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## WS 2016/2017

### 18.10.2016 um 16:15 Uhr in 69/125:

#### Michela Di Marca (University of Genova, Italy)

##### (Set-theoretic) complete intersections and dual graphs

I n this talk I will examine certain properties of the dual graph of a projective line arrangement in *P ^{n} *that is defined by a complete intersection ideal (or more generally that is arithmetically Gorenstein). I will give some examples of union of lines in

*P*that lie on some smooth surface and form a complete intersection. Moreover, motivated by a result of Benedetti, Bolognese and Varbaro, I will explain some open problems about set-theoretic complete intersections, whose solution would provide a construtive method to build a complete intersection ideal (possibly non-radical) having a given connected graph as a dual graph.

^{3}### 08.11.2016 um 16:15 Uhr in 69/125:

#### Shreedevi K. Mausti (University of Genova, Italy)

##### Symbolic blow-up algebras of certain monomial curves

### 29.11.2016 um 16:15 Uhr in 69/125:

#### Olivier Benoist (Université de Strasbourg)

##### On Hilbert's 17th problem in low degree

Artin has solved Hilbert's 17th problem, showing that a nonnegative real polynomial in n variables is a sum of squares of rational functions, and Pfister proved that it is sufficient to use *2 ^{n}* squares. In this talk, we will investigate when Pfister's theorem may be improved. We will show that a nonnegative real polynomial of degree

*d*in

*n*variables is a sum of

*(2*squares of rational functions if

^{n})-1*d<2n*, and sometimes if

*d=2n*.

### 05.12.2016 um 15:15 Uhr in 93/E31:

#### Alessio D'Ali (University of Genova)

##### Resolutions of letterplace and co-letterplace ideals

Fløystad, Greve and Herzog recently introduced, extending ideas by Ene, Herzog and Mohammadi, a class of squarefree monomial ideals encoding the information about order-preserving maps between two finite posets. When the source poset (respectively, the target) is totally ordered, we say that we are considering a *letterplace* (respectively, *co-letterplace*) ideal. The graded Betti numbers of some very diverse monomial ideals (e.g. initial ideals of certain determinantal ideals, strongly stable ideals) can be recovered in this new setting. It is hence interesting to investigate the homological behaviour of general letterplace and co-letterplace ideals: we report on some results in this direction, trying to give some insight about the techniques used (mostly topological for the letterplace case, mostly algebraic for the co-letterplace one). This is joint work with G. Fløystad and A. Nematbakhsh.

### 06.12.2016 um 16:15 Uhr in 69/125:

#### Susanne Müller (Universität Mainz)

##### The *F*-pure threshold of quasi-homogenous polynomials

The *F*-pure threshold of a polynomial * f* over a field of characteristic

*p>0*is a numerical invariant which measures the severity of the singularities of the associated projective hypersurface

*X*. This invariant is the characteristic

*p*analogue of the log canonical threshold in characteristic zero. In this talk, I will explain, after giving a general introduction to this circle of ideas, how to compute the

*F*-pure threshold of quasi-homogeneous polynomials. I consider the case of a Calabi-Yau hypersurface

*X*given by a quasi-homogeneous polynomial

*f*and relate the F-pure threshold of

*f*to a numerical invariant of

*X*, namely the order of vanishing of the so-called Hasse invariant. Furthermore, I give a connection between the height of the Artin-Mazur formal group associated to

*X*and the

*F*-pure threshold of

*f*.

### 17.01.2017 um 16:15 Uhr in 69/125:

#### Felicitas Lindner (Philipps Universität Marburg)

##### Ideals of polynomial rings in infinitely many variables and of the exterior algebra of infinite-dimensional vector spaces

We will study special ideals and ideal chains in polynomial rings *R* whose variables are indexed by an infinite indexing set *I,* namely ideals and ideal chains that are invariant under the action of some monoid *P* that changes variable indices. The talk is based on work by Hillar/Sullivant and Nagel/Römer, who showed that for certain indexing sets *I* and monoids *P*, *P*-invariant ideals in *R* are generated by finitely many *P*-orbits, and *P*-invariant ideal chains in *R* stabilize up to the action of $P$ and have rational Hilbert series. I will present examples for indexing sets *I* and monoids *P* where neither of these results apply. Furthermore, I will describe a general approach to transfer finiteness and rationality results from *R* to the exterior algebra of vector spaces generated by a basis indexed by *I.*

### 24.01.2017 um 16:15 Uhr in 69/125:

#### Binh Hong Ngoc (Universität Osnabrück)

##### Mixed multiplicities of monomial ideals

Study of mixed multiplicities arised from Teissier's paper on complex analytic hypersurfaces with isolated singularities and thereafter has many applications in algebra and singularity theory. In this talk, I will show how we can compute mixed multiplicities of monomial ideals in terms of mixed volumes of some polytopes.

### 31.01.2017 um 16:15 Uhr in 69/125:

#### Liran Shaul (Universität Bielefeld)

##### Non-abelian derived completion of commutative (DG-)rings

Adic completion is a basic operation in commutative algebra with many applications in the category of commutative noetherian rings. In this talk I will explain how to derive this opertion, and construct a derived completion functor, defined on the homotopy category of commutative differential graded rings with an adic topology.