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Wintersemester 2015/2016

27.10.2015 um 16:15 Uhr in 69/125:

Prof. Dr. Tim Römer (Universität Osnabrück)

Commutative Algebra up to Symmetry

Ideal theory over a polynomial ring in infinitely many variables is rather complicated which is (beside other things) due to the fact that the ring is not Noetherian. Especially motivated by results from algebraic statistics one is interested in ideals in such a ring which are invariant under certain well-behaved group and monoid actions, respectively. We present some recent results and open questions on algebraic properties of these ideals and associated objects of interest. The talk is based on joint work with Uwe Nagel.

17.11.2015 um 16:15 Uhr in 69/125:

Thi Van Anh Nguyen (Universität Osnabrück)

On partial regularities

Castelnouvo-Mumford regularity is one of the most important invariants in commutative algebra. As one of the refinements for the Castelnouvo-Mumford regularity, Trung introduced so-called partial regularities and proved many interesting properties of these new invariants. We present an overview on this topic and some new results as well as related open questions.

01.12.2015 um 16:15 Uhr in 69/125:

Jonathan Steinbuch (Universität Osnabrück)

Continuous closure of ideals and sheaves

Continuous closure of an ideal generated by polynomials is the restriction to the polynomial ring of the ideal generated by said polynomials in the ring of continuous functions. A dominating aspect in research on this has been the quest for a description in purely algebraic terms, i.e. without the concept of continuity. We present part of Brenner's work on continuous closure of primary and monomial ideals, his attempt at algebraic description with a concept called axes closure, Epstein and Hochster's counterexample to that approach and finally Kollár's algebraic description via the theory of sheaves.

05.01.2016 um 16:15 Uhr in 69/125:

Prof. Dr. Wojciech Kucharz (Jagiellonian University, Krakau, z.Zt. Max-Planck-Institut, Bonn)

Linear equations on real algebraic surfaces

I will discuss the notion of continuous closure of ideals from the perspective of real algebraic geometry. We will consider a linear equation whose coefficients are continuous rational functions on a nonsingular real algebraic surface. If such an equation has a continuous solution, then it also has a continuous rational solution. This is known to fail in higher dimensions.

26.01.2016 um 16:15 Uhr in 69/125:

Dr. Kristian Moi (Universität Münster)

K-theory and hermitian forms

The Grothendieck group of a ring A is classical invariant which captures many interesting properties of the A and its modules. When A has an involution one has a lesser known invariant called the Grothendieck-Witt group which is defined as the group completion of non-degenerate hermitian forms over A with respect to "orthogonal sum". Each of these invariants is the 0-th group in a sequence of abelian groups which have important applications in algebra and topology. Recently Hesselholt and Madsen have defined real algebraic K-theory which is a generalization of the two theories with a very rich algebraic structure. I will first give an introduction to hermitian forms and real algebraic K-theory and then discuss some open questions regarding localization and the relation to Ranicki's L-theory.

02.02.2016 um 16:15 Uhr in 69/125:

Dr. Martin Frankland (Universität Osnabrück)

Introduction to André-Quillen cohomology

In the late 1960s, M. André and D. Quillen independently introduced a (co)homology theory for commutative rings, using methods of homotopy theory. In this expository talk, I will present the basics of this theory, and more generally of simplicial methods and homotopical algebra. I will discuss some examples and applications.