

1988  Dr. rer. nat., University of Marburg (advisor: Manfred Breuer)  1989  1995  Assistant Professor (C1, later C2), University of Augsburg  1994  Habilitation, University of Augsburg  1994  1995  Visiting Assistant Professor, The Ohio State University, Columbus, OH, USA (on leave from Augsburg)  1995  1999  Senior Assistant Professor (C2), HU Berlin  1999  Heisenberg Fellow, University of Bonn  1999  2000  Associate Professor, University of Arizona, Tucson, AZ, USA  2001  2005  Professor (C3), University of Cologne  2005  2007  Professor (C3), University of Bonn  Since 2007  Professor (W2), University of Bonn 


A large part of my research focuses around geometric differential operators (Dirac and Laplace operators) and their spectral theory. In particular I am interested in spectral invariants which may be extracted from the heat kernel (torsion, eta invariants, zetadeterminants, rho invariants).
I am also interested in Noncommutative Geometry a la Connes and I am working on heat invariants in the noncommutative setting. The noncommuative setting exhibits interesting phenomena which are generally not present in the commutative context.
In collaboration with Boris Vertman I have established a work programme on ''spectral geometry, index theory and geometric flows'' in the context singular spaces. One of the main objectives is to establish a heat resp. resolvent expansion for certain Laplace type operators on certain stratified spaces. This would have interesting applications for the understanding of various of the above mentioned spetral invariants.
A second long term project is in operator algebras: I am working on functional analytic problems related to the construction of the celebrated Kasparov product in the unbounded picture of KKtheory.


Research Area A In [1], we study the Laplacian on singular algebraic curves. The main result gives a complete asymptotic expansion of the heat trace. The paper [2] deals with the etainvariant and its behavior under analytic surgery. The main result provides a gluing formula in which the Maslov index of boundary data plays a crucial role. The paper [3] studies the regularity structure of boundary value problems for Dirac type operators from a functional analytic perspective. There is a one to one correspondence between regular boundary value problems and Lagrangian subspaces in a certain symplectic Hilbert space of boundary data. The more recent paper [4] gives a complete account of the Calderon projector and its role in the theory of boundary value problems for first order (nonDirac type) elliptic differential operators.
The main technical tool of local index theory, the heat trace, leads to more rigid spectral invariants (zetadeterminants) which are also of certain interest in quantum physics. In [5], zetadeterminants are calculated in a singular onedimensional situation by employing classical techniques from the theory of SturmLiouville operators. This paper (and its predecessors), though onedimensional, could not have been written without my background on the heat equation. It is motivated by the problem of extending the celebrated CheegerMüller Theorem on the equality of the analytic and combinatorial torsion to manifolds with singularities. The calculation of determinants is an important case study. An upcoming paper with Vertman will shed some new light on the conical case.
An important problem is to have criteria for essential selfadjointness (quantum completeness). [6] deals with this problem for Hamiltonian systems (SturmLiouville systems with highly degenerate coefficients).  Research Area C Structural questions about algebras of pseudodifferential operators are somewhat on the borderline between areas A and C. Differential operators embed nicely into an algebra of pseudodifferential operators. This is an algebra which contains also the parametrics to elliptic operators. This algebra has interesting structural properties. E.g. it has a unique trace which is sometimes called the WodzickiGuillemin residue trace and which plays an important role in renormalization theory, noncommutative geometry and in the asymptotic analysis of heat and resolvent traces. In [7], the residue trace is generalized to a larger class of pseudodifferential operators (logpolyhomogeneous), also it is shown that such operators do have a heat trace asymptotics where powers occur. [8], based on a lecture series delivered in Boston, surveys the fundamental results on (parametric) pseudodifferential operators, heat expansions the Wodzicki trace, and the Dixmier trace. The recent manuscript [9], jointly with C. NeiraJimenez, gives a refined classification of trace functionals on subalgebras of the algebra of classical pseudodifferential operators. A recent joint paper with Carolina NeiraJimenez gives a refined classification of trace functionals on subalgebras.
The paper with Moscovici and Pflaum “ConnesChern character for manifolds with boundary and eta cochains” (arXiv:0912.0194 [math.OA]) is in a sense an amalgamation of my interest in the heat trace on the one hand and my fascination for Connes Noncommutative Geometry on the other hand. The Chern character of a spectral triple (in a very precise sense it is a generalization of the classical Chern character) in the socalled JLO version is defined in terms of (a generalization of) the heat trace. Its limits as and contain interesting spectral and geometric information. In the above mentioned paper we give a detailed account of this Chern character for a manifold with boundary and we calculate its limits. A predecessor of this paper is [10]. 


