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Report 5/2006: Convex and Algebraic Geometry (January 29th – February 4th, 2006). (English) Zbl 1110.14300

Abstract: The subjects of convex and algebraic geometry meet primarily in the theory of toric varieties. Toric geometry is the part of algebraic geometry where all maps are given by monomials in suitable coordinates, and all equations are binomial. The combinatorics of the exponents of monomials and binomials is sufficient to embed the geometry of lattice polytopes in algebraic geometry. Recent developments in toric geometry that were discussed during the workshop include applications to mirror symmetry, motivic integration and hypergeometric systems of PDE’s, as well as deformations of (unions of) toric varieties and relations to tropical geometry.
Contributions:
Jürgen Hausen (joint with Klaus Altmann), Polyhedral divisors and algebraic torus actions (p. 259)
Robert Vollmert, Codimension one torus actions (p. 261)
Hendrik Süß (joint with Klaus Altmann and Jürgen Hausen), Glueing polyhedral divisors (p. 262)
Diane Maclagan (joint with Alastair Craw, Rekha R. Thomas), Moduli of representations of the McKay quiver (p. 264)
Alastair Craw (joint with Gregory G. Smith), Toric varieties are fine moduli spaces (p. 266)
Bernd Siebert (joint with Mark Gross), Toward a new, “tropical” construction of Calabi-Yau varieties (p. 269)
Ilia Itenberg (joint with Viatcheslav Kharlamov and Eugenii Shustin), Tropical enumeration of real rational curves (p. 271)
Eugenii Shustin (joint with Ilia Itenberg and Viatcheslav Kharlamov), Recursive formulas for tropical and algebraic Welschinger invariants (p. 273)
Alexander Klyachko, Quantum marginal problem, flag varieties, and representations of the symmetric group (p. 275)
Jan Stienstra, Crystals, quivers and dessins d’enfants (p. 279)
Tom Braden (joint with Nicholas Proudfoot), Intersection cohomology of hypertoric varieties and Gale duality (p. 281)
Alexey Bondal, Derived categories of toric varieties (p. 284)
R. Paul Horja (joint with Lev A. Borisov), The \(K\)-theory of toric Deligne-Mumford stacks and mirror symmetry (p. 286)
Christian Haase (joint with Andreas Paffenholz), Quadratic Gröbner bases for smooth \(3 \times 3\) transportation polytopes (p. 288)
Winfried Bruns (joint with Tim Römer), On the unimodality of \(h\)-vectors (p. 290)
Boris Pasquier, Fano horospherical varieties (p. 292)
Kazushi Ueda, Homological mirror symmetry and McKay correspondence (p. 294)
Markus Perling (joint with Lutz Hille), Searching for strongly exceptional sequences of line bundles on toric varieties (p. 296)
Gavin Brown (joint with Miles Reid), Diptych varieties and semistable Mori flips (p. 298)
Pedro Daniel González Pérez (joint with Helena Cobo Pablos), Arcs and jets on toric singularities and quasi-ordinary singularities (p. 302)
András Szenes (joint with Gergely Bérczi), Thom polynomials of singularities (p. 305)
Ezra Miller (joint with Laura Felicia Matusevich and Uli Walther), Homological methods for hypergeometric families (p. 307)
Milena Hering (joint with Hal Schenck, Greg Smith), Syzygies of toric varieties (p. 310)

MSC:

14-06 Proceedings, conferences, collections, etc. pertaining to algebraic geometry
52-06 Proceedings, conferences, collections, etc. pertaining to convex and discrete geometry
00B05 Collections of abstracts of lectures
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
52B20 Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry)
14J32 Calabi-Yau manifolds (algebro-geometric aspects)
14D06 Fibrations, degenerations in algebraic geometry
14L30 Group actions on varieties or schemes (quotients)
14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
33C70 Other hypergeometric functions and integrals in several variables
14J45 Fano varieties
16G20 Representations of quivers and partially ordered sets

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