My research expertise lies in the field of partial differential equations on spaces with singular geometry. In particular I currently work on topics in spectral geometry, parabolic Schauder theory with applications to non-linear geometric evolution equations, in the setup of singular spaces. One fundamental aspect of my research is the microlocal approach to the elliptic theory of edge degenerate differential operators. The analysis of regular-singular Sturm-Liouville operators also takes a central place within my research interests. One other activity area results from the ansatz to define the quantum field theoretical path integral als the regularized limit of its discrete realisation, using tools from discrete geometry and approximation theory. Below I outline several illustrative examples of the various topics of my research.
Analytic Torsion on Singular Manifolds
One of the key achievements in modern spectral geometry is the proof by Cheeger and Müller of the Ray-Singer conjecture, equating analytic and Reidemeister torsions of a closed odd-dimensional manifold equipped with a flat Hermitian vector bundle. Since one of these quantities is analytic and the other combinatorial, their equality has many important applications in fields ranging from topology and number theory to mathematical physics. The advancement from the setting of smooth compact manifolds to singular spaces is one of the central challenges of modern geometric analysis. A particular class of singularities wich are accessible by analytic techniques are the incomplete edges, which include isolated conical singularities as a special case.
- Analytic torsion of a bounded generalized cone, published in Comm.Math. Phys. (2009) provided the first computation of analytic torsion in the singular setup and triggered a sequence of publications, by de Melo, Hartmann and Spreafico, where analytic torsion has then been evaluated explicitly in the special case of a cone over a sphere.
- The metric anomaly of analytic torsion on manifolds with conical singularities, with Werner Müller (2012) identifies the so-called residual term in the formula for the analytic torsion of a cone as the Brüning-Ma metric anomaly of the cone at its boundary. This proves the conjecture of Hartmann and Spreafico. Moreover we establish invariance of analytic torsion under higher order deformations of the metric near the conical tip, reducing the discussion to exact cones.
- Analytic torsion on manifolds with edges, with Rafe Mazzeo, published in Advances in Math. (2012) initiated an investigation of analytic torsion on a class of stratified spaces beyond the ones with conical singularities, namely simple edge spaces. We show that analytic torsion of a simple edge space exists and depends only on the lower orders in the asymptotic structure of $g^M$ near the singular stratum of $M$. When the dimension of the edge is odd and the Riemannian metric on the boundary fibration is an even function of $x$, analytic torsion is in fact independent of the choice of admissible edge metric.
- The metric anomaly of analytic torsion at the boundary of an even dimensional cone, Ann. Glob. Analysis (2012) investigates the geometric structure of analytic torsion on an even dimensional truncated cone. In even dimensions, analytic torsion of a cone is completely determined by the topology of its cross section and yields a Cheeger-Müller type result in a trivial way.
Refined Analytic Torsion on Manifolds with Boundary
By construction, both the analytic Ray-Singer and the combinatorial Reidemeister torsions provide canonical norms on the determinant line of cohomology. There have been various approaches to obtain a canonical construction of analytic and Reidemeister torsions as elements instead of norms of the determinant line of the cohomology. These constructions seek to refine the notion of analytic and Reidemeister torsion norms on that determinant line, which basically corresponds to fixing a complex phase in the family of complex vectors of length one. In case of the Reidemeister torsion this has been done by Farber and Turaev. Refinement of analytic torsion has been studied by Braverman and Kappeler, as well as by Burghelea and Haller, who subsequently compared the two notions.
- Refined analytic torsion on manifolds with boundary, published in Geometry and Toplogy (2009) extends the refined analytic torsion construction to compact Riemannian manifolds with boundary. The main challenge is a choice of correct boundary conditions for the odd signature operator that respect the grading of the de Rham complex. Our work triggered consecutive results by Huang and Su. An alternative independent construction on manifolds with boundary has been proposed by Huang and Lee.
- Refined torsion as analytic function on the representation variety and applications, with M. Braverman (2013) The fundamental property of the refined analytic torsion is that it defines an analytic function on the complex representation variety. We extend this result to the refined analytic torsion on manifolds with boundary, as constructed above. As an application we obtain an alternative proof of the gluing formula by Brüning and Ma for the Ray Singer analytic torsion norm for certain non-unitary representations.
- A new proof of a Bismut-Zhang formula for some class of representations, with M. Braverman (2013) Bismut and Zhang computed the ratio of the Ray-Singer and the combinatorial torsions corresponding to non-unitary representations of the fundamental group. Using the fact that the refined analytic torsion is a holomorphic function on the space of representations, we show that for certain non-unitary representations the Bismut-Zhang formula follows rather easily from the Cheeger-Mueller theorem.
