Oberwolfach Seminar: Tropical algebraic geometry

October 10th - 16th, 2004
Ilia Itenberg, Université Louis Pasteur, Strasbourg
Gregori Mikhalkin, University of Toronto
Eugenii Shustin, Tel Aviv University
The subject of the tropical algebraic geometry is algebraic varieties defined over the tropical (idempotent) semi-ring of the real numbers equipped with the two operations: "product", defined as the usual sum, and "sum", defined as the maximum. The tropical geometry is directly related to many branches of mathematics such as algebraic geometry, symplectic geometry, topology, complex analysis, dynamical systems, and combinatorics. The interest to this area has grown over the last years due to the recently found, fascinating applications of the tropical geometry to the classical real and complex enumerative problems, and, in particular, to computation of Gromov-Witten invariants. The key notion of the theory is an amoeba of an algebraic variety, i.e., the image of the algebraic variety by a coordinate-wise valuation map. We introduce tropical varieties as amoebas of algebraic varieties over fields with a real non-Archimedean valuation, describe their geometry and their relation to amoebas of complex algebraic varieties and to combinatorics of Newton polytopes. The main link between the tropical and complex (or real) algebraic varieties is based on a presentation of tropical varieties as limits of the algebraic ones, and, on the other side, algebraic varieties as deformations of the tropical ones. We discuss this link both from the symplectic and algebraic-geometric point of view. As an application, we consider the enumeration of nodal curves on toric surfaces associated with convex lattice polygons, the tropical count of the Welschinger invariant and enumeration of real rational curves, as well as the tropical aspects of the Viro patchworking construction.
A basic knowledge in algebraic geometry, symplectic geometry, and toric varieties would be helpful for a better understanding of techniques used in applications.
I. Itenberg: Amibes des variétés algébriques et dénombrement de courbes [d'après G. Mikhalkin]. Séminaire N. Bourbaki 2002/03, exp. 921, Juin 2003.

I. Itenberg, V. Kharlamov, and E. Shustin: Welschinger invariant and enumeration of real rational curves. Int. Math. Res. Notices 49 (2003), 2639--2653.

G. Mikhalkin: Counting curves via the lattice paths in polygons. C. R. Acad. Sci. Paris, Sér. I, 336 (2003), no. 8, 629--634.

G. Mikhalkin: Enumerative tropical algebraic geometry. Preprint arXiv:math.AG/0312530.

G. Mikhalkin: Real algebraic curves, the moment map and amoebas. Ann. of Math. (2) 151 (2000), no. 1, 309--326.

G.Mikhalkin: Amoebas of algebraic varieties. Preprint arXiv:math.AG/0108225.

J. Richter-Gebert, B. Sturmfels, and T. Theobald: First steps in tropical geometry. Preprint arXiv:math.AG/0306366.

E. Shustin: Patchworking singular algebraic curves, non-Archimedean amoebas and enumerative geometry. Preprint arXiv:math.AG/0211278.

O. Viro: Dequantization of real algebraic geometry on logarithmic paper. European Congress of Mathematics, Vol. I (Barcelona, 2000), Progr. Math., 201, Birkhäuser, Basel, 2001, pp. 135--146.

Deadline for applications:
August 27, 2004
The seminars take place at the Mathematisches Forschungsinstitut Oberwolfach. The number of participants is restricted to 24. Applications including

should be sent as hard copy or by e-mail (.ps or .pdf file) to:

Prof. Dr. Gert-Martin Greuel
Universität Kaiserslautern
Fachbereich Mathematik
Erwin Schrödingerstr.
67663 Kaiserslautern, Germany

Mathematisches Forschungsinstitut Oberwolfach   updated: June 4th, 2004