Welcome to Math S101!

Spaces, Mappings and Mathematical Reasoning: 

An Introduction to Proof


Math S101 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 – topology, 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!We are also hopeful that along the way you will discover that doing mathematics is a creative (and enjoyable!) process.

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.Multivariable calculus (such as Math S21A) or even single variable calculus (such as Math S1A or B) can be taken concurrently with this course.Note, however, that Math S101 is not intended for students who have taken Math 23, Math 25 or Math 55.


Andy Engelward, Danny Goroff and Emily Riehl (Course Assistant)

offices:  rooms 435 (Andy) and 427 (Danny) in the Science Center

phone numbers:  (617) 495-4744 (Andy) and (617) 495-2168 (Danny)

emails:  engelward@math.harvard.edu, goroff@math.harvard.edu and eriehl@fas.harvard.edu

Office Hours:  Thursday 11am - noon (room 435)

Class Times:  Tuesdays, Wednesdays and Thursdays from 9:30 – 11:00 and a weekly problem session, TBD

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 course (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 one or two homework sets due per week during the course.In order to make sure that everyone is keeping up during the semester, there will usually be no late homework accepted.Course assistants will normally grade your homework and return it to you within several days.Your lowest homework score from the semester will be dropped before grades are calculated.


Weekly Problem Sessions, Group Projects: Doing mathis 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.Several times during the semester we will ask for the group you’re working with 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 three weeks into the course and then a take home midterm about two thirds the way through the semester. There will also be a standard, comprehensive three hour final to be held during the regular exam period in August.

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

Homework – 25%Group Work – 10%

Main topics for the course:


Point Set Topology:Closure Operators, Continuity, Connectedness, Mappings,

and the Brouwer Fixed Point Theorem