Math 101 is intended for students who are interested in learning more about math beyond calculus. The main goal of the course is to introduce the practice of higher mathematics, with minimal prerequisites. We will begin from a very basic standpoint and develop tools we need along the way to work through a number of theorems and proofs from the main branches of mathematics – geometry, analysis and algebra. You will learn about what it means to "do" math by learning how to turn your own observations and conjectures into formal statements backed up by rigorous arguments – proofs!
 
 

Prerequisites: As stated in the course catalog there are in principle no real prerequisites for this course other than a serious interest in mathematical reasoning. During the semester we will touch on topics from calculus, and so some exposure to calculus would be helpful, but we won’t require it. Math 21 or even Math 1 can be taken concurrently with this course. Note, however, that Math 101 is not intended for students who are taking or have taken Math 25 or Math 55.
 
 

Teachers:

Andy Engelward and Course Assistant(s) TBA

office: room 435 in the Science Center phone: (617) 495-4744

email: engelward@math.harvard.edu

Office Hours: Fridays from 12 to 2 and by appointment
 
 

Class Times: Tuesdays and Thursdays from 10 – 11:30 in room 507 and a weekly problem session, TBA
 
 

Course Website: http://www.math.harvard.edu/~engelwar/Math101

The website will be used for most of our course administration issues, including homework assignment postings, posting of handouts, notes, problem session times, etc.
 
 

Textbook: Proof, Logic and Conjecture by Robert Wolf is available at the Coop (or online if you’d prefer). We will use this textbook more as a reference throughout the semester, e.g. we will not be simply working our way through the book in order. Readings and some problems will be assigned out of the book, but will be taken from different sections of the book. The real textbook for the course will come as a series of notes that will be posted on the web as we progress through the semester (along with some handouts in class).
 
 

Homework: The best way to learn math is by doing math, and the homework assignments for this class will be an important part of the course. There will be roughly a homework set per week due during the semester, due at the beginning of class. In order to keep everyone up to speed and in the same place during the semester, there will usually be no late homework accepted. Course assistants will normally grade your homework and return it to you within a week. Your lowest homework score from the semester will be dropped before grades are calculated.
 
 

Weekly Problem Sessions, Group Projects: Doing math is often much more productive and fun, when done with other students. There are plenty of times when it pays off to mull over something on your own, and this is important, but there are also times when talking through problems with other mathematicians can make a huge difference in your understanding of a problem. During the semester we will ask you to participate in some group projects during the weekly problem sessions, usually during the last part of the problem session. Topics to work on will be handed out and you can then begin working on them together with a few other students. To simplify the logistics, we will go ahead and assign groups in the beginning of the semester, but in the second half of the semester we will let you form your own groups if you’d like. About four times during the semester we will ask for your group to eventually turn in the work you’ve been doing together for a grade. Everyone in the group will receive the same grade for the group effort.
 
 

Tests: We will have two midterm tests, one in-class test roughly five weeks into the course (e.g. around March 8^th or 10^th), and then a take home midterm about two thirds the way through the semester, due in April. There will also be a standard, comprehensive three hour final to be held during the regular exam period in May.
 
 

Grades: Midterm 1 – 15%           Take Home Midterm – 20%                  Final – 30%

                                 Homework – 25%                   Group Work – 10%
 
 

Topics for the Semester:

Point Set Topology:
Closure Operators, Continuity, Connectedness, Homeomorphisms, The Game of Hex and the Brouwer Fixed Point Theorem

Group Theory:
Basic Group Theory, Symmetry Groups, Impossibility Theorems, and some other topics if time permits