GRK 1692:  Curvature, Cycles, and Cohomology

• Fachliche Zuordnung: Mathematik
• Förderung: Förderung von 2010 bis 2019
• Projektkennung: Deutsche Forschungsgemeinschaft (DFG) - Projektnummer 169166737

• Erstellungsjahr: 2020

Zusammenfassung der Projektergebnisse

Das Graduiertenkolleg an der Mathematischen Fakultät der Universität Regensburg wurde fakultätsübergreifend von Wissenschaftlern aus den drei Schwerpunkten der Fakultät • Angewandte Analysis; • Globale Analysis, Geometrie und Topologie; • Arithmetisclie Geometrie getragen. Ausgehend von der Idee, dass das Erlernen von Elementen der allen beteiligten Gebieten gemeinsamen grundlegenden Sprache und Techniken eine fruchtbare Interaktion zwischen den Doktoranden in den verschiedenen Spezialgebieten ermöglichen kann, wurde das Ausbildungs- und Forschungsprogramm konzipiert und umgesetzt. Dieses Umfeld führte zu interessanten Ergebnissen der Forschungsaktivitäten der Doktoranden des Gradiertenkollegs in den beteiligten Spezialgebieten. Aus dem Bereich der Arithmetischen Geometrie wurden Ergbenisse zur arithmetischen Schnittheorie im nicht-archimedischen Fall, zur Theorie der Höhen und zur Theorie der Auflösung von Singularitäten erhalten. Die Beiträge aus dem Bereich der Globalen Analysis, Geometrie und Topologie sind vielfältig. Sie reichen von Ergebnissen über die Topologie des Raumes der Riemannschen Metriken mit vorgegegeben Spektraleigenschaften des assoziierten Diracoperators, der Konstruktion von groben Homologietheorien bis zur Berechnung von Invarianten der Topologie von 3-dimensionalen Mannigfaltigkeiten. Zu diesem Bereich zählen auch die für die mathematische Physik interessanten Fortschritte zum Fermionic-Projector-Ansatz. Die zentrale Frage der im Bereich der angewandten Analysis betrachteten Projekt sind Existenz, Eindeutigkeit und qualitatives Verhalten der Lösungen von geometrischen Evolutionsgleichungen, insbesondere für Netzwerke von Kurven in der Ebene und Flächen im Raum, aber auch für an Spinoren gekoppelte Evolutionen von Riemannschen Metriken und Abbildmigen. Neben der fachwissenschaftlichen Ausbildung hat das Graduiertenkolleg durch Aktivitäten im Bereich von Schlüsselqualifikationen, Teamarbeit und durch Vermittlung von allgemeinen Aspekten wissenschaftlicher Arbeit mit Themen wie „Gute wissenschaftliche Praxis" oder „Karrieremöglichkeiten für promovierte Mathematiker (innen)“ zur Entwicklung der Doktoranden beigetragen.

Projektbezogene Publikationen (Auswahl)

• Minimal elastic networks, Indiana Univ. Math. J.
  A. DairAcqua, M. Novaga, A. Pluda
  
• Generic metrics and the mass endomorphism on spin threemanifolds, Ann. Glob. Anal. Geom., 37, pp. 163-171, 2010
  A. Hermann
  
• A Dynamical Bogomolov Property, preprint 2011
  L. Pottmeyer
  
• Homotopy Theoretic Models of Type Theory, in: Typed Lambda Calculi and Applications. Lecture notes in Computer Science 6690, pp. 45-60, Springer, 2011
  P. Arndt, K. Kapulkin
  
• A universal deformation ring with unexpected Kruli dimension. Math. Z, 275, pp. 647-652, 2012
  J. Sprang
  
• Classification of traces and associated determinants on odd-class operators in odd dimensions, SIGMA Symmetry, Integrability and Geometric: Methods and Applications, 8, paper 023, 2012
  C. Neira-Jimenez, M. F. Ouedraogo
  
• Dirac eigenspinors for generic metrics, PhD thesis, Universität Regensburg, 2012
  A. Hermann
  
• Displays and gauges, PhD thesis, Universität Regensburg, 2012
  M. Wid
  
• Higher congruence companion forms, Acta Arith., 156, pp. 159-175, 2012
  R. Adibhatla, J. Manoharmayum
  
• Characteristic polyhedra of idealistic exponents with history, PhD thesis, Universität Regensburg, 2013
  B. Schober
  
