Tropical geometry and cluster theory are new and rapidly developing areas of modern mathematics. They have enjoyed spectacular progress in the last years. The recent appearance of tropical geometry and cluster algebras was motivated by their deep relations to many branches of pure and applied mathematics. These include algebraic geometry, symplectic geometry, mirror symmetry, representation theory, complex analysis, dynamical systems, combinatorics, logic, computer algebra, statistical models, and mathematical biology.
Among the fascinating applications of tropical geometry which are closest to the project, one can mention the proof by G. Mikhalkin of M. Kontsevich’s enumerative conjecture giving rise to new methods of calculation of Gromov-Witten invariants, several important results concerning Welschinger invariants (real analogs of genus zero Gromov-Witten invariants) and enumeration of real rational curves, the works of M. Kontsevich and Y. Soibelman on homological mirror symmetry and Donaldson-Thomas invariants, and the work of R. Kenyon and A. Okounkov on mathematical models for dimers accumulation.
Cluster algebras were invented by S. Fomin and A. Zelevinsky as a tool for the study of Lusztig-Kashiwara’s canonical bases and Lusztig’s theory of total positivity. A first major application to canonical bases was Geiss-Leclerc-Schroer’s construction of large parts of the dual semicanonical basis (Inv. Math. 2006). Of major importance are several unexpected applications, like the discovery of the “generalized associahedra” in combinatorics, and Fomin-Zelevinsky’s proof (Ann. Math. 2003) of Zamolodchikov’s conjecture on the periodicity of a discrete integrable system which arose in conformal field theory in the nineties. The most spectacular application to geometry so far has been the invention of a completely new algebro-geometric approach to Teichmuller theory by V. Fock and A. Goncharov, which naturally lead them to higher Teichmuller theory (Publ. IHES 2003). Kontsevich-Soibelman’s recent discovery of the fact that cluster transformations govern the behaviour of Donaldson-Thomas invariants under wall-crossing is bound to bring further striking applications in algebraic geometry.
The investigations cited above show deep intertwining of tropical geometry and theory of cluster algebras, and indicate the necessity of cooperative efforts for further developments of the both topics.
The ANR-network includes two research groups located at IRMA (Strasbourg) and IMJ (Paris). The postdoctoral position will be held in one of these two institutions. The successful candidate will be strongly encouraged for short and long visits to the other institution.
This is a full research postdoctoral fellowship. The research area for the position is tropical geometry and/or cluster algebras. Candidates are expected to have a good knowledge in one of these fields.
Candidates should send an application including a detailed curriculum vitae, a list of publications and a research statement before December 31, 2010 to the following address
Prof. V. Kharlamov IRMA, Université de Strasbourg 7, rue René-Descartes 67084 Strasbourg Cedex FranceThey should simultaneously send their application as a unique pdf file to the electronic addresses of V. Kharlamov and B. Keller (obtained by concatenating kharlam respectively keller with an ampersand followed by math.u-strasbg.fr respectively math.jussieu.fr). Candidates should also ask for two letters of recommendation to be sent directly to these addresses.
Other postdoctoral grants:
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