# Berlin-Leipzig Seminar on Algebra, Geometry and Combinatorics

## Abstracts TU Berlin - December 2017

Please also see the programm of the seminar.

Alessio D'Ali (MPI Leipzig)**What is... a gaussoid?**

Gaussoids are combinatorial gadgets introduced in 2007 by Lněnička and Matúš as an attempt to encode which patterns of conditional independence can arise between a finite number of Gaussian random variables. The aim of this talk is to give an algebraic interpretation of these objects, drawing a parallel to the well-known concept of matroid. This is joint work with Tobias Boege, Thomas Kahle and Bernd Sturmfels.

Elisa Gorla (Université de Neuchâtel)**Gröbner bases in theory and practice **

Gröbner bases are both a fascinating topic in its own right and a very useful computational tool with many applications. After briefly introducing Gröbner bases, I will discuss some applications to coding theory and cryptography. Then I will present some recent joint results with Aldo Conca and Emanuela De Negri on universal Gröbner bases.

Mohab Safey El Din (Sorbonne Universités, Université Pierre et Marie Curie - Paris 6)**The quest of asymptotically fast algorithms for polynomial system solving over the reals**

Polynomial systems arise in many areas of engineering sciences. Most of the time, the end-user expects some information about the real solution set of the system under study. Algorithmic problems encompass which appear frequently is to decide the existence of real solutions, compute sample points in each connected component of the set under study, compute a description of this set on some affine space or answer connectivity queries. All these problems are NP-hard and, actually the best known complexities to solve them are exponential in the number of variables. Moreover, because of the non-linearity of the considered systems, reliable issues are important. In this talk, I will give an overview of computer algebra algorithms for solving these problems with a focus on those which conciliate practical efficiency with asymptotically optimal complexity.

Mahsa Sayyary (MPI Leipzig)**Real Space Sextics and their Tritangents**

The intersection of a quadratic surface and a cubic surface in 3-space is canonical curve of genus four. It has120 complex tritangent planes. We study these tritangents for non-empty real curves. These have at most 5 connected components. We show that the number of planes that are tangent at three real points vary widely; both 0 and 120 are attained . This solves a problem suggested by Arnold Emch in 1928.

Lorenzo Venturello (Universität Osnabrück)** Balanced Combinatorial Manifolds with Boundary**

A simplicial complex is called balanced if its underlying graph can be minimally colored and these objects, introduced by Stanley, carry a lot of interesting combinatorial and algebraic properties. In 2015 Izmestiev, Klee and Novik introduced a set of flips that suffices to relate any two PL homeomorphic balanced combinatorial manifolds, preserving balancedness at each step. Starting from their result we present a balanced version of a theorem by Pachner for manifolds with boundary, showing that any two balanced combinatorial manifolds with boundary can be connected by a sequence of shellings and their inverses preserving the coloring at each step.

## General information

The seminar will take place on a regular basis at the

Max Planck Institute for Mathematics in the Sciences Leipzig,

FU Berlin**,**TU Berlin

## Next seminar

TU Berlin, December 17-18, 2017

Program

Abstracts

## Scientific Organizers

**MPI MIS Leipzig**

**FU Berlin**

**TU Berlin**

## Administrative Contact

**Saskia Gutzschebauch**

MPI für Mathematik in den Naturwissenschaften

Leipzig

Contact by Email