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## Sommersemester 2015

### 28.04.2015 um 16:15 Uhr in 69/125:

#### Prof. Dr. Sara Faridi (**Dalhousie University**, **Canada**)

##### Combinatorial methods for resolutions of monomial ideals

In this talk we will review various methods to study resolutions of monomial ideals using graphs and simplicial complexes. Some well known methods will be highlighted with new perspectives, and some specific methods to compute Betti numbers of monomial ideals will be given. In particular, we give a method to characterize all Betti numbers of simplicial forests.

### 19.05.2015 um 16:15 Uhr in 69/125:

#### Davide Alberelli (**Universität Osnabrück**)

##### Cohomology and Picard Group on Schemes of Binoids

Binoids are interesting combinatorial objects that allow us to describe some properties of monomial and binomial algebras. In this talk I will present our ongoing work about schemes of binoids, sheaves over them and cohomology of these sheaves. I will then present a result that links the Picard group, of line bundles on the scheme of binoids, to the first cohomology group of the sheaf of invertible elements. I will later focus on a particular case, giving the full description of the cohomology of the sheaf of invertibles on the punctured binoid scheme associated to a simplicial complex and the effective algorithm to compute this cohomology groups, implemented in GAP. Eventually I will give some insights on the link with monomial and binomial algebras, explaining some examples in which the combinatorics of binoids is not enough to describe the invariants of the corresponding algebras.

### 02.06.2015 um 16:15 Uhr in 69/125:

#### Hongyi Chu (**Universität Osnabrück**)

##### Building Higher Categories

In recent years people from different areas of mathematics and mathematical physics have increasingly become interested in the concept of higher categories. In my talk I will explain what these higher categories are and why they provide a natural language to describe structures such as cobordisms, which are fundamental for the understanding of TQFT and the proof of the Mumford conjecture. First, I will recall the definition of categories and introduce multicategories as their natural generalization. We will see that algebraic structures such as binoids or categories can be redefined by using the language of multicategories. Furtheremore, these structures even permit us to enrich categories in another category. At the end of the talk I will show how higher categories can be defined in this way.

### 23.06.2015 um 16:15 Uhr in 69/125:

#### Dr. Sara Saeedi Madani (Universität Osnabrück)

##### Binomial edge ideals

In this talk, we are going to give an overview of some properties of a class of binomial ideals associated to graphs, called ”binomial edge ideals”. We focus on some invariants arising from the minimal graded free resolution, in particular, the Castelnuovo-Mumford regularity of such ideals. Special attention is given to a conjectured upper bound for the regularity, due to Matsuda and Murai. (This talk is based on some joint works with Dariush Kiani.)

### 14.07.2015 um 16:15 Uhr in 69/125:

#### Alejo López Ávila (Universität Osnabrück)

##### Multiplicative structures in hermitian K-theory

The *K*-theory began with the definition of the Grothendieck group for an abelian monoid at the end of the 1950'ss. Subsequently, topological *K*-theory, a version built from vector bundles, was defined. Next, at the start of 1970's, Quillen gave two useful definitions for higher analogs of Grothendieck's construction, respectively. With the work of Witt in the classification of quadratic form over a field, Knebusch and Quillen defined the Witt group of a field and Grothendieck-Witt groups. We will start by explaining the original construction, group completion, as well as its topological counterpart built from vector bundles, ie, the zeroth algebraic and topological *K*-groups. During the talk we will see classic examples in both settings. Furthermore, we will explain how in the topological setting the zeroth *K*-group inherits a ring structure from the direct sum and the tensor product. Next, we will see the original definition of the Witt and Grothendieck-Witt groups for symmetric spaces, being the first groups examples of general Witt groups and the Hermitian *K*-theory. And the end, of the talk we will focus our attention on the ring structure over them.