Examination set up

In this document Download this document we describe the rules for the examination. In short there are three set of homeworks and a final exam. PhD-students taking the course FSF3705, must both get the grade E with the rules explained in the document and pass an oral presentation to the other PhD-students. The choice of topic and reading is done together with the examiner.

The final exam will have two parts; Problems and Theory. 

The problem part will have some problems that you should solve. If you have done the suggested exercises for each class and the hand-in-problems you should be well prepared.

The theory part will have theory questions, such as: state theorem X, prove the following Y or define what is ment by Z.
You should know all definitions in the chapters included in the course. Below are lists of theorems.

You should be able to state the following. That is, a question might be: State X'', where X is something from the list below (you should also explain the terminology involved) or  Define Y'', where Y is any central definition in the course.

- Gale's Evenness condition (Thm 0.7 in Ziegler)
- Main Theorem for Polytopes (Thm 1.1)
- Main Theorem for cones (Thm 1.3)
- Five properties of the face lattice of a polytope. (Thm 2.7)
- Representation theorem for polytopes (Thm 2.15)
- Upper Bound Theorem (thm 8.23)
- Kruskal-Katona Theorem (Thm 8.32)
- g-Theorem (Thm 8.35)
- The definitions of regular subdivision, height/lifting function, Delauney triangulation, flip, and secondary cones.
- The (strong and weak) duality Theorems in linear optimization.
- Theorem on the structure of triangulations and regular subdivisions (Thm 5.9.1).
- Give an example of a non-regular triangulation.
- Definitions of Ehrhart polynomial of a lattice polytope, Ehrhart series and h^*-polynomial.
- Ehrhart's theorem (Thm 3.8 in Beck-Robins).
- Stanley's nonnegativity theorem (Thm 3.12 in Beck-Robins).
- Pick's theorem and interpreting the coefficients of the Ehrhart polynomial of a lattice polygon.
- Reciprocity theorem (Thm 4.1 in Beck-Robins).
- Definitions of Gorenstein and reflexive polytopes, with characterizations.
- Definition of Birkhoff polytope and connection with semimagic squares counting.

 

To be extended during the course.

You should be able to prove the following.  (You don't need to know the numbers in the book of the theorems.)  If there are several ways to prove a statement you may choose either. You may use the same lemmas used in the book, but not stronger results.

- Gale's Evenness condition (Thm 0.7 in Ziegler)
- $F$$\diamond$ is a face of the the polar polytope $P\Delta$ if $F$ is a face of $P$. (Thm 2.12)
- Brugesser-Mani Theorem (Thm 8.12)
- Euler-Poincaré Formula for polytopes
- Dehn-Sommerville Equations (Thm 8.21)
- A way to show that every polytope/point configuration has a triangulation.
- A proof for the dimension formula for secondary polytopes (Thm 5.1.10).
- Ehrhart's theorem (Thm 3.8 in Beck-Robins).
- Stanley's nonnegativity theorem (Thm 3.12 in Beck-Robins).
- Reciprocity theorem (Thm 4.1 in Beck-Robins).
- The h^*-polynomial of a Gorenstein polytope is palindromic (Thm 4.6 in Beck-Robins, only one direction but for all Gorenstein polytopes)

To be extended during the course.