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## WS 2013/2014

### 05.11.2013 um 16:15 Uhr in 69/125:

#### Dr. Bogdan Ichim (University of Bucharest, Romania)

##### An algorithm for computing the multigraded Hilbert depth of a module

A method for computing the multigraded Hilbert depth of a module was presented in the paper "A method for computing the multigraded Hilbert depth of a module" (joint work with J.Moyano). We improve the method and we introduce an effective algorithm for performing the computations. In a particular case, the algorithm may also be easily adapted for computing the Stanley depth of the module. We further present interesting examples which were found with the help of an experimental implementation of the algorithm. Thus, we completely solve several open problems proposed by Herzog.

### 26.11.2013 um 16:15 Uhr im 69/125:

#### Dr. Christian Bey (Universität Osnabrück)

##### An inequality in Ehrhart theory

### 14.01.2014 um 16:15 Uhr in 69/125:

#### Dr. Jan Uliczka (Universität Osnabrück)

##### Duality and syzygies for semimodules over numerical semigroups

*Let* Γ = <α, β> be a numerical semigroup. In this article we consider the dual ∆^{*} of a Γ-semimodule ∆; in particular we deduce a formula that expresses the minimal set of generators of ∆^{*} in terms of the generators of ∆. As applications we compute the minimal graded free resolution of a graded F[t^{α},t^{β}]-submodule of F[t], and we investigate the structure of the selfdual Γ-semimodules, leading to a new way of counting them.

### 28.01.2014 um 16:15 Uhr in 69/125:

#### Dr. Gharchia Abdellaoui (Universität Osnabrück)

##### On topology of moduli spaces of framed sheaves

In the first part of this talk I will introduce the moduli space *M(r,n)* of framed sheaves on the projective plane and perform a study of some of its topological properties. I will show that *M(r,n)* admits cell decompositions determined by a torus action. Using these decompositions we prove that *M(r,n)* is homotopy equivalent to a compact proper subvariety.

In the second part of the talk I will discuss a generalization to framed sheaves on toric surfaces under some assumptions

### 11.02.2014 um 16:15 Uhr in 69/125:

#### Prof. Dr. Vijaylaxmi Trivedi (TIFR Mumbai)

##### Frobenius pull backs of vector bundles for higher dimensional varieties

We know that a Frobenius pull back of a semistable bundle need not remain semistable.. However, if *X* is a nonsingular projective curve of genus *g* and dened over a eld of characterstic *p > 0*, then Shepherd-Barron and X. Sun proved (independently), that for a semistable vector bundle *V* of rank *r*, the instability degree of* F⋅V* is bounded by *2(g-1)(r-1)*.

This bound on the instability is useful in keeping a check on some of the behaviour of a vector bundle after Frobenius pullbacks.

For example one can prove that, for any vector bundle *V* and for large *p* (in terms of degree of *X* and rank of *V*), the Harder-Narasimhan ltration of *F⋅V* is a renement of the Frobenius pull back of the Harder-Narasimhan ltration of *V*. We give counterexamples to prove that some such conditions on p are necessary.

We extend such results to vector bundles over higher dimensional varieties. To prove these, we answer a question/conjecture of X. Sun (though for *p* bigger than rank of *E +* dimension of *X*), which is an anaolgue of the above mentioned result of Shepherd-Barron and X. Sun in higher dimension.

### 11.02.2014 um 17:45 Uhr in 69/125:

#### Prof. Dr. Vasudevan Srinivas (TIFR Mumbai)

##### Algebraic versus topological entropy for surfaces over finite field

<pre>For an automorphism of an algebraic surface, we consider some properties of eigenvalues of the induced linear transformation on l-adic cohomology, motivated by some results from complex dynamics, which are related to the notion of entropy. This is a report on joint work with H'el`ene Esnault.