• Classification of traces and hypertraces on spaces of classical pseudodifferential operators, Noncommut Geom., 7, pp. 457-498, 2013
  M. Lesch, C. Neira-Jimenez
  
• Continuity of Dirac Spectra. Ann. Glob. Anal. Geom., 44, pp. 541-563, 2013
  N. Nowaczyk
  
• Fields with the Bogomolov Property, PhD thesis, Universität Regensburg, 2013
  L. Pottmeyer
  
• Heights and totally real numbers, Rendiconti Lined - Matematica e Applicazioni, 24, pp. 471-483, 2013
  L. Pottmeyer
  
• Minimal number of points with bad reduction for elliptic curves over P. Rocky Mountain Journal of Mathematics, 43, No. 6, 2013
  J. Sprang
  
• Modularity of certain mod p^n Galois representations
  R. Adibhatla
  
• Volume comparison for hypersurfaces in Lorentzian manifolds and singularity theorems, Ann. Glob. Anal. Geom., 43, pp. 233-251, 2013
  J.-H. Treude, J.D.E. Grant
  
• Volume-Preserving Mean Curvature and Willmore Flows with Line Tension, PhD thesis, Universität Regensburg, 2013
  L. Müller
  
• Comparison Theorems between Overconvergent Cohomology and Rigid Cohomology with Coefficient. Oberwolfach Report 32/2014, 1784-1786
  V. Ertl
  
• Deformations of Generalized Complex Branes, preprint, 2014
  B. Collier
  
• Isometric embeddings in bounded cohomology, J. Topol. Anal. 6 (2014), no. 1, 125
  M. Bucher, M. Burger, A. lozzi, R. Frigerio, C. Pagliantini, M.B. Pozzetti
  
• Lokale Schnitttheorie an riicht-archimedischen Stellen für Produkte semistabiler Kurven, PhD thesis, Universität Regensburg. 2013
  J. Kolb
  
• Mass endomorphism, surgery and perturbations, Ann. Inst. Fourier (Grenoble) 64 (2014), no. 2, 467-487
  B. Ammann, M. Dahl, A. Hermann, E. Humbert
  
• Optimal control of Allen-Cahn systems, preprint, Trends in PDE constrained optimization, 11-26, Internat. Ser. Numer. Math., 165, Birkhäuser/Springer, Cham., 2014
  L. Blank, M. H. Farshbaf-Shaker, C. Hecht, J. Michl, C. Rupprecht
  
• Zero sets of eigenspinors for generic metrics. Comm. Anal. Geom. 22 (2014), no. 2, 177-218
  A. Hermann
  
• A characterization of ordinary modular eigenforms with CM, Arithmetic and geometry, 24-35, London Math. Soc. Lecture Note Ser., 420, Cambridge Univ. Press, Cambridge, 2015
  R. Adibhatla and P. Tsaknias
  
• A refinement of Hironaka’s additive group schemes for an extended invariant. Dissertation, Universität Regensburg (2015)
  B. Dietel
  
• Causal fermion systems as a candidate for a unified physical theory. Journal of Physics: Conference Series. Vol: 626. No. 1. lOP Publishing, 2015
  F. Finster, J. Kleiner
  
• Formal asymptotic limit of a diffuse-interface tumor-growth model. Mathematical Models and Methods in Applied Sciences 25, 06, 1011-1043 (2015)
  D. Hilhorst. J. Kampmann, T. N. Nguyen, K. G. Van Der Zee
  
• Heights of points with bounded ramification, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 14 (2015), no. 3, 965-981
  L. Pottmeyer
  
• On convergence of solutions to equilibria for fully nonlinear parabolic problems with nonlinear boundary conditions, J. Evol. Equ. 15 (2015), no. 4, 913-959
  H. Abels, N. Arab, H. Garcke
  
• Stability of Spherical Caps under the Volume-Preserving Mean Curvature Flow with Line Tension, Nonlinear Anal. 117 (2015), 8-37
  H. Abels, H. Garcke, L. Müller
  
• Uniformly finite homology and amenable groups, Algebr. Geom. Topol. 15 (2015), no. 1, 467-492
  M. Blank, F. Diana
  
