Prof. Dr. Eva Maria Feichtner
Prof. Dr. Dmitry Feichtner-Kozlov

Oberseminar Algebra, Geometrie und Topologie

Tuesday, 2:15 pm
MZH 7050


Please check shortly for the program of the upcoming term.

04.05.10 Dmitry Feichtner-Kozlov:
Stellar Equipartitions
11.05.10 no seminar - instead, please consider to attend
The North German Plain Goes Tropical
18.05.10 Giacomo d'Antonio:
TBA

Last terms' schedules

14.04.08 Eva Maria Feichtner:
A combinatorial introduction to tropical geometry I
The core undertaking of tropical geometry is to transform algebro-geometric objects into piecewise linear ones, while retaining much of the original algebraic information. This, in particular, opens problems on algebraic varieties to a completely new set of techniques from the discrete geometric realm. Tropical geometry draws from the results of geometric combinatorics and, at the same time, exhibits a wealth of interesting structures to be explored from the discrete geometric viewpoint.
This is the beginning of a series of (at least) three lectures, where we will provide an introduction to tropical geometry from the combinatorial point of view. We will keep the exposition self-contained by introducing the necessary concepts from geometric combinatorics as they cross our way.
21.04.08 Martin Stolz (U Bonn):
Unitäre Spektren
Die stabile Homotopiekategorie bildet die kanonische Plattform um Phänomene der stabilen Homotopietheorie zu untersuchen. Sowohl für die Verbindung zur instabilen Theorie, als auch für fortgeschrittene Problemstellungen ist ein hochstrukturiertes Modell für diese Kategorie überaus hilfreich. Die Kategorie der Unitären Spektren ist solch ein Modell, das zusätzlich zu den guten Eigenschaften klassischerer Ansätze insbesondere für komplexe Strukturen einfachere Einbettungen ermöglicht. Wichtigstes Beispiel hierfür ist das komplexe Kobordismusspektrum MU.
25.04.08 Danijela Vasiljevic (Faculty of Mathematics, Belgrade, Serbia): Observe unusual time slot: Fr 12:15 !
Computational Homology of Simplicial Complexes
Simplicial complexes (which are constructed from graphs) are used to study complex networks. We show that this approach has many advantages in comparison with the standard graph approach. Topological invariants of simplicial complexes give new and important information about the networks structure and functionality. We present recent results from the field of persistent homology and relate them to the graph complexes. Apart from the mathematical investigation we study the algorithms for homology computation and compare them via different parameters. We give a brief discussion about existing software for homology computation.
28.04.08 Anders Jensen (TU Berlin):
Computing tropical varieties
The tropicalization of an algebraic variety defined by a polynomial ideal I, also called the tropical variety of I, plays an important role in tropical geometry. One way to compute it is to consider it as a polyhedral subcomplex of the Gröbner fan of I. The maximal cones in the Gröbner fan of I index the reduced Gröbner bases of I and may be computed by applying Gröbner basis conversion techniques. Afterwards one can pick out just those cones of the Gröbner fan that are in the tropical variety of I. In this talk we show how the method can be refined by applying a connectivity result for tropical varieties of prime ideals and an algorithm for constructing tropical bases of curves. The presented algorithms have been implemented in the software package Gfan.
This is joint work with T. Bogart, K. Fukuda, D. Speyer, B. Sturmfels and R. Thomas
05.05.08 Jorik Mandemaker (Radboud University Nijmegen):
Racks and quandles and their applications
Quandles are an algebraic structure coming from taking a group and throwing away the multiplication but retaining the conjugation information. Quandles also appear in a natural way in the study of knots. Quandles and the more general racks can be used in various ways to define invariants of knots, surface knots and higher-dimensional equivalents.
12.05.08 no seminar: Pentecost
19.05.08 Doris Hein (U Dortmund):
The Conley conjecture - a proof following V. Ginzburg
26.05.08 Juliane Lehmann:
Nested set complexes of posets
The notion of nested set complexes of lattices can be generalized to posets in the following way: Let P be a bounded poset with building set G; that is, for each element x of P the interval [0,x] is isomorphic to the product of the intervals [0,x_1],...,[0,x_t], where x_1,...,x_t are the maximal elements of the intersection of G and [0,x]. Then a finite subset N of G is nested in G, if the join of each incomparable subset A of G with |A|>1 exists and is not in G. The nested sets in G form an abstract simplicial complex, coinciding with the (reduced) order complex of P if G is the maximal building set. Stepwise extending the building set induces topological changes in the nested set complex that turn out to be either subdivisions or conings. If P is a lattice, only subdivisions occur; but the conditions can also be met by non-lattices, which allows applications like a structural proof of results on the Bier construction or on the complex of k-trees.
09.06.08 - no seminar - due to
Bremen Lectures in Mathematics, Sergey Yuzvinsky (U Oregon, USA), June 10& 11, 2008.
07.07.08 - no seminar - due to
Mathematics Colloquium, Sefi Ladkani (U Oregon), July 8, 2008.
Joint Mathematics Colloquium (UB/IUB), Bernd Sturmfels (UC Berkeley/TU Berlin), July 10, 2008.

