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## SS 2017

### 16.03.2017 um 16:15 Uhr in 69/125:

#### Charles Jordan (Hokkaido University, Japan)

##### Parallel Vertex and Facet Enumeration With mplrs

We describe mplrs, a new parallel vertex/facet enumeration program based on the reverse-search code lrs. The implementation uses MPI and can be used on single machines, clusters and supercomputers. It is a C wrapper derived from the earlier parallel implementation plrs (written by G. Roumanis). We describe the budgeted parallel tree search implemented in mplrs and compare performance with other sequential and parallel programs for vertex/facet enumeration. In some instances, mplrs achieves almost linear scaling with more than 1000 cores.

The approach used in mplrs can be easily applied to various other problems; if time permits we will briefly mention results along these lines. This is joint work with David Avis.

### 04.04.2017 um 16:15 Uhr in 69/125:

#### Davide Bolognini (Universitá Genova)

##### Binomial Edge Ideals of Bipartite Graphs

We present a combinatorial classification of all bipartite graphs G whose binomial edge ideal J_G is Cohen-Macaulay. The connected components of such graphs can be obtained by gluing a finite number of basic blocks with two operations. In this context, we prove the converse of a well-known result due to Hartshorne, showing that the Cohen-Macaulayness of these ideals is equivalent to the connectedness of their dual graphs. As application, we verify that, for this class of ideals, a "Hirsch style" conjecture by Benedetti-Varbaro holds. Moreover, we study interesting properties also for non-bipartite graphs and in the unmixed case, constructing classes of bipartite graphs with J_G unmixed and not Cohen-Macaulay. All the results are independent of the field.

This is a joint work with A. Macchia and F. Strazzanti.

### 02.05.2017 um 16:15 Uhr in 69/125:

#### Sebastian Gutsche (RWTH Aachen)

##### A constructive model for coherent sheaves over a toric variety in CAP

The software project CAP - Categories, algorithms, and programming is a realization of categorical programming written in GAP. CAP makes it possible to compute complicated mathematical structures, e.g., spectral sequences. This can be achieved using only a small set of basic algorithms given by the existential quantiers of Abelian categories, e.g., composition, kernel, direct sum. In this talk I will explain the concept of categorical programming and describe the framework for computable categories implemented in CAP. As an example of the exibility of this framework, I will describe a way to model the category of coherent sheaves over a toric variety, using a computable model of the Serre quotient category of a category of graded modules in CAP and combinatorial computations in Normaliz. To emphasize the computational capabilities of CAP I will compute the bidualizing Grothendieck spectral sequence of a graded module and a coherent sheaf. We will see that using categorical programming in CAP, the same algorithm can be applied to both contexts.

This is joint work with Sebastian Posur.

### 30.05.2017 um 16:15 Uhr in 69/125:

#### Yue Ren (Max-Planck-Institut, Leipzig)

##### Computing tropical varieties using Newton polygon method

Computing tropical varieties is an algorithmically challenging task, requiring sophisticated techniques from computer algebra and convex geometry. We will give a brief overview of the techniques developed by Bogart-Jensen-Speyer-Sturmfels-Thomas for tropical varieties over the complex Puiseux series and the mathematics behind it. We will highlight the mathematical challenges in generalizing those techniques and the practical bottlenecks in their implementation. In particular,we will address those bottlenecks and show how they can be avoided using well-known methods in computer algebra. This is joint work with Thomas Markwig and Tommy Hofmann.

### 08.06.2017 um 16:15 Uhr in 69/E23:

#### Husney Parvez Sarwari (TIFR Mumbai India)

*K*-theory of monoid algebras and a question of Gubeladze

We will discuss some questions of Gubeladze on homotype type property of monoid *R*-algebra *R[M*], where *R* is a regular ring and * M* is a commutative cancellative torsion-free monoid. At the end, we will give a application of our result to a question of Lindel on Serre dimension of monoid algebras.

### 13.06.2017 um 16:15 Uhr in 69/125:

#### Christiane Görgen (Max-Planck-Institut, Leipzig)

##### Discovery of statistical equivalence classes using computer algebra

Discrete statistical models supported on labelled event trees can be specified using so-called interpolating polynomials which are generalizations of generating functions. These admit a nested representation. A new algorithm exploits the primary decomposition of monomial ideals associated with an interpolating polynomial to quickly compute all nested representations of that polynomial. It hereby determines an important subclass of all trees representing the same model. To illustrate this method we analyze the full polynomial equivalence class of a staged tree representing the best fitting model inferred from a real-world dataset.

### 20.06.2017 um 16:15 Uhr in 69/125:

#### Emanuele Ventura (Max-Planck-Institut, Leipzig)

*Real rank geometry of ternary forms*

The problem of expressing a homogeneous polynomial as a sum of powers of linear forms is very classical and goes back to the work of Sylvester, Hilbert, and Scorza among others. The real rank of a homogeneous polynomial is the smallest number of linear real forms such that the polynomial admits such a representation. The space parametrizing all real decompositions of a polynomial as a minimal sum of powers is a semialgebraic set sitting inside the classical varieties of sums of powers. We will discuss these real geometric objects for general plane curves of small degrees. This is a joint work with M. Michalek, H. Moon and B. Sturmfels.

### 27.06.2017 um 16:15 Uhr in 69/125:

#### Mario Kummer (Max-Planck-Institut, Leipzig)

##### Separating Morphisms from Real Algebraic Curves

Given a real algebraic curve we consider the set of all morphisms to the projective line with the property that the preimage of every real point consists entirely of real points. It turns out that this generalises the notion of interlacing polynomials on the real line to projective curves. Using this theory, we will answer a question raised by Shamovich and Vinnikov on hyperbolic curves as well as a question by Fiedler-LeTouzé on totally real pencils on plane curves. This is joint work with Kristin Shaw.