• A coupled surface-Calm-Hilliard bulk-diffusion system modeling lipid raft formation in cell membranes. Mathematical Models and Methods in Applied Sciences 26, 06, 1149-1189 (2016)
  H. Garckc, J. Karnpmann, A. Rätz, M. Röger
  
• About the mass of certain second order elliptic operators, Adv. Math. 294 (2016), 596-633
  A. Hermann, E. Humbert
  
• Arithmetic Divisors on Products of Curves over non-Arcliimedean Fields, Dissertation, Universität Regensburg (2016)
  P. Vollmer
  
• Differential cohomology theories as sheaves of spectra, J. Homotopy Relat. Struct. 11 (2016), no. 1, 1-66
  U. Bunke, T. Nikolaus, M. Völkl
  
• Full faithfulness for overconvergent F-de Rhain-Witt connections. Comptesrendus - Mathematique vol. 354, no. 7, pp. 653-658, (2016)
  V. Ertl
  
• Integral foliated simplicial volume of hyperbolic 3-manitoids. Groups Geom. Dyn. 10 (2016), no. 3, 825-865
  C. Löh, C. Paigliantini
  
• Local Well-Posedness for Volume-Preserving Mean Curvature and Willmore Flows with Line Tension, Math. Nachr. 289 (2016), no. 2-3, 136-174
  H. Abels, H. Garcke, L. Müller
  
• Noether-like theorems for causal variational principles. Calculus of Variations and Partial Differential Equations 55.2 (2016): 35
  F. Finster, J. Kleiner
  
• On the structure of the absolute Galois group of local fields of residue characteristic 2, Dissertation, Universität Regensburg (2016)
  F. Schneider
  
• Optimal control of elastic vector-valued Allen- Cahn variational inequalities, SIAM J. Control Optim. 54 (2016), no. 1, 129-152
  M. H. Farshbaf-Shaker, C. Hecht
  
• Quantum Mathematical Physics: A Bridge Between Mathematics and Physics. Birkhäuser, 2016
  Finster, J. Kleiner, C. Röken, J. Tolksdorf
  
• The fermionic projector in a time-dependent external potential: Mass oscillation property and hadamard states, J. Math. Phys. 57, 072303 (2016)
  F. Finster, S. Murro and C. Röken
  
• The spinorial energy functional: solutions of the gradient flow on Berger spheres. In: Ann. Global Anal. Georn. 49 (2016) pp. 329-348
  J. Wittmann
  
• A Hamiltonian formulation of causal variational principles. Calculiis of Variations and Partial Differential Equations 56.3 (2017): 73
  F. Finster, J. Kleiner
  
• A new class of Fermionic Projectors: Möller operators and mass oscillation properties, Lett Math. Phys. 117, 2433-2451 (2017)
  N. Drago and S. Murro
  
• Dynamics of Causal Fermion Systems-Field Equations and Correction Terms for a New Unified Physical Theory. Dissertation Regensburg. 2017
  J. Kleiner
  
• Homogenization of layered materials with rigid components in single-slip finite crystal plasticity. Calc. Var., 5, 75 (2017)
  F. Christowiak, C. Kreisbeck
  
• Linking forms revisited. Pure and Applied Mathematics Quaterly (2017)
  A. Conway, S. Friedl, G. Herrmann
  
• Non-existence of natural states for Abelian Chern-Simons theory, J. Geom. Phys. 116, 119-123 (2017)
  C. Dappiaggi, H. Gimperlein, S. Murro and A. Schenkel
  
• Quantum states on the algebra of Dirac fields: a functional analytic approach, Dissertation, Universität Regensburg (2017)
  S. Murro
  
• Short time existence of the heat flow for Diracharmonic maps on closed manifolds, In: Calc. Var. Partial Differential Equations 56 (2017), Art.169, 32
  J. Wittmann
  
• Some minimization problems for planar networks of elastic curves. Geom. Flows, 3 (2017), pp. 105-124
  A. Dall’Acqua, A. Pluda
  
• The fermionic signature operator and quantum states in Rindler space-time, J. Math. Anal. Appl. 454, 385-411 (2017)
  F. Finster, S. Murro and C. Röken
  
• Torsion in the homology of finite covers of 3-manifolds, Preprint (2017)
  S. Priedl, G. Herrmann
  
• Wavefront sets and polarizations on supermanifolds, J. Math. Phys. 58, 023504 (2017)
  C. Dappiaggi, H. Gimperlein, S. Murro and A. Schenkel
  
