**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: Eulerâ€™s formula (Counting faces), Graphs of polytopes (Steinitz, Balinski's theorem). (d) Duality and Polarity

**GRADING, ORGANIZATION, and EXPECTATIONS**:

- There are 100 points possible in the course.
There will be
**2 midterms exams**, each exam will count 40 points. Each exam will have a take-home portion (20) and an in-class portion (20). Writing quality is particularly expected in the take-home portion. THERE ARE NO MAKE-UP EXAMS but instead, to allow for eventualities, I will drop the lowest of the two midterm grades for the calculation of the final grade.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. - Homework will be assigned each day but will
**not**be collected. Instead homework problems will be used in questions in the take-home portion of each midterm. All Homework will be posted online.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. - I will assign grades based on an statistical information of the
points obtained by all students (I compute the mean, standard
deviation, etc. and set letter grades according with those numbers),
but 60 points or higher are normally expected to pass the class.
I will handle all grades via the myucdavis grade
system. This means that if you are registered students at UC Davis you
can access grade information for this class via the internet (check
https://my.ucdavis.edu/ for details). I will not disclose your grade
in any other form.
- MAT 25 and 67 or equivalent preparation is a pre-requisite. If in doubt, please ask me. You are expected to work outside the classroom solving exercises, reading the book, thinking about the theorems, etc. I estimate a minimum of 2 hours work at home per lecture. The most important thing is what YOU learn by working the exercises. Math is not an spectator sport! It is easy to fall behind, please be careful! I am here to help you and I hope you enjoy the course.

Jesus De Loera 2009-12-16