

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 One of the main technical tools in spectral geometry is the asymptotic expansion of the heat trace of an elliptic operator. In [1], we study the Laplacian on singular algebraic curves and prove a complete asymptotic expansion of the heat trace. [2] studies regularity, Fredholmness and the heat expansion for general Fuchs type differential operators. These are the natural operator occuring in the context of conical singularities.
A striking new development is the discovery of Connes and Moscovici that the second heat coefficient of the Laplacian on the noncommutative torus exhibits universal one and two variable functions with deep, not yet fully understood connections, to classical special functions. In [3] this is worked out for the Laplacian on all vector bundles (Heisenberg modules) over the noncommutative torus. More combinatorial aspects and an explanation of the universal functions in terms of divided differences is the content of [4].
Several other papers deal with more rigid spectral invariants (e.g. zetadeterminants, etainvariants). In [5,6,7], zetadeterminants are calculated in singular onedimensional situations motivated model operators occuring in the context of conical or hyperbolic singularities. This 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.  Research Area C Though I would consider myself a mathematician, I am dealing with mathematical structures which are of some relevance in Mathematical Physics:
KKtheory and Ktheory play a prominent role in recently developed mathematical models of topological insulators. The main feature of Kasparov’s KKtheory is the intersection product (aka Kasparov product), which also plays a prominent role in the bulkedge correspondence of topological insulators. The Kasparov product is of intimidating generality and its construction is intimidating as well. In the last years I have worked intensely on various aspects of the unbounded picture of the Kasparov product [8,9]. Currently, I am working with Bram Mesland on a constructive version of the intersection product in the unbounded picture, building on recent work by MeslandRennie and BrainMeslandvan Suijlekom. [8,9] were written while J. Kaad was a one year HCM Postdoc (Research Area C, mentor: M. Lesch) at the Mathematical Institute.
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 parametrices of 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 [10], 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 log tpowers occur. [11], essentially the PhD thesis of C. Neira Jimenez, gives a refined classification of residue traces trace functionals on subalgebras of the algebra of classical pseudodifferential operators.
The paper [11] 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 t to 0 and t to infinity 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.
The papers [3,4], discussed under research area A, also belong to the context of noncommutative geometry and are therefore at the borderline between A and C. 


[ 1] Jochen BrÃ¼ning, Matthias Lesch
On the spectral geometry of algebraic curves J. Reine Angew. Math. , 474: : 2566 1996[ 2] 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[ 3] Matthias Lesch, Henri Moscovici
Modular curvature and Morita equivalence Geom. Funct. Anal. , 26: (3): 818873 2016 DOI: 10.1007/s0003901603756[ 4] Matthias Lesch
Divided differences in noncommutative geometry: rearrangement lemma, functional calculus and expansional formula J. Noncommut. Geom. , 11: (1): 193223 2017 DOI: 10.4171/JNCG/1116[ 5] Luiz Hartmann, Matthias Lesch, Boris Vertman
Zetadeterminants of SturmLiouville operators with quadratic potentials at infinity J. Differential Equations , 262: (5): 34313465 2017 DOI: 10.1016/j.jde.2016.11.033[ 6] Matthias Lesch, Boris Vertman
Regularizing infinite sums of zetadeterminants Math. Ann. , 361: (34): 835862 2015 DOI: 10.1007/s0020801410787[ 7] Matthias Lesch, Boris Vertman
Regular singular SturmLiouville operators and their zetadeterminants J. Funct. Anal. , 261: (2): 408450 2011 DOI: 10.1016/j.jfa.2011.03.011[ 8] Jens Kaad, Matthias Lesch
Spectral flow and the unbounded Kasparov product Adv. Math. , 248: : 495530 2013 DOI: 10.1016/j.aim.2013.08.015[ 9] Jens Kaad, Matthias Lesch
A local global principle for regular operators in Hilbert C^{*}modules J. Funct. Anal. , 262: (10): 45404569 2012 DOI: 10.1016/j.jfa.2012.03.002[ 10] Matthias Lesch
On the noncommutative residue for pseudodifferential operators with logpolyhomogeneous symbols Ann. Global Anal. Geom. , 17: (2): 151187 1999 DOI: 10.1023/A:1006504318696[ 11] Matthias Lesch, Carolina Neira JimÃ©nez
Classification of traces and hypertraces on spaces of classical pseudodifferential operators J. Noncommut. Geom. , 7: (2): 457498 2013 DOI: 10.4171/JNCG/123[ 12] Alexander Gorokhovsky, Matthias Lesch
On the spectral flow for Dirac operators with local boundary conditions Int. Math. Res. Not. IMRN (17): 80368051 2015 DOI: 10.1093/imrn/rnu188[13] Matthias Lesch
A gluing formula for the analytic torsion on singular spaces Anal. PDE , 6: (1): 221256 2013 DOI: 10.2140/apde.2013.6.221 [ 14] 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 DOI: 10.1090/S006592662012006563[ 15] Paul Kirk, Matthias Lesch
The Î·invariant, Maslov index, and spectral flow for Diractype operators on manifolds with boundary Forum Math. , 16: (4): 553629 2004 DOI: 10.1515/form.2004.027[ 16] Jochen BrÃ¼ning, Matthias Lesch
On the Î·invariant of certain nonlocal boundary value problems Duke Math. J. , 96: (2): 425468 1999 DOI: 10.1215/S0012709499096138





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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