**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.

**Teachers:**

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