Profile
Profile

Prof. Dr. Matthias Lesch

E-mail: ml(at)matthiaslesch.de
Phone: +49 228 73 7641
Homepage: http://www.math.uni-bonn.de/people/lesch/
Room: 1.033
Location: Mathematics Center
Institute: Mathematical Institute
Research Areas: Research Area C (Leader)
Research Area A
Date of birth: 24.Nov 1961

Academic Career

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

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
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 eta-invariant 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 (non-Dirac type) elliptic differential operators.
The main technical tool of local index theory, the heat trace, leads to more rigid spectral invariants (zeta-determinants) which are also of certain interest in quantum physics. In [5], zeta-determinants are calculated in a singular one-dimensional situation by employing classical techniques from the theory of Sturm-Liouville operators. This paper (and its predecessors), though one-dimensional, could not have been written without my background on the heat equation. It 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. An upcoming paper with Vertman will shed some new light on the conical case.
An important problem is to have criteria for essential self-adjointness (quantum completeness). [6] deals with this problem for Hamiltonian systems (Sturm-Liouville 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 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 [7], 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 <br>log t-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. Neira-Jimenez, gives a refined classification of trace functionals on subalgebras of the algebra of classical pseudodifferential operators. A recent joint paper with Carolina Neira-Jimenez gives a refined classification of trace functionals on subalgebras.
The paper with Moscovici and Pflaum “Connes-Chern 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 so-called JLO version is defined in terms of (a generalization of) the heat trace. Its limits as t<br>to 0 and t<br>to<br>infty 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].

Selected Publications

[1] Jochen Brüning, Matthias Lesch
On the spectral geometry of algebraic curves
J. Reine Angew. Math. , 474: : 25--66
1996
[2] 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
2004
[3] Jochen Brüning, Matthias Lesch
On boundary value problems for Dirac type operators. I. Regularity and self-adjointness
J. Funct. Anal. , 185: (1): 1--62
2001
[4] Bernhelm Booß -Bavnbek, Matthias Lesch, Chaofeng Zhu
The Calderón projection: new definition and applications
J. Geom. Phys. , 59: (7): 784--826
2009
[5] Matthias Lesch, Boris Vertman
Regular singular Sturm-Liouville operators and their zeta-determinants
J. Funct. Anal. , 261: (2): 408--450
2011
[6] Matthias Lesch, Mark Malamud
On the deficiency indices and self-adjointness of symmetric Hamiltonian systems
J. Differential Equations , 189: (2): 556--615
2003
[7] Matthias Lesch
On the noncommutative residue for pseudodifferential operators with log-polyhomogeneous symbols
Ann. Global Anal. Geom. , 17: (2): 151--187
1999
[8] Matthias Lesch, Henri Moscovici, Markus J. Pflaum
Regularized traces and K-theory invariants of parametric pseudodifferential operators
Traces in number theory, geometry and quantum fields
Aspects Math., E38 : 161--177
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): 457--498
2013
[10] Matthias Lesch, Henri Moscovici, Markus J. Pflaum
Relative pairing in cyclic cohomology and divisor flows
J. K-Theory , 3: (2): 359--407
2009
[11] Luiz Hartmann, Matthias Lesch, Boris Vertman
Zeta-determinants of Sturm-Liouville operators with quadratic potentials at infinity
J. Differential Equations , 262: (5): 3431--3465
2017
[12] Matthias Lesch, Henri Moscovici
Modular curvature and Morita equivalence
Geom. Funct. Anal. , 26: (3): 818--873
2016
[14] Jens Kaad, Matthias Lesch
Spectral flow and the unbounded Kasparov product
Adv. Math. , 248: : 495--530
2013
[15] Matthias Lesch
A gluing formula for the analytic torsion on singular spaces
Anal. PDE , 6: (1): 221--256
2013
[16] 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
2012
ISBN: 978-0-8218-7296-3
[17] Jens Kaad, Matthias Lesch
A local global principle for regular operators in Hilbert C*-modules
J. Funct. Anal. , 262: (10): 4540--4569
2012
[18] 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
1997
ISBN: 3-8154-2097-0
[19] Jochen Brüning, Matthias Lesch
On the η-invariant of certain nonlocal boundary value problems
Duke Math. J. , 96: (2): 425--468
1999

Publication List

MathSciNet Publication List (external link)

Awards

1995

Gerhard-Hess Award, German Research Foundation (DFG)

1999

Heisenberg fellowship, German Research Foundation (DFG)

2000

NSF Grant DMS 0072551

Offers

2007

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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