Gitterpolytope, insbesondere Triangulierungs- und Überdeckungseigenschaften. Beziehungen zu torischen Varietäten und String Theorie

• Applicant: Professor Dr. Christian Haase
• Subject Area: Mathematics
• Term: from 2005 to 2011
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 5440849

• Final Report Year: 2014

Final Report Abstract

During its final funding period the Emmy Noether group “Lattice Polytopes” moved from FU Berlin to Goethe University Frankfurt. The group continued and extended projects it begun during the first funding period such as the study of polytopes admitting unimodular triangulations or the exploration of permutation polytopes. Additionally, new projects were started such as a new collaboration with algebraic geometer Sandra Di Rocco from Stockholm to explore the relation between Ehrhart Theory and the Adjunction Theory of toric varieties. An almost tight Cayley decomposition theorem for polytopes of small effective threshold was proved. This lead ultimately to the resolution of Fujita’s Spectrum Conjecture in the toric case by Andreas Paffenholz. Another example of a new project is the joint work with Gregg Musiker and Josephine Yu developing a theory of linear systems on tropical curves. To a large extent it parallels the classical theory but in some aspects intriguingly diverts from the classical path. Benjamin Nill together with Benjamin Lorenz proved a finiteness theorem for stringy Hodge numbers of nef-partitions. The experimental framework built in previous projects was extended and put to good use. The stringy finiteness result above was first observed experimentally; along the way, previously unknown Hodge numbers for Calabi-Yau three-folds were found. Classification results for smooth polytopes lead to counterexamples in symplectic geometry. Classes of facet defining inequalities for permutation groups could be identified. While many of the originally proposed “entry point” and “capstone” projects have been solved (generation of smooth 3-polytopes, toric ideals of transportation polytopes, reflexive dimension of segments, and a combinatorial proof of the 24-Theorem for reflexive 3-polytopes, adjunction for parameterized surfaces, Ehrhart interpretation of Tuttepolynomials, stabilizer conjecture for dihedral groups, . . . ), completion of the unimodular triangulation survey was delayed several times by additional results. Also, smooth F4-root polytopes still wait to be triangulated unimodularly. Very new results of Haase and Santos on 3-dimensional GL(Z)-scissor’s congruence suggest that the Dehn-like invariant hoped for in the proposal might not exist.

Publications

• On permutation polytopes. Advances in Mathematics 222: 431-452 (2009)
  Barbara Baumeister, Christian Haase, Benjamin Nill and Andreas Paffenholz
  
• Few smooth d-polytopes with N lattice points. Israel Journal of Mathematics
  Tristram Bogart, Christian Haase, Milena Hering, Benjamin Lorenz, Benjamin Nill, Andreas Paffenholz, Günter Rote, Francisco Santos and Hal Schenck
  
• Bounds on the Coefficients of Tension and Flow Polynomials. Journal of Algebraic Combinatorics 33(3), 465-482 (2011)
  Felix Breuer and Aaron Dall
  
• Ehrhart theory, Modular flow reciprocity, and the Tutte polynomial. Mathematische Zeitschrift 270(1), 1-18 (2012)
  Felix Breuer and Raman Sanyal
  (See online at https://doi.org/10.1007/s00209-010-0782-6)
• Linear systems on tropical curves. Mathematische Zeitschrift 270: 1111-1140 (2012)
  Christian Haase, Gregg Musiker and Josephine Yu
  
• Polyhedral adjunction theory. Algebra and Number Theory 7(10): 2417-2446 (2013)
  Sandra Di Rocco, Christian Haase, Benjamin Nill and Andreas Paffenholz