MAT246H1 Y: Concepts in Abstract Mathematics

The coordinator values rigour and depth over the number of subjects covered.

The goal of the problem sets is to test and expand the students’ understanding of the material.  In the tests, students are expected to demonstrate that they worked individually and comprehended the  problem  sets.   In  particular,  students are not expected to come up with entirely new solutions on the spot.

3 Evaluation and Course Policies

.  Attendance in lectures and tutorials is very important:  there is a clear correlation between students who show up for class and those who do well on the final. To encourage atten- dance every week, in one of the two lectures, a short quiz will be given.  Students will only have a few minutes to submit their solution. While lecture attendance is not mandatory, it is strongly encouraged and the quiz is designed to reward those who do attend.  The two lowest quiz marks are dropped. There are no late submissions and there is no way to make up for a missed quiz.

.  Every student starts the term with 2 points in communication.  Points, up to a total of 5, can be earned by writing the first clear answer (not necessarily correct or complete, but useful hints for other students, or clarifications, etc.) on Piazza, being the first to inform the coordinator of mistakes in the course content (typos in the notes, missing hypotheses in the assignments, etc.) .  Points can be lost by breaking the course communication policies (asking questions via email when the answer  is found  in the syllabus, already asked on Piazza, etc.) .  This  is to discourage chaos and confusion  in the course.  It is possible to get a perfect grade with 0 communication points.  The coordinator keeps track of these points and shared only on the last week of classes.

.  There are six problem sets.  In each of these,  it is clearly indicated which problems need to be turned in to get a full mark, and these are always solvable with the material covered in the lectures.  Students can cite theorems from the lecture notes.  Material from other sources is not allowed, as the course is self-contained.  Extra  problems are given but not marked.  Solving these should render the final significantly easier and offer students the opportunity to step beyond the required curriculum of the course.  The lowest problem set mark is dropped. There are no late submissions and there is no way to make up for a missed problem set.  Any legible format (LATEX, handwritten and scanned/photographed, Word doc, etc.)  is acceptable.

.  There is an in-person comprehensive final exam.  No aids and no formula sheet.

4 Subjects

1. Foundations.

.  Elementary set theory: subsets, operations, functions, images of functions, injectivity, surjectivity, equivalence relations.

.  Logic:  basic proving techniques.

.  The natural numbers:  induction and well-ordering.

2. The Integers.

.  Construction through equivalence relation.

.  Divisibility and primes.

.  B´ezout’s shortcut to the fundamental theorem of arithmetic.

.  Modular arithmetic.

.  Linear Diophantine equations:  Chinese remainder theorem.

.  Fermat’s little theorem.

3. Fields.

.  Finite fields.

.  Construction of the rationals through equivalence relation.

.  Construction of the reals as a complete field through Cauchy sequences.

4. Infinity.

. A naive approach to cardinal arithmetic.

.  Harder set theory questions and advanced proving techniques.

5. Topology. Basic point-set topology of the  reals with some very gentle abstraction to other areas.

6. Combinatorics. Students  can  choose  between  advanced  problem-solving  techniques, graph theory, computability theory (in the context of the first two subjects), and pos- sibly others.

5    Course Schedule and Zoom links

All activities are conducted over the following Zoom session (you might need to log in with your UofT email) .