MAT 114: CONVEX GEOMETRY
Instructor: Prof. Jesús A. De Loera.
Phone: (530)-554 9702
Meetings: MWF 10:00-10:50 AM, Physics 140.
Office hours: Wednesday 4:10pm-5:00pm and Friday 11:10pm-12:00pm
or by appointment. My office is 3228 Math. Sciences Building.
NOTE: Jan 22nd, 29th, March 5th, Office hours will be changed to other times.
The TA for this class is Mr. Steven Lu, his office hours are
11:00-11:50am on Wednesdays at 3125 MSB (email@example.com).
We will be glad to help you with any questions or problems you may have.
Text and References: There is not a single text for this class, but
there are several notes and reference material for the class.
The key material can be downloaded here for free! ( You are kindly asked not to waste paper if you intend to print them.)
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 convex sets
and how to answer fundamental questions about them, such as: What is
their volume? What is the smallest convex set that contains another
set? How wide is the convex set? etc. We will also care about computing with
convex sets and functions. The nice feature of this topic is that,
with very little, one can construct beautiful deep mathematics. We
will have a lot of fun!
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.
A version with larger font is also available.
- The second half of the course will be based on my own
notes ``Actually doing it: A hands on introduction to Convex Polyhedra''
- I will also post a few short video-lectures relevant for the course
- We will also use Polymake
for manipulating polyhedra using the computer. I wrote a short introduction for you.
- N.Lauritzen, Undergraduate Convexity: From Fourier and Motzkin to Kuhn and Tucker, World Scientific, 2013.
S. Lay, Convex sets and their applications, Dover, New York, 2007.
I.M. Yaglom and V.G.Boltyansky, Convex Figures,
Holt, Rinehart and Winston, 1961
D. Barnette, Map coloring, polyhedra, and the four-color
problem. The Dolciani Mathematical Expositions, 8. Math. Assoc. of America, Washington, D.C., 1983
G. Ziegler, Lectures on Polytopes, Springer Verlag,
New York, 1995.
Topics to be covered in 114 (breakup of topics is approximate):
(a) Convex sets, Basic linear geometry in Euclidean space,
(b) Linear, Affine, and Convex Combinations & Hulls.
(c) Families of Convex bodies: Balls and Ellipsoids, Polytopes and Polyhedra
(d) Supporting hyperplanes and Separation, Width and diameter.
(e) Faces and Extreme points, Krein-Milman theorem.
(f) Caratheodory, Helly and Radon (their famous theorems)
(a) Polarity and Duality.
(b) Structure of Polyhedra and Polytopes (Farkas, Weyl-Minkowski's theorem)
(c) Main constructions and Visualization (e.g. Prisms, Pyramids, Projections,
Schlegel Diagrams, Nets).
(d) Combinatorial & Computational Issues: Faces, Euler's formula,
Graphs of polytopes (Steinitz, Balinski's theorem).
ORGANIZATION, PRE-REQUISITES and EXPECTATIONS:
- There are 100 points possible in the course.
- There will be two midterms exams , each exam will count
35 points. The midterm exams will be Friday Jan 31st and Friday March 7th.
Each midterm exam will have a take-home portion (15) and an
in-class portion (20). Writing quality is particularly expected in
the take-home portion. THERE ARE NO MAKE-UP EXAMS but, to
allow for unfortunate eventualities, I will drop the lowest of the two
midterm grades for the calculation of the final grade.
The written take-home response must be crafted in LaTEX.
Here I provide you with a sample document
There will be a comprehensive final exam worth 40 points
will take place Thursday, March 20, 2014 at 6:00 pm
- Homework will be assigned generously 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 is
posted online here:
HOMEWORK PROBLEMS (first midterm)
HOMEWORK PROBLEMS (second midterm)
- Quizzes There will be 7 or 8 online quizzes, you will
have 24-36 hours to answer each of them via SMARTSITE. Each will be
worth 5 points, but the lowest two or three scores will be dropped for
a total of 25 points. NO MAKE UP QUIZZES AVAILABLE EVER!
- After collecting all the scores, I will assign grades based on the
statistical information of the points obtained by all students (I
compute the mean, standard deviation, etc. and set letter grades
according with those numbers).
Although I do not make predictions of grades with partial information,
usually a minimum of 60 points is 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.
- My teaching philosophy is simple: Whoever does the work does the
learning. I lecture and discuss things with you as most professors,
but it is only through your work that YOU can learn. I think of the
course as I climbing a mountain, I can't carry you there, you have to
put great personal effort to access knowledge and wisdom. I serve as your
personal guide, trying to guide you through the safest route to the summit.
I do not want anybody to fall, but ultimately it is your responsability that
leads to success.
- For the above reason:
- (MAT 25 and 22A) or (MAT 25 and 67) or
equivalent preparation are pre-requisites. If in doubt,
please ask me.
- I am here to help you, I love teaching this subject and I sincerely
hope you enjoy the course!