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#### Contact Details

Universität Osnabrück

Institut für Mathematik

Albrechtstr. 28a

D-49076 Osnabrück

Phone.: +49 541 969 2564

Fax: +49 541 969 2770

gkmath@uni-osnabrueck.de

#### Spokesman

Prof. Dr. Matthias Reitzner

Phone: +49 541 969 2239

Fax: +49 541 969 2770

matthias.reitzner@uni-osnabrueck.de

## Research Projects

**Binoids and their algebras**

A binoid is a simple combinatorial structure and their associated algebras give a common generalization of monoid rings and Stanley Reisner rings. In this project the focus will be on topological aspects, infinitesimal properties or on the representation theory of binoids.

*Advisors: Brenner, Bruns, Römer, Röndigs***Combinatorial characterization of stability properties of vector bundles**

A family of monomials in a polynomial ring yields a vector bundle on projective space. Its semistability can be detected by a combinatorial property of the exponents, which yields to the study of lattice point configurations. Possible projects are to determine under which numerical data such semistable configurations exist and how they behave asymptotically.

*Advisors: Brenner, Bruns, Reitzner, Spindler***Differential symmetric signature for monoid rings**

Monoid rings and affine toric varieties provide an important class of examples for the interplay between combinatorial and algebra-geometric structures. The module of differentials is the algebraic version of the tangent bundle of the toric varietiy. As these have singularities, this bundle is only locally free on a large open subset. The asymptotic behaviour of the symmetric powers of the module of Zariski-differentials as a measure for the singularity is in the focus of this project. When the defining cone of the monoid ring is simplicial, then this asymptotic is fairly well understood and expressed by the symmetric signature, which coincides with the F-signature, a notion coming from positve characteristic. But even for the simplest non-simplicial cones the asymptotic behaviour and its relation to the F-signature is completely open.

*Advisors: Brenner, Bruns, Juhnke-Kubitzke, Römer, Röndigs***Hilbert bases of simplicial cones**

The software Normaliz, developed at Osnabrück, computes Hilbert bases of rational cones. The project aims at the improvement of its algorithms by using stochastic methods. A central question is the distribution of lattice points in the fundamental paralleloptopes of simplicial cones.

*Advisors: Bruns, Reitzner, Römer***Optimization under uncertainties**

In practice, data are often not exactly known and can only be estimated by distribution functions (e.g. from intervals) for certain parameters. Then, the goal of optimization is to use this information such that better solutions are obtained than when uncertainties are ignored. While stochastic programming especially optimizes the expectation value, robustness concepts mainly deal with the worst case. The objective of this project is to develop the theoretical background and solution methods integrating data uncertainties in an appropriate way (stochastic optimization, different robustness concepts). The concepts are to be evaluated in connection with applications in practice.

*Advisors: Knust, Döring, Reitzner***Optimization problems with special structures**

Many optimization problems like the TSP or graph coloring are NP-hard in general. However, in practical applications often data are based on special structures (e.g., Monge matrices or graphs with special properties) which allow the development and usage of more efficient algorithms. The objective of this project is to study optimization problems with special structures (e.g., storage or scheduling problems) and develop efficient solution algorithms for them.

*Advisors: Knust, Juhnke-Kubitzke***Polynomial approximation of multivariate functions**

Approximating functions by polynomials is a standard procedure in applied analysis. Besides classical mathematical questions regarding best possible approximation in certain functions spaces, many practical problems ask for good approximations from only a couple function values. The goal of this project is the study of high dimensional approximation problems and their discretisation using algebraic techniques. For example, we are interested in the discretisation by lattice rules or the construction of interpolatory point sets.

*Advisors: Kunis, Römer***Face enumeration of manifolds**

Given a triangulable manifold it is interesting to know how many vertices a triangulation must have at least or more generally, which vectors might occur as f-vectors of such a triangulation. The answers to these questions should depend on topological properties of the manifold. In this project, we want to study these and related questions. We are also interested in the case that the triangulation carries additional combinatorial properties such as balancedness or flagness.

*Advisors: Reitzner, Juhnke-Kubitzke***Combinatorial structure of random polytopes**

A random polytopes is the convex hull of a random point set. Of interest are distributional properties of combinatorial functions of random polytopes, for example the f-vector, or the number of shells of the so-called 'convex-hull-peeling'.

*Advisors: Reitzner, Juhnke-Kubitzke, Bruns***Betti numbers for random simplicial complexes**

Choose random points in a state space and generate a random simplicial complex by connecting points if they are in a special spatial position. The topological properties of this random simplicial complex can be described by the Betti numbers. We are interested in distributional properties of these random Betti numbers.

*Advisors: Reitzner, Juhnke-Kubitzke, Römer, Röndigs***Combinatorial properties of random mosaics**

A random mosaic is the tessellation of space into convex polytopes. In this project the combinatorial structure of the zero cell and the typical cell shall be investigated.

*Advisors: Reitzner, Römer***Algebraic and combinatorial structures in statistics**

On one hand, the algebraic statistic uses algebraic and combinatorial methods in order to study statistical problems. On the other hand, results in statistics have given reason for new problems in algebra and discrete mathematics. It is the aim of this project to examine algebraic and combinatorial structures which are, for example, of a certain meaning for the understanding of statistical models.

*Advisors: Römer, Juhnke-Kubitzke, Reitzner***Hermitian K-theory for infinity categories**

In this project the existing machinery for hermitian K-theory of exact categories with weak equivalences and duality should be enlarged to infinity categories with duality. The aim is to obtain comparison results for the new theory as well as new examples such as hermitian K-theory of E-infinity ring spectra.

*Advisors: Röndigs, Brenner, Spitzweck>***Homotopy types of varieties**

In this project the existing machinery for hermitian K-theory of exact categories with weak equivalences and duality should be enlarged to infinity categories with duality. The aim is to obtain comparison results for the new theory as well as new examples such as hermitian K-theory of E-infinity ring spectra.

*Advisors: Röndigs, Brenner, Spitzweck***Random maps of simplicial complexes**

The aim is to investigate homotopical properties of random maps of simple simplicial complexes.

*Advisors: Röndigs, Reitzner***Derived motivic fundamental groups**

Construct and study derived motivic fundamental groups for e.g. Artin-Tate motives, Tate motives with integral coefficients on the projective line minus three points and on moduli spaces of curves. Define a Grothendieck-Teichmueller group and relate it to the motivic fundamental group with integer coefficients of the integers.

*Advisors: Spitzweck, Röndigs, Brenner***Comparison results in derived algebraic geometry**

The project aims to compare the frameworks of Lurie on the one hand and of Toen-Vezzosi on the other for derived algebraic geometry. I t should build upon work of Porta on the comparison for derived Deligne-Mumford stacks (using simplicial commutative rings), with a view towards objects within spectral algebraic geometry. On the way acquaintance with infinity categorical techniques should be obtained.

*Advisors: Spitzweck*

### Related Institutions

The centre for PhD students at the University of Osnabrueck (ZePrOs) is an institution, that unites all PhD students of all departments of the university as an umbrella organisation and interlinks the complete research orientated education of PhD students. ZePrOs offers all PhD students an especially fitted qualification range and individual assistance which aim at the optimisation of their scientific work and the achievement of job market relevant competences.