If you missed the organizational meeting you can still participate, just send us an email.
First (organizational) meeting is Tuesday, April 15, at 16:15 in the Seminar room of the Villa (Arnimallee 2). Topics will be assigned there.
In this seminar we will explore advanced notions and constructions from discrete geometry. In particular, we will explore connections to other areas. Students should have a good background in discrete geometry or discrete mathematics (such as ‘Discrete Geometry I’ or ‘Discrete Mathematics I’).
This will be a block seminar (all talks on 56 July).
Language: English/German
The organizational meeting will be in the first semester week.
If you are interested in the seminar signup at: https://lists.fuberlin.de/listinfo/DGseminar
Who  What  References 

Tom  Secondary polytopesGelfand, Kapranov, and Zelevinsky introduced a construction that maps each triangulation of a polytope P with n vertices into a point in R^n. The convex hull of these points is called the secondary polytope of P and has a remarkable combinatorial structure. The vertices of the secondary polytope are in correspondence with regular triangulations of P, and two of them are linked with an edge if and only if the corresponding triangulations are related by a bistellar flip. 

Nevena  Fiber polytopesFiber polytopes are an amazing generalization of secondary polytopes. While secondary polytopes concern regular subdivisions, those induced by the projection of a simplex onto the polytope, fiber polytopes reflect the combinatorial structure of coherent subdivisions induced by a general projection of polytopes P > Q. 

Andreas  HompolytopesAn hompolytope captures the set of all affine maps between two polytopes. While its facets have a well understood geometric explanation, understanding its vertices is still a big challenge. 

Hannah  The universal polytope and equidecomposabilityThe universal polytope of a polytope P is a polytope associated to all triangulations of P (not only the regular ones). A polytope is called equidecomposable if all its triangulations have the same number of simplices. The universal polytope is used to prove a combinatorial characterization of equidecomposable polytopes. 

Katy  AssociahedraThe associahedron is another remarkable polytope. Its vertices have different combinatorial interpretations: they correspond to the different ways to bracketing a string of letters, binary trees, triangulations of a polygons,... 

Julian  SylvesterGallaiThe SylvesterGallai theorem states that for any set of points in the plane, not all of them in a common line, there is a line that goes exactly through 2 points. This theorem admits a lot of interesting proofs. Once proved, a natural question is how many of these lines exist. 

Paul  Colorful combinatoricsCharathéodory's Theorem states that if a point x is in the convex hull of a set X in R^d, then there is a subset of X of at most d+1 points that contains x in its convex hull. It has a colorful version, where the points of X are colored and we require the subset to be rainbow. It can be used to prove Tverberg's theorem, which also admits a colorful version. 

Wayne  Rigidity and the Lower Bound TheoremThe lower bound theorem characterizes those simplicial polytopes with the minimal number of facets. Kalai's beautiful proof is based on Cauchy's rigidity theorem. 

Giulio  Permutation polytopesThe Birkhoff polytope is the convex hull of the nxn permutation matrices and has a nice halfspace representation as the set of doubly stochastic matrices. More generally, a permutation polytope is the convex hull of a subgroup of the nxn permutation matrices. These can be, however, much more complicated. 

Stefan  Geometric constructionsIn his study of projectively unique polytopes, McMullen introduced the polyhedral operations of subdirect sums and products. The latter can be further generalized as a special case of wedge products. 
