Prof. Dr. Matthias Lesch

E-mail: ml(at)
Phone: +49 228 73 7641
Room: 1.033
Location: Mathematics Center
Institute: Mathematical Institute
Research Areas: Research Area C (Leader)
Research Area A

Academic Career


Dr. rer. nat., University of Marburg (advisor: Manfred Breuer)

1989 - 1995

Assistant Professor (C1, later C2), University of Augsburg


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


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

Research Profile

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, zeta-determinants, 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 KK-theory.

Contribution to Research Areas

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. zeta-determinants, eta-invariants). In [5,6,7], zeta-determinants are calculated in singular one-dimensional situations motivated model operators occuring in the context of conical or hyperbolic singularities. This is motivated by the problem of extending the celebrated Cheeger-Müller 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:

KK-theory and K-theory play a prominent role in recently developed mathematical models of topological insulators. The main feature of Kasparov’s KK-theory is the intersection product (aka Kasparov product), which also plays a prominent role in the bulk-edge 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 Mesland-Rennie and Brain-Mesland-van 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 Wodzicki-Guillemin 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 (log-polyhomogeneous), also it is shown that such operators do have a heat trace asymptotics where log t-powers 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 so-called 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.

Selected Publications

[1] Jochen Brüning, Matthias Lesch
On the spectral geometry of algebraic curves
J. Reine Angew. Math. , 474: : 25--66
[2] Matthias Lesch
Operators of Fuchs type, conical singularities, and asymptotic methods
of Teubner-Texte zur Mathematik [Teubner Texts in Mathematics] : 190
Publisher: B. G. Teubner Verlagsgesellschaft mbH, Stuttgart
ISBN: 3-8154-2097-0
[3] Matthias Lesch, Henri Moscovici
Modular curvature and Morita equivalence
Geom. Funct. Anal. , 26: (3): 818--873
[4] Matthias Lesch
Divided differences in noncommutative geometry: rearrangement lemma, functional calculus and expansional formula
J. Noncommut. Geom. , 11: (1): 193--223
[5] Luiz Hartmann, Matthias Lesch, Boris Vertman
Zeta-determinants of Sturm-Liouville operators with quadratic potentials at infinity
J. Differential Equations , 262: (5): 3431--3465
[6] Matthias Lesch, Boris Vertman
Regularizing infinite sums of zeta-determinants
Math. Ann. , 361: (3-4): 835--862
[7] Matthias Lesch, Boris Vertman
Regular singular Sturm-Liouville operators and their zeta-determinants
J. Funct. Anal. , 261: (2): 408--450
[8] Jens Kaad, Matthias Lesch
Spectral flow and the unbounded Kasparov product
Adv. Math. , 248: : 495--530
[9] Jens Kaad, Matthias Lesch
A local global principle for regular operators in Hilbert C*-modules
J. Funct. Anal. , 262: (10): 4540--4569
[10] Matthias Lesch
On the noncommutative residue for pseudodifferential operators with log-polyhomogeneous symbols
Ann. Global Anal. Geom. , 17: (2): 151--187
[11] Matthias Lesch, Carolina Neira Jiménez
Classification of traces and hypertraces on spaces of classical pseudodifferential operators
J. Noncommut. Geom. , 7: (2): 457--498
[12] Alexander Gorokhovsky, Matthias Lesch
On the spectral flow for Dirac operators with local boundary conditions
Int. Math. Res. Not. IMRN (17): 8036--8051
[13] Matthias Lesch
A gluing formula for the analytic torsion on singular spaces
Anal. PDE
, 6: (1): 221--256
[14] Matthias Lesch, Henri Moscovici, Markus J. Pflaum
Connes-Chern character for manifolds with boundary and eta cochains
Mem. Amer. Math. Soc. , 220: (1036): viii+92
ISBN: 978-0-8218-7296-3
[15] Paul Kirk, Matthias Lesch
The η-invariant, Maslov index, and spectral flow for Dirac-type operators on manifolds with boundary
Forum Math. , 16: (4): 553--629
[16] Jochen Brüning, Matthias Lesch
On the η-invariant of certain nonlocal boundary value problems
Duke Math. J. , 96: (2): 425--468

Publication List

MathSciNet Publication List (external link)

ArXiv Preprint List (external link)



Gerhard-Hess Award, German Research Foundation (DFG)


Heisenberg fellowship, German Research Foundation (DFG)


NSF Grant DMS 0072551



Chair in Pure Math, Loughborough University, England, UK

Selected PhD students

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 Feyman-Kac formula for Schrödinger semigroups on vector bundles”,
now Postdoc, HU Berlin

Supervised Theses

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