[ 1] Jochen Brüning, Matthias Lesch
On the spectral geometry of algebraic curves J. Reine Angew. Math. , 474: : 2566 1996[ 2] Paul Kirk, Matthias Lesch
The ηinvariant, Maslov index, and spectral flow for Diractype operators on manifolds with boundary Forum Math. , 16: (4): 553629 2004[ 3] Jochen Brüning, Matthias Lesch
On boundary value problems for Dirac type operators. I. Regularity and selfadjointness J. Funct. Anal. , 185: (1): 162 2001[ 4] Bernhelm Booß Bavnbek, Matthias Lesch, Chaofeng Zhu
The Calderón projection: new definition and applications J. Geom. Phys. , 59: (7): 784826 2009[ 5] Matthias Lesch, Boris Vertman
Regular singular SturmLiouville operators and their zetadeterminants J. Funct. Anal. , 261: (2): 408450 2011[ 6] Matthias Lesch, Mark Malamud
On the deficiency indices and selfadjointness of symmetric Hamiltonian systems J. Differential Equations , 189: (2): 556615 2003[ 7] Matthias Lesch
On the noncommutative residue for pseudodifferential operators with logpolyhomogeneous symbols Ann. Global Anal. Geom. , 17: (2): 151187 1999[ 8] Matthias Lesch, Henri Moscovici, Markus J. Pflaum
Regularized traces and Ktheory invariants of parametric pseudodifferential operators Traces in number theory, geometry and quantum fields Aspects Math., E38 : 161177 Publisher: Friedr. Vieweg, Wiesbaden 2008[ 9] Matthias Lesch, Carolina Neira Jiménez
Classification of traces and hypertraces on spaces of classical pseudodifferential operators J. Noncommut. Geom. , 7: (2): 457498 2013[ 10] Matthias Lesch, Henri Moscovici, Markus J. Pflaum
Relative pairing in cyclic cohomology and divisor flows J. KTheory , 3: (2): 359407 2009[ 11] Luiz Hartmann, Matthias Lesch, Boris Vertman
Zetadeterminants of SturmLiouville operators with quadratic potentials at infinity J. Differential Equations , 262: (5): 34313465 2017[ 12] Matthias Lesch, Henri Moscovici
Modular curvature and Morita equivalence Geom. Funct. Anal. , 26: (3): 818873 2016[ 14] Jens Kaad, Matthias Lesch
Spectral flow and the unbounded Kasparov product Adv. Math. , 248: : 495530 2013[ 15] Matthias Lesch
A gluing formula for the analytic torsion on singular spaces Anal. PDE , 6: (1): 221256 2013[ 16] Matthias Lesch, Henri Moscovici, Markus J. Pflaum
ConnesChern character for manifolds with boundary and eta cochains Mem. Amer. Math. Soc. , 220: (1036): viii+92 2012 ISBN: 9780821872963[ 17] Jens Kaad, Matthias Lesch
A local global principle for regular operators in Hilbert C^{*}modules J. Funct. Anal. , 262: (10): 45404569 2012[ 18] Matthias Lesch
Operators of Fuchs type, conical singularities, and asymptotic methods of TeubnerTexte zur Mathematik [Teubner Texts in Mathematics] : 190 Publisher: B. G. Teubner Verlagsgesellschaft mbH, Stuttgart 1997 ISBN: 3815420970[ 19] Jochen Brüning, Matthias Lesch
On the ηinvariant of certain nonlocal boundary value problems Duke Math. J. , 96: (2): 425468 1999




1995  GerhardHess Award, German Research Foundation (DFG)  1999  Heisenberg fellowship, German Research Foundation (DFG)  2000  NSF Grant DMS 0072551 


2007  Chair in Pure Math, Loughborough University, England, UK 


Boris Vertman (2008): “The Analytic Torsion for Manifolds with Boundary and Conical Singularities”,
now Professor, University of Münster
Carolina Neira Jimenéz (2010): “Cohomology Of Classes Of Symbols And Classification Of Traces On Corresponding Classes Of Operators With Non Positive Order”,
now Assistant Professor, National University of Colombia
Batu Güneysu (2011): “On the FeymanKac formula for Schrödinger semigroups on vector bundles”,
now Postdoc, HU Berlin


 Master theses: 5, currently 3
 Diplom theses: 11
 PhD theses: 8, currently 2


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