Instructor: Prof. Jesús A. De Loera.
email: deloera@math.ucdavis.edu
http://www.math.ucdavis.edu/~deloera/TEACHING/MATH114
Phone: (530)-554 97 02
Meetings: MWF 10:00pm-10:50 AM, Room 293 Kerr Hall. Discussion section will take place Thursdays same time and place.
Office hours: Monday and Wednesday 11:10pm-1:00pm or by appointment. My office is 3228 Math. Sciences Building. The TA for this class is Mr. Brandon Crain, his office hours are 11:00-12noon on Thursdays at 2139 MSB (for contact bncrain@math.ucdavis.edu). We will be glad to help you with any questions or problems you may have!
Text: There are free notes and material for the class which can be downloaded here. You are kindly asked not to waste paper if you intend to print them.
The first half of the class will be based on the book Theory of Convex Sets by G.D. Chakerian and J.R. Sangwine-Yager. The second half of the course will be based on my own notes ``Actually doing it: A hands on introduction to Convex Polyhedra'' (available later on).
Other references include:
S. Lay, Convex sets and their applications, Dover, New York, 2007.
H. G. Eggleston, Convexity. Cambridge Tracts in Mathematics and Mathematical Physics, no. 47, Cambridge University Press, 1958
I.M. Yaglom and V.G.Boltyansky, Convex Figures, Holt, Rinehart and Winston, 1961
A. Barvinok, A course in Convexity, AMS, Providence, 2005.
D. Barnette, Map coloring, polyhedra, and the four-color problem. The Dolciani Mathematical Expositions, 8. Mathematical Association of America, Washington, D.C., 1983
J. Goodman and J. O'Rourke: Handbook of Discrete and Computational Geometry. CRC press, 1997.
G. Ziegler, Lectures on Polytopes, Springer Verlag, New York, 1995.
Description: This course is an undergraduate-level introduction to the geometry of convex sets and their applications. Convex sets are perhaps the simplest geometric objects, examples include balls, cubes, ellipses, polyhedra, and linear spaces. Their simplicity makes them very important and we will look at the structure of these objects and how to answer fundamental questions about them, such as: What is their volume? What is the smallest convex set that contains another set? etc. The nice feature of this topic is that, with very little, one can construct beautiful deep mathematics. We will have a lot of fun!!
Topics to be covered in 114 (breakup of topics is approximate):
(FIRST MIDTERM)
(a) Convex sets, Basic linear geometry in Euclidean space, (b) Linear, Affine, and Convex Hulls. (c) Families of Convex bodies: Balls and Ellipsoids, Polytopes and Polyhedra (d) Supporting hyperplanes and functions, Width and diameter. (e) Faces and Extreme points. (f) Caratheodory, Helly and Radon (their famous theorems)
(SECOND MIDTERM)
(a) Structure of Polyhedra and Polytopes (Farkas, Weyl-Minkowski's theorem) (b) Main constructions, Visualization (e.g. Projections, Schlegel Diagrams). Graphs of polytopes (Steinitz, Balinski's theorem), (c) Combinatorial Issues: Euler’s formula (Counting faces), Graphs of polytopes (Steinitz, Balinski's theorem). (d) Duality and Polarity
GRADING, ORGANIZATION, and EXPECTATIONS:
The dates of the exams are February 1st and March 8th. There will be a comprehensive final exam worth 40 points on the date announced by the registrar. I repeat: THERE ARE NO MAKE-UP EXAMS.
Finally, this class is a transition class in the math department program, thus it is meant to have a strong writing component. The last 20 points of the course will be awarded to each student for a careful final project . The project will be crafted in LaTEX (see example below). The student will select one section of the second half of the course and with the class notes as a base the student will add a careful in-depth exploration of the topic. The student must include the (correct!) solutions for all problems in the section of their choice. Students must submit a draft for comment before finals and deliver the final version of the project the day of the final.