FB 6 Mathematik/Informatik

Institut für Mathematik

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WS 2017/2018

02.10.2017 um 16:15 Uhr in 69/125:

Javier A. Carvajal-Rojas  (University of Utah)

Finite torsors over strongly F-regular singularities

We will present an extension of the work by K. Schwede, K. Tucker and myself on local étale fundamental groups of (strongly) F-regular singularities. We will discuss the existence of finite torsors over the regular locus of these singularities that do not come from restricting a torsor over the whole spectrum. In the process we will prove that canonical covers of F-regular (resp. F-pure) local rings are F-regular (resp. F-pure), as well as bounding the torsion of: (locally) the Picard group of F-regular singularities and (globally) the divisor class group of globally F-regular varieties.

05.12.2017 um 16:15 Uhr in 69/125:

Bogdan Ichim (University of Bucharest, Romania)

An introduction to voting theory

We describe several experimental results obtained in four candidates voting theory. These include the Condorcet and Borda paradoxes, as well as the Condorcet efficiency of plurality voting with runoff. The computations are done by Normaliz. It finds precise probabilities as volumes of polytopes and counting functions encoded as Ehrhart series of polytopes.

09.01.2018 um 16:15 Uhr in 69/125:

Dinh Le Van (Universität Osnabrück)

Hyperplane arrangements

16.01.2018 um 16:15 Uhr in 69/125:

Dr. Matteo Varbaro (University of Genova, Italy)

The nerve of a positively graded K-algebra

Given a Noetherian positively graded K-algebra R, the nerve (or Lyubeznik complex) of R, denoted by N(R), is the following simplicial complex:
- the vertices of N(R) correspond to the minimal primes P1,…,Ps of R;
- Pi1,…,Pir is a face of N(R) if and only if the radical of Pi1+…+Pir is different from the maximal irrelevant ideal R+. It is not difficult to see that Proj R is connected if and only if N(R) is connected.
In the talk I will discuss how topological properties of N(R) relate to algebraic properties of R.

23.01.2018 um 16:15 Uhr in 69/125:

Konrad Voelkel (Universität Osnabrück)

Wonderful Completions

We will learn about (partial, equivariant) compactifications/completions, in particular for torus actions. Amongst these, there are some truly wonderful ones, called wonderful completions in the literature. Relevant to Combinatorial Hodge Theory is a wonderful completion of a hyperplane arrangement complement. This is constructed by a sequence of blowups. It can be understood by comparison with a canonical hyperplane arrangement that yields the permutahedral variety, the main subject of Huh's thesis. I will aim to keep the talk as elementary as possible (all words mentioned will be explained).

30.01.2018 um 16:15 Uhr in 69/125:

Manh Toan Nguyen (Universität Osnabrück)

Hodge theory and applications

In this talk, I will give an introduction to the (classical) Hodge theory. This pakage provides rich structures on the singular cohomology ring of a compact Kähler manifold (e.g., a smooth complex projective variety). I will discuss about Harmonic forms, (p,q)-forms, Hodge decomposition, Hard Lefschetz theorem and Hogde-Riemann bilinear relations.
As an application, I will show how to obtain the Khovanskii-Teisser inequalities for nef line bundles on a complete algebraic variety from the Hodge index theorem for surfaces. This generalizes the Aleksandrov-Fenchel inequalities for mixed volumes of convex bodies.