SFB 701:  Spectral Structures and Topological Methods in Mathematics

Final Report Abstract

Spectral structures and topological methods were basic unifying concepts for scientists in different fields of theoretical and applied mathematics who cooperated in the CRC 701. New concepts of mathematical physics recently have had a significant impact in theoretical mathematics and stimulated new areas of mathematical research which have motivated research projects in this CRC like Seiberg-Witten invariants from topology, universal spectral distributions from quantum physics and their appearance in number theory, the application of concepts from quantum field theory to the theory of moduli spaces in algebraic topology, as well as concepts of quantum information relating to quantum groups. Conversely, modern methods in mathematics, in particular in topology, number theory and recently also stochastic analysis and free probability, which have been investigated in this CRC resulted not only in interesting mathematical results but have been relevant in theoretical physics and other applications of mathematics, such as fluid dynamics, crystallography, material sciences and mobile communications. In this CRC applied and theoretical mathematics have joined forces in order to realize the considerable potential of infra disciplinary research.

Publications

• Off-diagonal upper estimates for the heat kernel of the Dirichlet forms on metric spaces. Invent. Math. 174 (2008), 81–126
  A. Grigor’yan and N. Hu
  
• Diffraction of stochastic point sets: explicitly computable examples. Commun. Math. Phys. 293(3), (2010), 611–660
  M. Baake, M. Birkner, and R. V. Moody
  
• The circular law for random matrices. Ann. Probab. 38(4), (2010), 1444–1491
  F. Götze and A. Tikhomirov
  
• Global well-posedness of the energy-critical nonlinear Schrödinger equation with small initial data in H 1 (T3 ). Duke Math. J. 159(2), (2011), 329–349
  S. Herr, D. Tataru, and N. Tzvetkov
  
• Overconvergent de Rham-Witt cohomology. Ann. Sci. Éc. Norm. Supér. 44(2), (2011), 197–262
  C. Davis, A. Langer, and T. Zink
  
• The global random attractor for a class of stochastic porous media equations. Comm. PDE 36(3), (2011), 446–469
  W.-J. Beyn, B. Gess, P. Lescot, and M. Röckner
  
• Stratifying modular representations of finite groups. Ann. of Math. 174 (2012), 1643–1684
  D. J. Benson, S. B. Iyengar, and H. Krause
  
• Higher finiteness properties of reductive arithmetic groups in positive characteristic: the rank theorem. Ann. of Math. (2) 177(1), (2013), 311–366
  K.-U. Bux, R. Köhl, and S. Witzel
  
• Rate of convergence and Edgeworth-type expansion in the entropic central limit theorem. Ann. Prob. 41 (2013), 2479–2512
  S. G. Bobkov, G. Chistyakov, and F. Götze
  
• The image of Colmez’s Montreal functor. Publ. Math. Inst. Hautes Etudes Sci. 118 (2013), 1–191
  V. Paskunas
  
• Estimates of heat kernels for non-local regular Dirichlet forms. Trans. Amer. Math. Soc. 366 (2014), 6397–6441
  A. Grigor’yan, J. Hu, and K.-S. Lau
  
• Explicit rates of approximation in the CLT for quadratic forms. Ann. Probab. 42 (2014), 354–397
  F. Götze and A. Y. Zaitsev
  
• Local universality of repulsive particle systems and random matrices. Ann. Probab. 42(6), (2014), 2207–2242
  F. Götze and M. Venker
  
• On special zeros of p-adic L-functions of Hilbert modular forms. Invent. Math. 196 (2014), 69–138
  M. Spieß
  
• Well-rounded sublattices of planar lattices. Acta Arith. 166 (2014), 301–334
  M. Baake, R. Scharlau, and P. Zeiner
  
• An analytic approach to infinite-dimensional continuity and Fokker–Planck–Kolmogorov equations. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 14(3), (2015), 983–1023
  V. I. Bogachev, G. Da Prato, M. Röckner, and S. V. Shaposhnikov
  
• An operatorial approach to stochastic partial differential equations driven by linear multiplicative noise. JEMS 17(7), (2015), 1789–1815
  V. Barbu and M. Röckner
  
• Dynamical versus diffraction spectrum for structures with finite local complexity. Ergodic Theory Dynam. Systems 35(7), (2015), 2017–2043
  M. Baake, D. Lenz, and A. van Enter
  
