
Combinatorial Algebraic Topology (CALTOP)
Collaborators:
Dmitry FeichtnerKozlov (U Bremen, PI),
JeanMarie Droz (U Bremen, SNFpostdoc),
Roman Bruckner (U Bremen),
Ralf Donau (U Bremen), Gerrit Grenzebach (U Bremen)
Funding:
DFG, SNF
Combinatorial algebraic topology is a fascinating and dynamic field at
the crossroads of algebraic topology and discrete mathematics. The
subject of Combinatorial Algebraic Topology is in a certain sense a
classical one, as modern Algebraic Topology derives its roots from
dealing with various combinatorially defined complexes and with
combinatorial operations on them. Yet, the aspects of the theory which
we consider in our research group, and which we distinguish under our
title are far from classical and have been brought to the attention of
the general mathematical public fairly recently.
Discrete Structures in Algebra and Geometry (DSAG)
Collaborators:
EvaMaria Feichtner (U Bremen, PI),
Emanuele Delucchi (U Bremen), Giacomo d'Antonio (Uni Bremen)
Sergey Yuzvinsky (U Oregon),
Hal Schenck (UrbanaChampain),
Mike Falk (U Northern Arizona).
Funding:
DFG
Discrete structure often proves to lie at the heart of geometric
or topological matters. The theory of arrangements of hyperplanes
is a prominent example. Extracting the "right" data from a geometric
situation leads to concise and beautiful descriptions of topological
invariants. The study of this discrete core data for its own sake is
a challenging chapter of geometric combinatorics.
Topological Data Analysis (CALTOP/DSAG)
Collaborators:
Dmitry FeichtnerKozlov (U Bremen, PI), EvaMaria Feichtner (U Bremen),
Herbert Edelsbrunner (ISTA), Gunnar Carlsson (Stanford),
Dmitry Morozov (Lawrence Labs, Berkeley)
Funding:
ESF
Topological data analysis is an emerging field on the borderline
between the classical algebraic topology and data mining. The main
analysis tool is the persistence homology which allows to use
algebraic invariants to measure the significance of qualitative
features of the data sets. ALTA is active in organizing cutting edge
research conferences on this topic at international research centers (Fields
Institute, Banff Research Station). ALTA is also a node in the European
Science Foundation network "Applied Algebraic Topology".
Tropical Geometry (DSAG)
Collaborators:
EvaMaria Feichtner (U Bremen, PI),
Zur Izhakian (U Bremen, Humboldt Postdoctoral Fellow),
Tim Haga (U Bremen), Martin Dlugosch (U Bremen)
Louis Rowen (BarIlan University),
Hannah Markwig (U Saarbruecken),
Jan Draisma (TU Eindhoven),
Bart Frenk (TU Eindhoven)
Funding:
Alexander von Humboldt Foundation,
Netherlands Organisation for Scientific Research (NWO)
Tropical Geometry is an emerging field of mathematics on the crossroads
of algebra, analysis, combinatorics, geometry, topology and applications.
Algebraic varieties are replaced by polyhedral objects that retain much
of the information of the original variety. Hence, a completely new toolbox
is created for longstanding problems in algebraic geometry.
Topological methods in Distributing Computing (CALTOP)
Collaborators:
Dmitry FeichtnerKozlov (U Bremen, PI),
Maurice Herlihy (Computer Science, Brown U, U.S.A.),
Sergio Rajsbaum (Computer Science, UNAM, Mexico)
Funding:
Elsevier
There has recently been a lot of activity applying the methods
of algebraic topology to the theoretical distributed computing. We are
working on studying the emerging mathematical models. One goal of this
heavily interdisciplinary project is to write a textbook on this subject.
ALTA is also involved in organizing international conferences
(such as at Schloss Dagstuhl) on this topic.
Algebrogeometric Methods in Mathematical Physics
Collaborators:
EvaMaria Feichtner, Claus Lämmerzahl (ZARM, U Bremen),
Victor Enolski (Kiew), Emma Previato (Boston U)
Funding:
DFG
The field of Tropical Geometry has recently produced dicretegeometric
analogues of some of the central theorems of Algebraic Geometry, e.g.,
RiemannRoch theorem, AbelJacobi inversion. The project envisions an
integration of the tropical language into the treatment of integrable
systems as they appear in General Relativity, nonAbelian gauge theory
and nonlinear quantum theory. ALTA is an associated partner
of the DFG graduate school "Models of Gravity," Bremen/Oldenburg.
