Mathematics Beyond Calculus

This half-course aims to lead you to broaden your perspective on what it means to do mathematics. You will encounter mathematics that usually lies beyond the introductory calculus and linear algebra sequence. You will gain familiarity with the formulation of a mathematical problem, develop strategies for investigating prob- lems, and explore the mathematical literature. In addition, you will improve your ability to think rigorously and communicate mathematical information.\ud You will apply these modes of thought to the study of mathematical knots. A mathematical knot is like the knots you use to tie your shoes, with two differences: first, a mathematical knot has no ends, that is, it is a knotted-up circle. Second, a mathematical knot has no thickness. Two knots are defined to be equivalent if one can be deformed into the other without cutting the string. A natural question — at least for a mathematician — is whether there exists a general and effective method for determining if two knots are, or are not, equivalent.\ud You will explore this question (and more) by using knot diagrams, exploring interesting families of knots, and defining “invariants” of knots. An invariant is an object you assign to a knot that does not change when the knot is deformed. That object can be as simple as an integer or a polynomial, though they can get much more complicated. If two knots have different invariants, then it is impossible to deform one into the other