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## SS 2018

### 17.04.2018 um 16:15 Uhr in 69/125:

#### Van Duc Trung (University of Genova, Italy)

##### The initial ideal of generic sequences and Fröberg's Conjecture

Let *K* be an infinite field and let *I = (f _{1},...,f_{r})* be an ideal in the polynomial ring

*R = K[x*generated by generic forms of degrees

_{1},...,x_{n}]*d*. In the case

_{1},...,d_{r}*r=n*, following an effective method by Gao, Guan and Volny, we give a description of the initial ideal of

*I*with respect to the degree reverse lexicographic order. From this description, we prove a conjecture stated by Pardue in 2010 under a suitable condition on

*d*. This holds partial solutions to a longstanding conjecture stated by Fröberg (1985) on the Hilbert series of

_{1},..., d_{n}*R/I*in the case

*r ≤ n+2*and over an infinite field of any characteristic.

### 22.05.2018 um 16:15 Uhr in 69/125:

#### Lorenzo Venturello (Universität Osnabrück)

##### Graded Betti numbers of balanced simplicial complexes

We discuss bounds for the graded Betti numbers of the Stanley-Reisner ring of a simplicial complex. Using lexicographic ideals, Migliore and Nagel, and Murai provide bounds for the case of simplicial polytopes and of Cohen-Macaulay complexes. In this talk I will consider balanced simplicial complexes, i.e., complexes whose underlying graph is minimally colorable, and I will present upper bounds for the general balanced and the Cohen-Macaulay balanced case. Finally, I will derive explicit formulas for so-called cross-polytopal stacked spheres, whose face numbers are minimal among all the balanced spheres on a fixed number of vertices.

### 19.06.2018 um 16:15 Uhr in 69/125:

#### Alexandros Grosdos Koutsoumpelias (Universität Osnabrück)

##### The Algebra of Local Mixtures

We study local mixtures of Dirac distributions and show that they possess nice algebraic properties, highlighting connections between algebra, statistics and geometry. In particular, we study the moment variety, the algebraic variety defined by the moments of these distributions, and characterize it by providing defining equations. Further, we consider mixtures of these distributions and investigate the problem of recovering the parameters of such a distribution from its moments.

This is based on an upcoming paper with Markus Wageringel.

### 26.06.2018 um 16:15 Uhr in 69/125:

#### Bernd Schober (Gottfried Wilhelm Leibniz Universität Hannover)

##### A polyhedral characterization of quasi-ordinary polynomials

The objects that we study in this talk are irreducible polynomials with coefficients in the ring of formal power series in several variables over a field of characteristic zero. Such a polynomial is called quasi-ordinary if its discriminant is a monomial times a unit. The goal of my talk is to present a construction of an invariant which detects whether a given polynomial is quasi-ordinary. The construction uses a weighted version of Hironaka's characteristic polyhedron and successive embeddings of the singularity defined by the polynomial in affine spaces of higher dimensions. In this context we will meet Teissier's notion of overweight deformations of toric varieties which appear in his program for locally resolving singularities with a single toric morphism. Finally, I will briefly mention an extension to the positive characteristic case.