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
  (See online at https://doi.org/10.37236/4610)
• Analytical index and eta forms for Dirac operators with onedimensional kernel over a hypersurface, 2015
  Anja Wittmann
  (See online at https://doi.org/10.48550/arXiv.1503.02002)
• 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
  (See online at https://doi.org/10.6094/UNIFR/10375)
• 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
  (See online at https://doi.org/10.1007/s11512-014-0211-x)
• Comparison of the categories of motives defined by Voevodsky and Nori, Dissertation Freiburg 2016, Freidok plus (2016)
  Daniel Harrer
  (See online at https://doi.org/10.6094/UNIFR/11178)
• 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
  (See online at https://doi.org/10.6094/UNIFR/11096)
• Hitchin and calabi-yau integrable systems, Freidok plus (2016)
  Florian Beck
  (See online at https://doi.org/10.6094/UNIFR/11668)
• Homotopy theory for rigid analytic varieties, Freidok plus (2016)
  Helene Sigloch
  (See online at https://doi.org/10.6094/UNIFR/11742)
• Koszul-Dualitat von komplexen Gruppen, Freidokplus (2016)
  Michael Rottmaier
  (See online at https://doi.org/10.6094/UNIFR/11236)
• Low degree Hodge theory for klt varieties
  Martin Schwald
  (See online at https://doi.org/10.48550/arXiv.1612.01919)
• The chiral de Rham complex of tori and orbifolds, Freidok plus (2016)
  Felix Grimm
  (See online at https://doi.org/10.6094/UNIFR/11501)
• The e-Invariant and Transfer Map, Ph.D. Thesis
  Yi-Sheng Wang
  (See online at https://doi.org/10.6094/UNIFR/12680)
• An approximation of the e-invariant in the stable homotopy category
  Yi-Sheng Wang
  (See online at https://doi.org/10.48550/arXiv.1707.03453)
• Categories of vector spaces and Grassmannians
  Yi-Sheng Wang
  (See online at https://doi.org/10.48550/arXiv.1711.03059)
• Fat realization and Segal’s classifying space
  Yi-Sheng Wang
  (See online at https://doi.org/10.48550/arXiv.1710.03796)
• Hodge theoretic methods in the study of symplectic varieties. Freiburg im Breisgau: Univ. Freiburg, Fakultät für Mathematik und Physik (2017)
  Martin Schwald
  (See online at https://doi.org/10.6094/UNIFR/12695)
• Homotopy Classification of Line Bundles Over Rigid Analytic Varieties, 2017
  Helene Sigloch
  (See online at https://doi.org/10.48550/arXiv.1708.01166)
• Coherent Sheaves on Calabi-Yau manifolds, Picard-Fuchs equations and potential functions, Feburary 2018, Freiburg
  Natalie Peternell
  (See online at https://doi.org/10.6094/UNIFR/17082)
• 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
  (See online at https://doi.org/10.6094/UNIFR/149320 https://doi.org/10.6094/UNIFR/149507)
• Geometric realization and its variants
  Yi-Sheng Wang
  (See online at https://doi.org/10.48550/arXiv.1804.00345)
• Graded and geometric parabolic induction for category O, (2018), Freidok plus (2017)
  Jens Eberhardt
  (See online at https://doi.org/10.48550/arXiv.1603.00327 https://doi.org/10.6094/UNIFR/12750)
• On absolute linear Harbourne constants, Finite Fields Appl. 51 (2018), 371-387
  Marcin Dumnicki, Daniel Harrer, Justyna Szpond
  (See online at https://doi.org/10.1016/j.ffa.2018.03.001)
• On the motivic Tamagawa number of number fields, Dissertation 2018, Freidok May 2018
  Maximilian Schmidtke
  (See online at https://doi.org/10.6094/UNIFR/15821)
• Aspects of Calabi-Yau integrable and Hitchin systems, SIGMA Symmetry Integrability Geom. Methods Appl. 15 (2019), Paper No. 001
  Florian Beck
  (See online at https://doi.org/10.3842/SIGMA.2019.001)
