Tobias Hartung

My primary research interest is functional analysis, especially unbounded operators and relations. Revolving around that, basically everything from topology and algebra via spectral theory to well-posedness and causality (wherever causality makes any kind of sense, of course) of differential equations/inclusions of all shapes, colors, and flavors looks interesting to me.

In particular, I am interested in:

• Functional Analysis: unbounded operators, Fourier Integral Operators, topological vector spaces, partial differential equations
• Mathematical Physics: Feynman path integrals (and their connection to Fourier Integral Operators), mathematical properties of lattice quantum field theory

My Ph.D. supervisors were Simon Scott, Yuri Safarov, and Eugene Shargorodsky. In my thesis I investigate Fourier Integral Operators; in particular, I computed the Laurent expansion of $\zeta$-functions of families of Fourier Integral Operators, computed the kernel singularity structure of Fourier Integral Operators, generalized the Kontsevich-Vishik trace to Fourier Integral Operators, and studied integration techniques in algebras of Fourier Integral Operators.

At NIC/DESY Zeuthen and HU Berlin we are looking at quasi-Monte Carlo methods and recursive numerical integration in lattice field theories, hoping to get an edge over "standard" Monte Carlo methods.

My Erdős number is $\le5$.

• '$\zeta$-functions of Fourier Integral Operators' (2015)

• 'Well-posedness and causality of the non-Newtonian Navier-Stokes equations on 3-dimensional Riemannian $C^{1,1}$-manifolds with respect to strong, local-in-time solutions' (2014)

• T. Hartung 'Integrals of families of Fourier Integral Operators'

• J. Hancox, T. Hartung '$\zeta$-regularization and the heat-trace on some compact quantum semigroups'
  arXiv:1801.03982

• T. Hartung, S. Scott 'A generalized Kontsevich-Vishik trace for Fourier Integral Operators and the Laurent expansion of $\zeta$-functions'
  arXiv:1510.07324 [math.AP]

• T. Hartung, K. Jansen, H. Leövey, J. Volmer 'Avoiding the sign-problem in lattice field theory' in B. Tuffin, P. L’Ecuyer (eds.) Monte Carlo and Quasi-Monte Carlo Methods - MCQMC, Rennes, France, July 2018 (2019)
  

• T. Hartung, K. Jansen 'Integrating Gauge Fields in the $\zeta$-formulation of Feynman's path integral' (accepted, upcoming volume of Applied Numerical and Harmonic Analysis)
  arXiv:1902.09926

• T. Hartung, K. Jansen 'Zeta-regularized vacuum expectation values' Journal of Mathematical Physics 60, 093504(2019)
  arXiv:1808.06784

• T. Hartung 'Feynman path integral regularization using Fourier Integral Operator $\zeta$-functions' in Böttcher A., Potts D., Stollmann P., Wenzel D. (eds) The Diversity and Beauty of Applied Operator Theory. Operator Theory: Advances and Applications, vol 268. Birkhäuser, 261-289 (2018)

• T. Hartung, K. Jansen, H. Leövey, J. Volmer 'Improving Monte Carlo integration by symmetrization' in Böttcher A., Potts D., Stollmann P., Wenzel D. (eds) The Diversity and Beauty of Applied Operator Theory. Operator Theory: Advances and Applications, vol 268. Birkhäuser, 291-317 (2018)

• T. Hartung 'Regularizing Feynman Path Integrals using the generalized Kontsevich-Vishik trace' Journal of Mathematical Physics 58, 123505 (2017)
  arXiv:1703.03451 [math-ph]

• A. Ammon, A. Genz, T. Hartung, K. Jansen, H. Leövey, J. Volmer 'Applying recursive numerical integration techniques for solving high dimensional integrals' PoS LATTICE2016 (2017) 335
  arXiv:1611.08628 [hep-lat]