• A Blow-up Criterion for the Curve Diffusion Flow with a Contact Angle, SIAM J. Math. Anal.
  H. Abels, J. Butz
  
• Calibrations in families for minimal networks. Proceedings: 89th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM)
  M. Carioni, A. Pluda
  
• Elastic networks, statics and dynamics. Proceedings: 11th Mathematical Society of Japan Seasonal Institute, held in Sapporo in July 2018
  M. Novaga, A. Pluda
  
• Integral Comparison of Monsky-Washnitzer and overconvergent de Rham-Witt cohomology. Proceedings of the AMS, Series B, vol. 5, pp. 64-72, (2018)
  V. Ertl, J. Sprang
  
• Milnor K-theory of complete discrete valuation rings with finite residue fields, J. Pure AppL Algebra 222 (2018), no. 6, 1355-1371
  C. Dahlhausen
  
• Minimal kernels of Dirac operators along maps
  J. Wittmann
  
• Phase separation on biological membranes. Dissertation, Universität Regensburg (2018)
  J. Kampmann
  
• Sutured manifolds and L²-Betti numbers (2018)
  G. Herrmann
  
• The Blanchfield pairing of colored links, Indiana Univ. Math. J. 67 (2018)
  A. Conway, S. Friedl, E. Toffoli
  
• The Curve Diffusion Flow with a Contact Angle, Dissertation, Universität Regensburg (2018)
  J. Butz
  
• The heat flow for Diracharmonic maps, Dissertation, Universität Regensburg (2018)
  J. Wittmann
  
• The leading coefficient of the A²-Alexander torsion. Preprint (2018)
  F. Ben Aribi, S. Friedl, G. Herrmann
  
• Well-Posedness of a Navier-Stokes/Mean Curvalture Flow system, AMS Contemporary Mathematics 710 (Mathematical Analysis in Fluid Mechanics: Selected Recent Results), 1-24 (2018)
  H. Abels, M. Moser
  
• Willmore flow of networks, proceeding of the 89th GAMM annual meeting
  H. Garcke, J. Menzel, A. Pluda
  
• Willmore flow of networks. Proceedings: 89th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM)
  H. Garcke, J. Menzel, A. Pluda
  
• A class of conserved surface layer integrals for causal variational principles. Calculus of Variations and Partial Differential Equations 58.1 (2019): 38
  F. Finster, J. Kleiner
  
• A new proof of a vanishing result due to Berthelot, Esnault, and Rülling. Journal of Number Theory
  V. Ertl
  
• Analysis of a Cahn-Hilliard-Brinkman model for tumour growth with chemotaxis. J. Differential Equations, 266(9), pp. 5998-6036, 2019
  M. Ebenbeck, H. Garcke
  
• Continuous K-Theory and Cohomology of Rigid Spaces, Preprint (2019)
  C. Dahlhausen
  
• Controlled objects as a symmetric monoidal functor, Preprint (2019)
  U. Bunke, L. Caputi
  
• Convergence of the Allen-Calm Equation to the Mean Curvature Flow with 90°-Contact Angle in 2D, Interfaces and Free Boundaries 21, 3, 313-365 (2019)
  H. Abels, M. Moser
  
• Coupled Evolution Equations for Immersions of Closed Manifolds and Vector Fields, Dissertation. Universität Regensburg (2019)
  C. Brand
  
• Generalized bornological coarse spaces and coarse motivic spectra
  D. Heiß
  
• Hochschild and cyclic homology for bornological coarse spaces. Dissertation, Universität Regensburg (2019)
  L. Caputi
  
• Lectures on curvature flow of networks, In: “Contemporary Research in Elliptic PDEs and Related Topics”, 369-417, Springer INDAM series vol 33, Springer, 2019
  C. Mantegazza, M. Novaga, A. Pluda
  
• Lectures on curvature flow of networks. In: '‘Contemporary Research-in Elliptic PDEs and Belated Topics”, 369-417, Springer INDAM scries vol 33, Springer, 2019
  C. Mantegazza, M. Novaga, A. Pluda
  
• Localization for coarse homology theories, Preprint (2019)
  U. Bunke, L. Caputi
  
• Non-linear Stability of Double Bubbles under Surface Diffusion, Preprint (2019)
  H. Garcke, M. Gößwein
  