• Calabi-Yau orbifolds over Hitchin bases, J. Geom. Phys. 136 (2019), 14-30
  Florian Beck
  (See online at https://doi.org/10.1016/j.geomphys.2018.10.010)
• Gradings and Soergel bimodules, Freiburg im Breisgau, Fakultat fur Mathematik und Physik (2019)
  Benjamin McDonnell
  (See online at https://doi.org/10.6094/UNIFR/151951)
• K-Motives and Koszul Duality, (2019)
  Jens Eberhardt
  (See online at https://doi.org/10.48550/arXiv.1909.11151)
• n invariants under degeneration to cone-edge singularities, Freiburg im Breisgau: Univ. Freiburg, Fakultät für Mathematik und Physik (2019)
  Nelvis Fornasin
  (See online at https://doi.org/10.6094/UNIFR/154617)
• Stable geodesics on a K3 surface, Freiburg im Breisgau: Univ. Freiburg, Fakultat fur Mathematik und Physik (2019)
  Jørgen Olsen Lye
  (See online at https://doi.org/10.6094/UNIFR/150790)
• The Becker-Gottlieb Transfer: a Geometric Description, Oberwolfach Preprints; 2019,13
  Yi-Sheng Wang
  (See online at https://doi.org/10.14760/OWP-2019-13)
• Construction of six functor formalisms, Dissertation 2019, Freidok May 2020
  Rene Recktenwald
  (See online at https://doi.org/10.6094/UNIFR/165780)
• Fujiki relations and fibrations of irreducible symplectic varieties, Epijournal de Geometrie Algebrique, Volume 4 (2020), Article no. 7
  Martin Schwald
  (See online at https://doi.org/10.46298/epiga.2020.volume4.4557)
• Isolated singularities of flat metrics on Riemann surfaces. Proc. Amer. Math. Soc. 148 (2020), no. 9, 4057-4064
  Jin Li, Bin Xu
  (See online at https://doi.org/10.1090/proc/15044)
• On the definition of irreducible holomorphic symplectic manifolds and their singular analogs
  Martin Schwald
  (See online at https://doi.org/10.48550/arXiv.2003.06004)
• Topological K-theory with coefficients and the e-invariant, Rocky Mountain J. Math, 50, 1 (2020), 281—318
  Yi-Sheng Wang
  (See online at https://doi.org/10.1216/rmj.2020.50.281)
• Algebraic K-theory and Grothendieck-Witt theory of monoid schemes, Mathematische Zeitschrift
  Jens Eberhardt, Oliver Lorscheid and Matthew B. Young
  (See online at https://doi.org/10.48550/arXiv.2009.12636)
• Balanced Product Quantum Codes, IEEE Transactions on Information Theory, vol. 67, no. 10, pp. 6653-6674 (2021)
  Nikolas Peter Breuckmann, Jens Eberhardt
  (See online at https://doi.org/10.1109/TIT.2021.3097347)
• Cotorsion Pairs and Quillen Adjunctions
  Rene Recktenwald,
  (See online at https://doi.org/10.48550/arXiv.2105.10531)
• Folding of Hitchin systems and crepant resolutions, Int. Math. Res. Not., 2021
  Florian Beck, Ron Donagi, Katrin Wendland
  (See online at https://doi.org/10.1093/imrn/rnaa375)
• Motivic Springer Theory, Indagationes Mathematicae
  Jens Eberhardt and Catharina Stroppel
  (See online at https://doi.org/10.48550/arXiv.2109.00305)
• Quantum Low-Density ParityCheck Codes, PRX Quantum, (2021)
  Nikolas Peter Breuckmann, Jens Eberhardt
  (See online at https://doi.org/10.1103/PRXQuantum.2.040101)
• Real Springer fibers and odd arc algebras, J. London Math. Soc., 103: 1415-1452, (2021)
  Jens Eberhardt, Grégoire Naisse and Arik Wilbert
  (See online at https://doi.org/10.1112/jlms.12413)
• Springer Motives, Proc. Amer. Math. Soc. 149, (2021)
  Jens Eberhardt
  (See online at https://doi.org/10.1090/proc/15290)
• A note on semiorthogonal decompositions for Fano fibrations. Geometriae Dedicata 216, 20 (2022)
  Pedro Nùnez
  (See online at https://doi.org/10.1007/s10711-022-00680-z)
• 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
  (See online at https://doi.org/10.1016/j.jpaa.2022.107018)