Have a great summer break !



28.07.08 Tim Haga (U Bremen):
Die Schälung eines Polytops
Ein polytopaler Komplex C ist eine Menge von Polytopen für die gilt, dass der Schnitt zweier Polytope entweder die leere Menge oder eine Seite jedes der beiden Polytope ist. Die Polytope aus C, die nicht in Polytopen höheren Dimensionen enthalten sind, heißen Facetten. Dabei heißt ein Komplex rein, wenn alle Facetten die gleiche Dimension d besitzen. Nummeriert man die Facetten, so heißt diese Nummerierung eine Schälung, wenn folgende Bedingung erfüllt ist: Der Komplex C ist 0-dimensional, oder für jede Facette F (mit Ausnahme der Ersten) ist der Schnitt mit den Vorhergehenden ein reiner, schälbarer Komplex der Dimension d-1. Wir zeigen in diesem Vortrag, daß Polytope immer schälbar sind. Die Schälbarkeit von Polytopen spielt eine wichtige Rolle beim Beweis des Upper Bound Theorem für Polytope.
Dies ist der Abschlussvortrag zur Bachelorarbeit von Tim Haga.
04.08.08 Ralf Donau (U Bremen):
Finite topological spaces (following J.P. May)
Every finite T_1 space is discrete, but finite T_0 spaces for example, which correspond to finite posets, are very interesting. We will see that any finite topological space is homotopy equivalent to a T_0 space. So we can study finite spaces up to homotopy equivalence by studyinh finite T_0 spaces up to homotopy equivalence.
11.08.08 Björn Walker (U Bremen):
Finite spaces and simplicial complexes (following J.P. May)
Finite T_0 spaces and simplicial complexes have a close relationship in terms of homotopy theory. There is a weak homotopy equivalence between a finite T_0 space and its order complex, as well as between a finite simplicial complex and its face poset. A modified version of the simplicial approximation theorem can be proofed with this. That is, every continuous map between order complexes of finite T_0 spaces can be approximated (which means up to homotopy) by a continuous map of finite spaces
28.10.08 Frederik von Heymann (FU Berlin):
Lattice points below rational lines in IR^2
Barvinok's Algorithm is a very effective tool to count the lattice points of a polytope. It heavily uses generating functions and also the fundamental parallelepipeds of rational cones.
After a short introduction to the basic concepts we will investigate an effective method to describe the lattice points of these parallelepipeds. In the two-dimensional case, we achieve this by giving a recursion for the lattice points "directly" below an arbitrary rational line. Also, we will talk about number-theoretic aspects of the problem and possible generalisations to higher dimensions.
04.11.08 Christina Krause (U Oldenburg):
Towards minimal tropical bases for regular matroids
It is unknown to date whether regular matroids have a unique minimal tropical basis. Yu and Yuster ("Representing tropical linear spaces by matroids", arXivmath/0611579v1) outline two ways to describe tropical linear spaces in terms of their circuits. While a matroid is associated to a tropical linear space it may be possible to attack the problem from two sides by comparing these different representations. If one can find out what consequences result on the circuits from taking minors or composition that might be a first step towards the construction of minimal tropical bases of regular matroids.
This is a project report. We will provide the necessary background as we go along.
11.11.08 no seminar due to hiring talks

18.11.08 no seminar due to hiring talks

25.11.08 Martin Raussen (U Alborg, Denmark):
Colloquium Pre-talk: The Vietoris-Begle Theorem
06.01.09 Emanuele Delucchi (SUNY Binghamton, USA):
Colloquium Pre-talk: Salvetti complex and friends
19.01.09 Jan Draisma (TU Eindhoven, Netherlands):
Colloquium Pre-talk: Finite generation of symmetric ideals
18.02.09 Gerrit Grenzebach: Observe unusual time slot: Wed 12:00 !
Der Satz von Ado für Super-Lie-Algebren


Kontakt:

Eva-Maria Feichtner
eMail: emf@math.uni-bremen.de
Tel: 0421 218 63691

Dmitry Feichtner-Kozlov
eMail: dfk@math.uni-bremen.de
Tel: 0421 218 63681


last updated: April 30, 2010.