• A. Ammon, T. Hartung, K. Jansen, H. Leövey, J. Volmer 'New polynomially exact integration rules on U(N) and SU(N)' PoS LATTICE2016 (2017) 334
  arXiv:1610.01931 [hep-lat]

• A. Ammon, T. Hartung, K. Jansen, H. Leövey, J. Volmer 'Overcoming the sign problem in 1-dimensional QCD by new integration rules with polynomial exactness' Physical Review D 94, 114508 (2016)
  arXiv:1607.05027 [hep-lat]

• A. Ammon, A. Genz, T. Hartung, K. Jansen, H. Leövey, J. Volmer 'On the efficient numerical solution of lattice systems with low-order couplings' Computer Physics Communications 198 (2016) 71-81
  DESY 15-033, arXiv:1503.05088 [hep-lat]

• A. Ammon, T. Hartung, K. Jansen, H. Leövey, A. Griewank, M. Müller-Preussker 'Applicability of Quasi-Monte Carlo for lattice systems' PoS LATTICE2013 (2014) 040
  HU-EP-13/69, SFB/CPP-13-98, DESY 13-225, arXiv:1311.4726 [hep-lat]

• 'Recursive Numerical Integration for Lattice Systems with Low-order Couplings'
  SIAM: Uncertainty Quantification 2018 (April 19, 2018)

• 'Heat traces and $\zeta$-regularization in compact quantum semigroups'
  Young Functional Analysts' Workshop 2018 (March 22, 2018)

• 'Path integral regularization and the generalized Kontsevich-Vishik trace'
  Geometric and Singular Analysis 201 (February 21, 2018)

• 'Path integral regularization and the generalized Kontsevich-Vishik trace'
  Analysis and Geometry in Cargèse (November 21, 2017)

• 'Feynman path integral regularization using Fourier integral operator $\zeta$-functions'
  International Workshop on Operator Theory and its Applications 2017 (August 17, 2017)

• 'Fun with Feynman integrals'
  Young Researchers in Mathematics 2017 (August 2, 2017)

• 'Semigroups, Generators, and Forms: the triumvirate of (linear) PDE'
  PostGraduate Forum, Lancaster University (June 12, 2017)

• 'Feynman path integral regularization using Fourier Integral Operator $\zeta$-functions'
  KCL Analysis Seminar (May 11, 2017)

• 'Fun with Feynman Integrals: A $\zeta$-regularization approach to Feynman's path integral'
  Oberseminar Analysis, Leibniz Universität Hannover (april 19, 2017)

• 'Alternative Methods: QMC, symmetrization'
  Lattice QCD at the physical pion mass: results, challenges and modern techniques (April 12, 2017)

• 'Fun with Feynman integrals'
  Young Functional Analysts' Workshop 2017 (March 30, 2017)

• 'Integrals in Algebras of Fourier Integral Operators'
  Geometric and Singular Analysis 2017 (February 20, 2017)

• 'Integrals in Algebras of Fourier Integral Operators'
  Young Researchers in Mathematics 2016 (August 1, 2016)

• 'New polynomially exact integration rules on U(N) and SU(N)'
  Lattice 2016 (July 25, 2016)

• 'Integrals in Algebras of Fourier Integral Operators'
  Young Functional Analysts' Workshop 2016 (April 8, 2016)

• '$\zeta$-functions of Fourier Integral Operators: the generalized Kontsevich-Vishik trace'
  Geometric and Singular Analysis 2016 (March 10, 2016)

• '$\zeta$-functions of Fourier Integral Operators'
  Young Researchers in Mathematics 2015 (August 18, 2015)

• '$\zeta$-regularization of gauged families of poly-log-homogeneous distributions and Fourier Integral Operator traces'
  Young Functional Analysts' Workshop 2015 (June 12, 2015)

• 'On Well-posedness and Causality of Navier-Stokes equations on 3-manifolds'
  Topological Analysis Seminar, King's College London (October 31, 2013)