FB 6 Mathematik/Informatik

Institut für Mathematik

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Wintersemester 2014/15

04.11.2014 um 16:15 Uhr in 69/125:

Dr. Marc Goerigk (Universität Kaiserslautern)

Challenges in Robust Optimization

Robust optimization has become one of the most frequent topics at operations research conferences around the world. Its approach is tempting: A huge choice of alternative formulations are available, usually with only small data requirements. We begin this talk with an overview on current formulations and ideas for robust optimization. Starting from there, we highlight some problems and challenges we encountered: Is robustness always the right approach? How good is my robust solution, actually? And which approach should I choose? While the answers to such questions are not always clear, we would like to create awareness that they are crucial for the future development of the field.

18.11.2014 um 16:15 Uhr in 69/125:

Richard Sieg (Universität Osnabrück)

Stanley depth of the upperhalf of the Kozul complex

Let R = K[X1, ... , Xn] be a polynomial ring over some field K. The Stanley depth of an R-module M is a combinatorial invariant coming from certain decompositions. We show that the k-th syzygy module of the residue class field K of R has Stanley depth n-1 for ⌊n/2⌋ ≤ k < n, as it had been conjectured by Bruns et. al. in 2010. In particular, this gives the Stanley depth for a whole family of modules whose graded components have dimension greater than 1. So far, the Stanley depth is known only for a few examples of this type. Our proof consists in a close analysis of a matching in the Boolean algebra.

18.11.2014 um 17:30 Uhr in 69/125:

Dr. Javier J. Gutiérrez (Radboud Universiteit Nijmegen)

On autoequivalences of the (∞,1)-category of ∞-operads

Higher operad theory can be formalized by means of different approaches, including simplicial operads, ∞-operads (Lurie), dendroidal sets (Moerdijk--Weiss) and complete dendroidal Segal spaces (Cisinski--Moerdijk). Each of these theories is organized in a Quillen model category and these are connected by Quillen equivalences.By using techniques introduced by Toën, Lurie and Barwick and Schommer-Pries, we show that the (∞,1)-category of autoequivalences of the (∞,1)-category of ∞-operads is a contractible ∞-groupoid. More precisely, we prove that the quasi-category of autoequivalences of Ω-spaces is a contractible Kan complex. This implies that if there is a way to compare two models for ∞-operads, then this can be done in an essentially unique way. Similarly, we show that the (∞,1)-category of autoequivalences of the (∞,1)-category of non-symmetric ∞-operads is the discrete category on the cyclic group ℤ/2ℤ of order two, the non-trivial element being the ``mirror autoequivalence''. Our calculations are based on the model of complete dendroidal Segal spaces introduced by Cisinski--Moerdijk.

02.12.2014 um 16:15 Uhr in 69/125:

Dr. Sean Tilson (Universität Osnabrück)

Homotopy theory and brave new algebra

Homotopy theorists try to gain geometric information and insight through the use of algebraic invariants. Specifically, these invariants are useful in determining whether or not two spaces can be equivalent. We will begin with an example to demonstrate the usefulness of cohomology and some o the extra structure it possesses (cup products and Steenrod squares). This extra structure provides a very strong invariant of the space. As these invariants are representable functors, this extra structure is coming from the representing object. Indeed, cohomology theories possess products and power operations (lifts of the Frobenius) when they are represented by objects called commutative ring spectra. We then shift focus to studying commutative ring spectra on their own and try to detect what maps of commutative ring spectra might look like.

09.12.2014 um 16:15 Uhr in 69/125:

Dr. Emanuele Delucchi (Universität Fribourg, Schweiz)

Toric arrangements – towards a comprehensive combinatorial theory

Recent work of De Concini, Procesi and Vergne on vector partition functions gave a new impulse to the study of toric arrangements from an algebraic, topological and combinatorial point of view.
In this context, many new discrete structures have appeared in the literature, each describing some aspect of the theory (i.e., either the arithmetic-algebraic one or the topological one) and trying to mirror the combinatorial framework which revolves around arrangements of hyperplanes.
I will give a quick overview of the state of the art and, taking inspiration from some recent results of topological flavor such as the computation of the cohomology algebra of a toric arrangement's complement ("toric Orlik-Solomon algebra"), I will try to outline a possible approach towards unifying these different objects.

16.12.2014 um 16:15 Uhr in 69/125:

Gregor Hendel (Zuse Institut Berlin)

Solving Constraint Integer Programs with SCIP

Constraint Integer Programming (CIP) denotes the minimization of a linear objective function under side constraints and integrality restrictions for (a subset of) the problem variables. An important subclass of CIP are Mixed Integer Programs (MIPs) which involve only linear side constraints. Many combinatorial optimization problems can be formulated as MIPs. One of the most famous combinatorial optimization problems is the Traveling Salesman Problem, which requires to find a tour with minimum cost through a complete graph. SCIP is a software framework for Solving Constraint Integer Programs and also one of the fastest MIP solvers available in source code.
The aim of this talk is to give a detailed introduction to SCIP and its extendability for custom projects falling in the area of Mixed Integer Programming and beyond. First, I will present an outline of the branch-and-cut procedure for solving MIPs to proven optimality. I will then give examples and background knowledge of existing solving techniques such as branching variable selection, cutting plane algorithms, preprocessing techniques and primal heuristic algorithms. At the example of the Traveling Salesman Problem, I will explain how to extend the SCIP framework to specific optimization problems.

16.12.2014 um 17:30 Uhr in 69/125:

Dr. Gharchia Abdellaoui (Universität Osnabrück)

Irreducibility of certain moduli spaces of framed sheaves

The moduli space of framed sheaves on projectives surfaces was _rst constructed by Huybrechts and Lehn. The smoothness of this moduli space is shown under certain assumptions but its irreducibility is still not known. In this talk, I will introduce the notion of framed sheaves on projective surfaces and their moduli spaces. Afterwards, I will show the irreducibility of this moduli space for a certain class of toric surfaces.

20.01.2015 um 16:15 Uhr in 69/125:

Sascha Bachmann (Universität Osnabrück)

Concentration Inequalities for Poisson U-Statistics

In recent years, the investigation of U-Statistics associated to Poisson point processes has been an active area of research. In this talk, concentration inequalities for certain Poisson U-Statistics will be presented. In particular, I will focus on edge counts in random geometric graphs. In these graphs, the vertices are given by a Poisson point process and any two vertices are connected by an edge whenever their distance does not exceed some fixed positive real number. The talk is based on joint work with Giovanni Peccati.