• Nonsmooth pseudodifferential boundary value problems on manifolds, J. Pseudo-Differ. Oper. Appl. 10, 415-453 (2019)
  H. Abels, C. Neira-Jimenez
  
• On continuous K-thoory and cohomology of rigid spaces, Dissertation. Universität Regensburg (2019)
  C. Dahlhausen
  
• On different notions of calibrations for minimal partitions and minimal networks. Advanced in Calculus of Variations
  M. Carioni, A. Pluda
  
• On the Sharp Interface Limit of a Model for Phase Separation on Biological Membranes. Preprint (2019)
  H. Abels, J. Kampmann
  
• On the Surface Diffusion Flow with Triple Junctions in Higher Space Dimensions, Preprint (2019)
  H. Garcke, M. Gößwein
  
• Rho Invariants for Manifolds with Boundary and Low-Dimensional Topology, Dissertation, Universität Regensburg (2019)
  E. Toffoli
  
• Rigid analytic reconstruction of Hyodo-Kato theory. Preprint
  V. Ertl, K. Yamada
  
• Sharp Interface Limit for a Stokes / Cahn-Hilliard System, Dissertation, Universität Regensburg (2019)
  A. Marquardt
  
• Short Time Existence for the Curve Diffusion Flow with a Contact Angle, J. Differential Equations 268 (2019), no. 1, pp. 318-352
  H. Abels, J. Butz
  
• Stability analysis for stationsiry solutions of the Mullins-Sekerka flow with boundary contact. Preprint (2019)
  H. Garcke, M. Rauchecker
  
• Surface diffusion flow of triple junction clusters in higher space dimensions, Dissertation, Universität Regensburg (2019)
  M. Gößwein
  
• Sutured manifolds, L²-Betti numbers and an upper bound on the leading, coefficient. Dissertation, Universität Regensburg (2019),
  G. Herrmann
  
• Syntornic cohomology and padic motivic cohomology. Algebraic Geometry, vol. 6, no. 1, pp. 100-131, (2019)
  V. Ertl, W.K. Niziol
  
• The Banach manifold C(M,N), Differential Geom. Appl. 63 (2019), pp. 166-185
  J. Wittmann
  
• The oriented mailing problem and its convex relaxation (2019). preprint
  M. Carioni, A. Marchese, A. Massaccesi, A. Pluda, R.. Tione
  
• Well-Poscdness and Qualitative behaviour of the Mullins-Sekerka problem with ninety degree contact angle, Preprint (2019)
  H. Abels, M. Rauchecker, M. Wilke
  
• Willmore Flow for planar networks, J. Differential Equations 266 (2019), no. 4, pp. 2019-2051
  H. Garcke, J. Menzel, A. Pluda
  
• Willmore flow for planar networks, J. Differential Equations, 266 (2019) no. 4, pp. 2019-2051
  H. Garcke, J. Menzel, A. Pluda
  
• Witt differentials in the h-topology. Journal of Pure and Applied Algebra, Vol. 223, no. 12, pp. 5285-5309, (2019)
  V. Ertl, L.E. Miller
  
• Witt dilTerentiala and the h-topology. Oberwolfach Report 6/2019
  V. Ertl
  
• Asymptotic Rigidity of Layered Structures and Its Application in Homogenization Theory. Arch Rational Mech Anal, 235, 51-98 (2020)
  F. Christowiak, C. Kreisbeck
  
• Calibrations for minimal networks in a covering space setting, ESAIM: COCV
  M. Carioni, A. Pluda
  
• Causal variational principle's in the sigma-locally compact setting: Existence of minimizers. Preprint (2020)
  F. Finster, C. Langer
  
• On a Model for Phase Separation on Biological Membranes and its Relation to the Ohta-Kawasaki Equation. European Journal of Applied Mathematics, 31, 02, 297-338 (2020)
  H. Abels, J. Karnpmann
  
• Progress and Visions in Quantum Theory in View of Gravity. Birkhäuser, 2020
  F. Finster, D. Giulini, and J. Kleiner and J. Tolksdorf
  
• Sharp Interface Limit of a Stokes/Cahn-Hiiliard System, Part II: Approximate Solutions, Preprint (2020)
  H. Abels, A. Marquardt
  
• Sharp Interface Limit of a Stokes/Cahn-HiUiard System, Part I: Convergence Result, Preprint (2020)
  H. Abels, A. Marquardt