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Combinatorial Algebraic Topology (CALTOP)

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Collaborators:
Dmitry Feichtner-Kozlov (U Bremen, PI), Jean-Marie Droz (U Bremen, SNF-postdoc), 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)

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Collaborators:
Eva-Maria Feichtner (U Bremen, PI), Emanuele Delucchi (U Bremen), Giacomo d'Antonio (Uni Bremen)
Sergey Yuzvinsky (U Oregon), Hal Schenck (Urbana-Champain), 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)

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Collaborators:
Dmitry Feichtner-Kozlov (U Bremen, PI), Eva-Maria 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)

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Collaborators:
Eva-Maria Feichtner (U Bremen, PI), Zur Izhakian (U Bremen, Humboldt Postdoctoral Fellow), Tim Haga (U Bremen), Martin Dlugosch (U Bremen)
Louis Rowen (Bar-Ilan 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)

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Collaborators:
Dmitry Feichtner-Kozlov (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.




Algebro-geometric Methods in Mathematical Physics

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Collaborators:
Eva-Maria Feichtner, Claus Lämmerzahl (ZARM, U Bremen),
Victor Enolski (Kiew), Emma Previato (Boston U)

Funding:
DFG
The field of Tropical Geometry has recently produced dicrete-geometric analogues of some of the central theorems of Algebraic Geometry, e.g., Riemann-Roch theorem, Abel-Jacobi inversion. The project envisions an integration of the tropical language into the treatment of integrable systems as they appear in General Relativity, non-Abelian gauge theory and non-linear quantum theory. ALTA is an associated partner of the DFG graduate school "Models of Gravity," Bremen/Oldenburg.





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