In this talk we will show that the sphere packing problem in dimensions 8 and 24 can be solved by a linear programing method. In 2003 N. Elkies and H. Cohn proved that the existence of a real function satisfying certain constrains leads to an upper bound for the sphere packing constant. Using this method they obtained almost sharp estimates in dimensions 8 and 24. We will show that functions providing exact bounds can be constructed explicitly as certain integral transforms of modular forms. Therefore, we solve the sphere packing problem in dimensions 8 and 24.
A projective variety is called hyperbolic if there is a certain family of linear spaces all of whose intersections with the variety are real. Hyperbolicity can be witnessed by a certain type of determinantal representation of its Chow form in the Grassmannian. After some general discussion, we will focus on reciprocal linear spaces, i.e. the Zariski closure of the image of a linear space under coordinate-wise inversion. This is based on joint work with Cynthia Vinzant.
A construction known in combinatorial optimization yields an unbounded convex polyhedron from any directed graphs whose edges are equipped with weights. The feasible points are the potentials of the digraph. We will explore geometric and combinatorial properties of these polyhedra. For instance, they are related to braid arrangements and order polytopes. Moroever, they play a key role in tropical combinatorics.
Gel'fand, Kapranov and Zelevinsky show in their famous book the following: The coisotropic hypersurfaces in Grassmannians are exactly the higher associated hypersurfaces to projective varieties. This talk investigates these objects: We will study their dimensions and degrees as well as their duality behaviour. I will also shortly present my Macaulay2 package Coisotropy.m2 for computations with coisotropic hypersurfaces.
Classical linear inequality systems can be examined well by linear programming. The tropical counterpart is more difficult to handle. Polyhedral methods offer new tools for tropical linear inequality systems. We formulate network problems as tropical inequality systems. In this way, we can apply the formerly developed methods.
Independece is one of the most importent concepts in mathematics. It is fundermental in (linear) algebra, graph theorie and other subfields. Matroids encode the abstract combinatrorial core of indepence. It is conjectured by Crapo, Rota and others that almost all matroids are paving. We present a class of matroids, which genealize paving matroids and furthermore allow us to study classical matroid properties in terms of polyhedral geometry. This follows from the fact that these matroids arise naturally by subdividing the hypersimplex with compatible splits.
In 1906 Steinitz gave a complete characterisation of the set of f-vectors of 3-polytopes. Since then people are trying to find a characterisation for the 4-dimensional case. However, there is still no such description that is complete. There is also no guess for a description of the set of f-vectors of 3-spheres (which have the boundary complexes of 4-polytopes as special cases), nor did we know up to now whether the set of f-vectors of 3-spheres is the same or strictly larger than the one of 4-polytopes - although most spheres are not polytopal (i.e. can be realised as the boundary complex of a convex polytope). In this talk I will present and explain an algorithm to enumerate 3-spheres for a given f-vector (f_0,f_1,f_2,f_3). Furthermore, I will present some of the enumeration results that finally prove that the set of f-vectors of 3-spheres is strictly larger than the set of f-vectors of 4-polytopes. This is joint work with Günter M. Ziegler.
This talk is part of the BMS Friday and takes place in a different venue!
Eigenvectors of square matrices are central to linear algebra. Eigenvectors of tensors are a natural generalization. The spectral theory of tensors was pioneered by Lim and Qi a decade ago, and it has found numerous applications. We present an introduction to this theory, with focus on results on eigenconfigurations due to Abo, Cartwright, Robeva, Seigal and the speaker. We also discuss a count of singular vectors due to Friedland and Ottaviani.