GRK 1821:  Cohomological Methods in Geometry

• Subject Area: Mathematics
• Term: from 2012 to 2021
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 201167725

• Final Report Year: 2022

Final Report Abstract

The research program is rooted in geometry. The research projects cover a wide range of subjects from mathematical physics to number theory, however, the methods used to study these distinct sets of problems are closely related. The most visible unifying technique used in all our projects is cohomology, a versatile tool central to all geometric disciplines, posed to gain yet more importance in years to come. Nearly all of our projects use Hodge theory, Dirac operators, deformation theory, Lie groups or algebraic geometry. This leads to synergies, of which we are taking advantage. The interplay between abstract algebra and concrete geometry is a Leitmotiv of our research. The participating researchers work in different branches of geometry. Yet, they find a common theme in the methods they use. This set-up is ideal for the training of doctoral researchers. They, as well as our postdocs and the advisers themselves profit immensely from the interactions between neighboring fields.

Publications

• Computing the Tutte polynomial of a matroid from its lattice of cyclic flats. Electron. J. Combin. 21 (2014) no. 3, Paper 3.47
  Jens Eberhardt
  
• Analytical index and eta forms for Dirac operators with onedimensional kernel over a hypersurface, 2015
  Anja Wittmann
  
• The Harder-Narasimhan filtrations and rational contractions. Freiburg im Breisgau: Univ. Freiburg, Fakultat fur Mathematik und Physik (Diss.). ii, 55 p. (2015)
  Thiam-Sùn Pang
  
• Unicity of graded covers of the category O of Bernstein-Gelfand-Gelfand, Arkiv for Matematik, 53 No. 2 (2015) 383-400
  Michael Rottmaier and Wolfgang Soergel
  
• Comparison of the categories of motives defined by Voevodsky and Nori, Dissertation Freiburg 2016, Freidok plus (2016)
  Daniel Harrer
  
• Eta-forms and adiabatic limits for fibrewise Dirac operators with varying kernel dimension, Freiburg im Breisgau: Univ. Freiburg, Fakultät fur Mathematik und Physik (2016)
  Anja Wittmann
  
• Hitchin and calabi-yau integrable systems, Freidok plus (2016)
  Florian Beck
  
• Homotopy theory for rigid analytic varieties, Freidok plus (2016)
  Helene Sigloch
  
• Koszul-Dualitat von komplexen Gruppen, Freidokplus (2016)
  Michael Rottmaier
  
• Low degree Hodge theory for klt varieties
  Martin Schwald
  
• The chiral de Rham complex of tori and orbifolds, Freidok plus (2016)
  Felix Grimm
  
• The e-Invariant and Transfer Map, Ph.D. Thesis
  Yi-Sheng Wang
  
• An approximation of the e-invariant in the stable homotopy category
  Yi-Sheng Wang
  
• Categories of vector spaces and Grassmannians
  Yi-Sheng Wang
  
• Fat realization and Segal’s classifying space
  Yi-Sheng Wang
  
• Hodge theoretic methods in the study of symplectic varieties. Freiburg im Breisgau: Univ. Freiburg, Fakultät für Mathematik und Physik (2017)
  Martin Schwald
  
• Homotopy Classification of Line Bundles Over Rigid Analytic Varieties, 2017
  Helene Sigloch
  
• Coherent Sheaves on Calabi-Yau manifolds, Picard-Fuchs equations and potential functions, Feburary 2018, Freiburg
  Natalie Peternell
  
• Cohomological descent for logarithmic differential forms in the log etale topology. Freiburg im Breisgau: Univ. Freiburg, Fakultat fur Mathematik und Physik (Diss.). 100 p. (2018)
  Elmiro Vetere
  
• Geometric realization and its variants
  Yi-Sheng Wang
  
• Graded and geometric parabolic induction for category O, (2018), Freidok plus (2017)
  Jens Eberhardt
  
• On absolute linear Harbourne constants, Finite Fields Appl. 51 (2018), 371-387
  Marcin Dumnicki, Daniel Harrer, Justyna Szpond
  
• On the motivic Tamagawa number of number fields, Dissertation 2018, Freidok May 2018
  Maximilian Schmidtke
  
• Aspects of Calabi-Yau integrable and Hitchin systems, SIGMA Symmetry Integrability Geom. Methods Appl. 15 (2019), Paper No. 001
  Florian Beck
  
• Calabi-Yau orbifolds over Hitchin bases, J. Geom. Phys. 136 (2019), 14-30
  Florian Beck
  
• Gradings and Soergel bimodules, Freiburg im Breisgau, Fakultat fur Mathematik und Physik (2019)
  Benjamin McDonnell
  
• K-Motives and Koszul Duality, (2019)
  Jens Eberhardt
  
• n invariants under degeneration to cone-edge singularities, Freiburg im Breisgau: Univ. Freiburg, Fakultät für Mathematik und Physik (2019)
  Nelvis Fornasin
  
• Stable geodesics on a K3 surface, Freiburg im Breisgau: Univ. Freiburg, Fakultat fur Mathematik und Physik (2019)
  Jørgen Olsen Lye
  
• The Becker-Gottlieb Transfer: a Geometric Description, Oberwolfach Preprints; 2019,13
  Yi-Sheng Wang
  
• Construction of six functor formalisms, Dissertation 2019, Freidok May 2020
  Rene Recktenwald
  
• Fujiki relations and fibrations of irreducible symplectic varieties, Epijournal de Geometrie Algebrique, Volume 4 (2020), Article no. 7
  Martin Schwald
  
• Isolated singularities of flat metrics on Riemann surfaces. Proc. Amer. Math. Soc. 148 (2020), no. 9, 4057-4064
  Jin Li, Bin Xu
  
• On the definition of irreducible holomorphic symplectic manifolds and their singular analogs
  Martin Schwald
  
• Topological K-theory with coefficients and the e-invariant, Rocky Mountain J. Math, 50, 1 (2020), 281—318
  Yi-Sheng Wang
  
• Algebraic K-theory and Grothendieck-Witt theory of monoid schemes, Mathematische Zeitschrift
  Jens Eberhardt, Oliver Lorscheid and Matthew B. Young
  
• Balanced Product Quantum Codes, IEEE Transactions on Information Theory, vol. 67, no. 10, pp. 6653-6674 (2021)
  Nikolas Peter Breuckmann, Jens Eberhardt
  
• Cotorsion Pairs and Quillen Adjunctions
  Rene Recktenwald,
  
• Folding of Hitchin systems and crepant resolutions, Int. Math. Res. Not., 2021
  Florian Beck, Ron Donagi, Katrin Wendland
  
• Motivic Springer Theory, Indagationes Mathematicae
  Jens Eberhardt and Catharina Stroppel
  
• Quantum Low-Density ParityCheck Codes, PRX Quantum, (2021)
  Nikolas Peter Breuckmann, Jens Eberhardt
  
• Real Springer fibers and odd arc algebras, J. London Math. Soc., 103: 1415-1452, (2021)
  Jens Eberhardt, Grégoire Naisse and Arik Wilbert
  
• Springer Motives, Proc. Amer. Math. Soc. 149, (2021)
  Jens Eberhardt
  
• A note on semiorthogonal decompositions for Fano fibrations. Geometriae Dedicata 216, 20 (2022)
  Pedro Nùnez
  
• Group completion in the K-theory and Grothendieck-Witt theory of proto-exact categories, Journal of Pure and Applied Algebra 226 (2022)
  Jens Eberhardt, Oliver Lorscheid and Matthew B. Young