• Dynamical Widom- Rowlinson model and its mesoscopic limit. J. Stat. Phys. 158(1), (2015), 57–86
  D. Finkelshtein, Y. Kondratiev, O. Kutoviy, and M. J. Oliveira
  
• From random Poincaré maps to stochastic mixedmode-oscillation patterns. J. Dynam. Differential Equations 27 (2015), 83–136
  N. Berglund, B. Gentz, and C. Kuehn
  
• A categorification of non-crossing partitions. J. Eur. Math. Soc. (JEMS) 18(10), (2016), 2273–2313
  A. Hubery and H. Krause
  
• Arithmetic groups, base change, and representation growth. Geom. Funct. Anal. 26(1), (2016), 67–135
  N. Avni, B. Klopsch, U. Onn, and C. Voll
  
• Heat kernel estimates for subordinate Brownian motions. Proc. Lond. Math. Soc. (3) 113(5), (2016), 627–648
  A. Mimica
  
• Similarity classes of integral p-adic matrices and representation zeta functions of groups of type A2 . Proc. Lond. Math. Soc. (3) 112(2), (2016), 267–350
  N. Avni, B. Klopsch, U. Onn, and C. Voll
  
• Spatial decay of rotating waves in reaction diffusion systems. Dynamics of Partial Differential Equations 13 (2016), 191–240
  W.-J. Beyn and D. Otten
  
• Strong uniqueness for SDEs in Hilbert spaces with nonregular drift. Ann. Probab. 44(3), (2016), 1985–2023
  G. Da Prato, F. Flandoli, M. Röckner, and A. Veretennikov
  
• The braided Thompson’s groups are of type F∞ . J. Reine Angew. Math. 718 (2016), 59–101. With an appendix by Zaremsky
  K.-U. Bux, M. G. Fluch, M. Marschler, S. Witzel, and M. C. B. Zaremsky
  
• The cubic Dirac equation: small initial data in H 2 (R2 ). Comm. Math. Phys. 343(2), (2016), 515–562
  I. Bejenaru and S. Herr
  
• Homologies of digraphs and Künneth formulas. Comm. Anal. Geom. 30(5), (2017), 969–1018
  A. Grigor’yan, Y. Muranov, and S.-T. Yau
  
• Intrinsic scaling properties for nonlocal operators. J. Eur. Math. Soc. (JEMS) 19(4), (2017), 983–1011
  M. Kassmann and A. Mimica
  
• Local Functional Equations for Submodule Zeta Functions Associated to Nilpotent Algebras of Endomorphisms. Int. Math. Res. Not. (IMRN), (2017)
  C. Voll
  
• Long-time behavior, invariant measures, and regularizing effects for stochastic scalar conservation laws. Comm. Pure Appl. Math. 70(8), (2017), 1562–1597
  B. Gess and P. E. Souganidis
  
• Lower estimates of heat kernels for non-local Dirichlet forms on metric measure spaces. J. Funct. Anal. 272 (2017), 3311–3346
  A. Grigor’yan, J. Hu, and E. Hu
  
• On global well-posedness and scattering for the massive Dirac- Klein-Gordon system. J. Eur. Math. Soc. (JEMS) 19(8), (2017), 2445–2467
  I. Bejenaru and S. Herr
  
• On the Drinfeld moduli problem of p-divisible groups. Camb. J. Math. 5(2), (2017), 229–279
  M. Rapoport and T. Zink
  
• Uniform analytic properties of representation zeta functions of finitely generated nilpotent groups. Trans. Amer. Math. Soc. 369(9), (2017), 6327–6349
  D. H. Dung and C. Voll
  
• Heat kernel estimates for an operator with a singular drift and isoperimetric inequalities. J. Reine Angew. Math. 736 (2018), 1–31
  A. Grigor’yan, S.-X. Ouyang, and M. Röckner
  
• Hybrid Iwasawa algebras and the equivariant Iwasawa main conjecture. Amer.q J. Math. 140 (2018), 245–276
  H. Johnston and A. Nickel
  
• On the geometric side of the Arthur trace formula for the symplectic group of rank 2. Mem. Amer. Math. Soc., Vol. 844. AMS, Providence, R.I., (2018)
  W. Hoffmann and S. Wakatsuki
  
• Stratification for module categories of finite group schemes. J. Amer. Math. Soc. 31(1), (2018), 265–302
  D. Benson, S. B. Iyengar, H. Krause, and J. Pevtsova
  

Completed projects