this assignment is to learn to communicate about mathematical ideas to an audience unfamiliar with the ideas you’re discussing. You should think of the audience as being your fellow math majors: mathematically skilled but unfamiliar with the specific topic and field you are working in.
 Choose a theorem, result, or technique from a 300 or 400level math class you have taken or are taking. You will write a paper of approximately three pages in which you clearly state the result, explain how the result was obtained, and tell your reader why this result is relevant and important.
 Mathematical results from courses in other departments may be approved on a casebycase basis, but please consult with me to confirm.
 Rubric for the theorem paper is here.
 You need to tell me the paper you have chosen and write an abstract for your summary paper by Thursday, March 28.
 Rough draft is due Thursday, February April 4.
 Final draft is due Thursday, April 18.
Final Presentation
 The goal of this assignment is to practice communicating mathematical ideas to an audience via a verbal/slideshow presentation. The audience will be your fellow math majors. Assume they are mathematically skilled, but unfamiliar with the specific topic you are discussing.
 Choose a theorem, result, or technique from a 300 or 400level math class you have taken or are taking. You will give a tenminute slideshow presentation in class in which you will explain the result, the basic ideas behind it, and the reason it is important and relevant.
 Mathematical results from courses in other departments may be approved on a casebycase basis, but please consult with m to confirm.)
 Slides from my inclass Beamer presentation are here. You can also download the TeX source code or the plain text source code.
 You can find the rubric for the talk here
Comps Part 1
The math comprehensive exam will consist of five sections given on five separate days, spread throughout the semester. Each section will cover one of the five fundamental courses. Dates for these exams are:
Thursday, February 7:  Calculus 1 
Thursday, February 21:  Calculus 2 
Thursday, March 7:  Multivariable Calculus 
Thursday, March 28:  Linear Algebra 
Thursday, April 11:  Discrete Mathematics 
There is a study guide available here that tells you what topics we think are most important. You can also download a complete (if somewhat dated) list of topics for each course:
We have a number of practice exams from previous years available.
 2004 and 2005
 Calculus 1 solutions
 Calculus 2 solutions
 Linear Solutions
 Multivariable Solutions
 Discrete solutions
 2006
 2008
Future Opportunities
I encourage everyone to look for a job/internship/research opportunity this summer. This will improve your resume, and also give you a better idea what sorts of things you might want to do after you graduate. Many of these opportunities will pay you reasonably well.
Occidental Undergraduate Summer Research
Occidental has an undergraduate research program to sponsor you doing research with a professor. You would need to find a professor to mentor you; the program comes with a $4500 stipend and subsidized (but not free) summer housing. The deadline is February 9.
Other Research Opportunities.
The NSF REU program funds experiences where a group of undergraduates from different institutions gather and do research on a math topic for ten weeks over the summer; typically the students also receive a stipend of several thousand dollars.
A list of programs running this summer is available here. Deadlines are typically in February and March. (Most NSF programs are only available to US permanent residents).
There are also a number of summer programs listed at mathprograms.org. You can look around without making an account; the list of undergraduate programs seems to be here.
Internships
The AMS has an info page rounding up several summer internship opportunities.
The MAA also has a page of internship opportunities. It also has a roundup pages on careers here.
SIAM has a page on internships and careers.
I’ll try to add more resources as I find them.