Syllabus The Nature of Mathematics Spring 2001 Professor Buck In this course we try to get an idea of how (some) mathematicians think about mathematics. The only prerequisite for the course is that the student agrees to make a concerted and consistent effort to laugh at the professor’s jokes. The topics we will study are briefly outlined below. We will take them in the order listed, unless we don’t. When we are discussing topology and knot theory and coordinate systems, students should bring to class some paper, markers of two different colors, tape, and scissors. The rest of the time you need only your good nature and your intellectual curiosity. There is no text for the course, but we will generate a set of class notes that will be on reserve at the library. Your grade for the course will depend on your efforts on the midterm exam, the final exam, and three papers. The mid-term will cover the first half of the course, the final the second, so now you don’t need to ask what will be on the exam. The dates for the exams are below and the due dates for the papers will be announced in class. Guidelines for the papers: The papers can be at most four pages. Good mathematics writing tends to be fairly concise, so you may have to distill several pages of notes to produce one page of good work. The paper should start off with a concise, complete introduction to a topic we have discussed in class. The second part of the paper should push the topic further, into an area not covered in class. Extend the results we discussed in class by considering new cases. Or find an interesting application. Or invent a new but related concept. Or ....? Concentrate on the mathematics, as opposed to cultural or historical facts about the topic. Include examples of the concept you are discussing. In fact, it is often helpful to include the simplest example you can think of. You are welcome to consult outside sources, but you don’t have to -- the best mathematics usually comes from simply thinking about something for a while. Note that if you consult web sources, you are responsible for ascertaining the accuracy. While the web can be a wonderful resource, there is a lot of nonsense out there too, and if what you hand in is nonsense you won’t get a good grade. In many cases you are probably better off with a book. Write about what you understand. For example, do not mention that Gauss invented algebraic geometry if you do not know what algebraic geometry is. Use neat, legible handwriting, making use of correction fluid for mistakes (as opposed to crossouts), or use machine preparation. Make sure there are no spelling or grammatical mistakes. Spend a little time on your figures and you will find you can do a nice job whether you consider yourself graphically inclined or not. 1. Topology and Knot Theory 8 lectures Manifolds: 1/16 1. Topology as essential shape. Topological equivalence. Definition of manifold, neighborhood. Manifolds of 1,2, and 3 dimensions. 1/18 2. The list of manifolds, homotopy theory, building manifolds by cutting and pasting 1/23 3. Manifolds with boundary, orientability, Euler’s formula Knot Theory: 1/25 4. Knot equivalence, knot types, some simple knots, prime knots and knot products 1/30 5. Planar diagrams of knots, Reidemeister moves, crossing number 2/1 6. Other measures of complexity: ropelength, stick number, energy 2/6 7. Applications of knot theory, DNA, chirality 2/8 8. Links, braids, the braid group 2. Coordinate Systems and Dimensions 4 lectures 2/13 1. Definition of coordinate system, Cartesian product, polar coordinates, metrics 2/15 2. Spherical coordinates, cylindrical coordinates, higher dimensional systems, fractal dimensions 2/20 3. Euclidean geometry, postulates and proof, the fifth postulate 2/22 4. Non-euclidean geometry 3. Variation 1 lecture 2/27 1. Area, volume, examples in physics and biology. 3/1 Mid-Term Exam 4. Infinity 2 Lectures 3/13 1. Cosmology, the physical world, infinity in extension and infinity in division -- Zeno’s paradoxes. Sizes of infinity. Proof that the rationals are countably infinite. Proof that the primes are too. 3/15 2. Proof that the reals are uncountably infinite 5. The Nature of Mathematical Knowledge 1 Lecture 3/20 1. Pure versus applied mathematics, Platonism and constructivism 6. Symmetry and Aesthetics 3 lectures 3/22 1. Symmetry and symmetry groups 3/27 2. Tilings of the plane, and of space 3/29 3. The mathematics of perspective 7. Mathematics and Calculating Machines 3 lectures 4/3 1. Abstract machines, descriptions, definitions, simple calculations 4/5 2. Doubling machines, multiplication 4/10 3. Mathematics software: symbolics, numerics, graphics 8. Networks 2 Lectures 4/17 1. Definition of network, nodes, edges, graphs, connectedness, path, distance. 4/19 2. Applications, including six degrees of separation, small worlds 9. Dynamical Systems, Chaos, the Mathematics of Time 2 Lectures 4/24 1. Introduction to iterated systems, examples, continuity and discreteness 4/26 2. Chaos |
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