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## SS 2014

### 13.05.2014 um 16:15 Uhr in 69/E15:

#### Davide Alberelli (Universität Osnabrück)

##### The ML-degree of Fermat hypersurfaces

In this joint work with D. Agostini, F. Grande e P. Lella, we study the critical points of the likelihood function over the Fermat hypersurface. The number of these critical points over a projective variety is a topological invariant of the variety and is called maximum likelihood degree (ML degree). We provide closed formulas for the ML degree of any Fermat curve in the projective plane and of Fermat hypersurfaces of degree 2 in any projective space. For other combinations of degree and dimension no closed formula is known, but there are algorithmic methods to compute the ML degree of a generic Fermat hypersurface. Some of them are developed throughout our paper, where the focus is centred on exploiting the symmetries of the varieties we are considering. A computational comparison of the different methods and a list of the maximum likelihood degrees of several Fermat hypersurfaces will be eventually presented at the end of the talk.

### 10.06.2014 um 16:15 Uhr in 69/125:

#### Prof. Dr. Serkan Hosten (San Francisco State University)

##### Tensors of nonnegative rank two, maximum likelihood estimation, and EM algorithm

A tensor of nonnegative rank two is a tensor that can be written as the sum of two rank one tensors with nonnegative entries. These form a semi-algebraic set. In statistics, mixtures of two independence models for discrete data can be thought of algebraically as this semi-algebraic set. Given data, an optimization problem called maximum likelihood estimation needs to be solved over this semi-algebraic set. I will present results on the boundary stratification of binary tensors of nonnegative rank two and discuss the algebraic complexity of the maximum likelihood estimation in this case. This leads us to study the behavior of a popular algorithm, known as the EM algorithm. We will look at the fixed point locus of this algorithm for binary tensors of at most rank two.

### 24.06.2014 um 16:15 Uhr in 69/125:

#### Dr. John Abbot (University of Genova)

##### The Buchberger-Moeller Algorithm and Extensions

The Buchberger-Moeller Algorithm (BMA) was published in 1982. Given a finite set of points (in affine space), it makes astute use of linear algebra to compute a reduced Groebner basis for the ideal of all polynomials vanishing at those points. I shall recall quickly the original algorithm, then show some extensions: e.g. to points in projective space, points with multiplicity, approximate points...

### 08.07.2014 um 16:15 Uhr in 69/125:

#### Prof. Dr. Uwe Nagel (University of Kentucky)

##### **Selbstduale Polyeder und Gorenstein-Ideale/**Self-dual polytopes and Gorenstein ideals

Die Beschreibung der Abbildungen in einer freien Auflösung eines polynomialen Ideals ist oft kompliziert. Allerdings ist eine geometrische Interpretation mit Hilfe von Zellkomplexen möglich, wenn man eine freie Auflösung mit zellulärer Struktur findet. Im Vortrag werden diese Ideen für die Stanley-Reisner-Ideale einiger simplizialer Polyeder illustriert. Im einfachsten Fall ist der auflösende Zellkomplex der Seitenkomplex eines konvexen Polyeders, der bemerkenswerte Eigenschaften hat. Diese Ergebnisse wurden zusammen mit Stephen Sturgeon erziel.

It is often difficult to describe the maps in a free resolution of a polynomial ideal. However, a geometric interpretation is possible if one finds a free resolution with a cellular structure. In the talk these ideas will be illustrated in the case of Stanley-Reisner ideals of some simplicial polytopes. In the simplest case the resolving cell complex is the complex of faces of a convex polytope with remarkable properties. This is based on joint work with Stephen Sturgeon.