| 14.04.08 |
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Eva Maria Feichtner:
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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.
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| 21.04.08 |
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Martin Stolz (U Bonn): |
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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.
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| 25.04.08 |
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Danijela Vasiljevic (Faculty of Mathematics, Belgrade, Serbia):
Observe unusual time slot: Fr 12:15 ! |
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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.
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| 28.04.08 |
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Anders Jensen (TU Berlin): |
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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
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| 05.05.08 |
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Jorik Mandemaker (Radboud University Nijmegen): |
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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.
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| 12.05.08 |
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no seminar: Pentecost |
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| 19.05.08 |
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Doris Hein (U Dortmund): |
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The Conley conjecture - a proof following V. Ginzburg
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| 26.05.08 |
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Juliane Lehmann: |
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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.
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| 09.06.08 |
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- no seminar - due to |
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Bremen Lectures in Mathematics,
Sergey Yuzvinsky (U Oregon, USA), June 10& 11, 2008.
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| 07.07.08 |
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- no seminar - due to |
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Mathematics Colloquium,
Sefi Ladkani (U Oregon), July 8, 2008.
Joint Mathematics Colloquium (UB/IUB),
Bernd Sturmfels (UC Berkeley/TU Berlin), July 10, 2008.
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| 28.07.08 |
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Tim Haga (U Bremen):
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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.
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| 04.08.08 |
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Ralf Donau (U Bremen):
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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.
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| 11.08.08 |
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Björn Walker (U Bremen):
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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
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| 28.10.08 |
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Frederik von Heymann (FU Berlin):
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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.
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| 04.11.08 |
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Christina Krause (U Oldenburg): |
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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.
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| 11.11.08 |
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no seminar due to hiring talks |
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| 18.11.08 |
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no seminar due to hiring talks |
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| 25.11.08 |
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Martin Raussen (U Alborg, Denmark): |
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Colloquium Pre-talk: The Vietoris-Begle Theorem
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| 06.01.09 |
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Emanuele Delucchi (SUNY Binghamton, USA): |
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Colloquium Pre-talk: Salvetti complex and friends
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| 19.01.09 |
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Jan Draisma (TU Eindhoven, Netherlands): |
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Colloquium Pre-talk: Finite generation of symmetric ideals
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| 18.02.09 |
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Gerrit Grenzebach:
Observe unusual time slot: Wed 12:00 ! |
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Der Satz von Ado für Super-Lie-Algebren
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