Parabolic Schauder Theory, Geometric Flows
Geometric flows are non-linear parabolic differential equations which describe the evolution of geometric structures. The analysis of geometric flows has been inspired by the Hamilton's Ricci flow and has seen tremendous progress since. It has led to important applications in geometry, topology, physics and nonlinear analysis. On singular incomplete spaces, as considered above, the curvature may be unbounded and the problem of posing Ricci flow in that setting is ill-posed without further geometric conditions. Even then, Ricci flow does not admit a unique solution and need not preserve the singular structure of the initial manifold.
- Yamabe flow on manifolds with edges, with Eric Bahuaud, accepted at Mathematische Nachrichten (2013) investigates the Yamabe flows preserving a class of Riemannian manifolds with edge singularities. In that setup even short time existence of the geometric flows may not hold, with the existence theory for parabolic equations in the singular setup to be developed yet. We develop existence theory for quasilinear heat-type equations and elaborate the geometric restrictions required to make sense of the flow on edges.
- Mapping properties of the heat operator on manifolds with edges, with Eric Bahuaud and Emily Dryden (2012) studies existence and regularity of solutions to semi-linear parabolic equations on non-compact spaces. A unique feature of our analysis is the parallel discussion of complete and incomplete edge spaces.
- The Cahn-Hilliard equation and the biharmonic heat kernel on edge manifolds, submitted (2013) constructs and studies the microlocal properties of the biharmonic heat kernel, as well as existence and regularity of solutions to certain non-linear fourth order geometric flows. The discussion is closely related to the analysis by Roidos and Schrohe of the Cahn-Hilliard equation spaces with isolated conical singularies.
Edge Degenerate Boundary Value Problems
Elliptic theory of edge degenerate operators has been developed in great depth by Schulze and his collaborators. The emphasis in those papers and monographs is on the operator-symbol quantization associated to spaces with both simple and iterated edge singularities. An alternative to the elliptic theory of edge differential operators had been undertaken by Mazzeo, which studies elliptic differential edge operators using the methods of geometric microlocal analysis.
- Elliptic theory of differential edge operators II, in preparation with Rafe Mazzeo (2013) introduces edge degenerate boundary value problems and proves their Fredholmness under bijectivity of an associated symbol.
- Heat-trace asymptotics for edge Laplacians with algebraic boundary conditions, submitted (2013) introduces a class of self-adjoint algebraic boundary conditions for the Hodge Laplacian on edges, constructs the corresponding heat kernel and establishes its heat trace asymptotics. This work is closely related to the thesis of Edith Mooers.
- The exotic heat-trace asymptotics of a regular-singular operator revisited, accepted at J. Math. Physics (2013) Heat trace asymptotics of certain regular-singular operators admits unusual exotic properties, observed by Falomir, Muschietti, Pisani and Seeley, as well as by Kirsten, Loya and Park. We explain how their results alternatively follow from the general heat kernel construction by Mooers, extending her statement beyond the statement of non-polyhomogeneity of the heat kernel.
Regular Singular Sturm-Liouville Operators
Spectral geometry on manifolds with isolated conical singularities is key in the general programme of geometric analysis on singular spaces, initiated by Cheeger. Separation of variables for the Laplacian in an open neighborhood of the conical singularity leads to an infinite sum of scalar regular singular Sturm Lioville operators H. Their spectal theory is closely intertwined with the problem of determining analytic torsion of a manifold with an isolated conical singularity. The physical relevance of H stems from the fact that it arises when separation of variables is used for the radial Schrödinger operator in the Euclidean space.
- Regular singular Sturm Liouville operators and their zeta determinants, with Matthias Lesch, J. Funct. Anal. (2011) We consider Sturm-Liouville operators on $[0,1]$ with general regular-singular potentials and separated boundary conditions. We establish existence and a formula for the associated zeta-determinant in terms of the Wronski- determinant of a fundamental system of solutions adapted to the boundary conditions. This generalizes the earlier work of Lesch, which treated general regular singular potentials but only the Dirichlet boundary conditions at the singular end, and the recent results by Kirsten-Loya-Park which studied general separated boundary conditions, however only special regular singular potentials.
- Regularized sums of zeta determinants with Matthias Lesch, in prepration (2013) Even though the Hodge Laplacian decomposes into an infinite sum of scalar regular singular operators in a neighborhood of the conical singularity, there is no obvious relation between its zeta determinant and the zeta determinants of the scalar components. In this work we establish this relation in the toy model of rotation surface by a careful analysis of a new multiparameter resolvent trace expansion. We prove that the zeta-determinant of the Laplacian on a surface of revolution is given by a regularized sum of zeta-determinants for the scalar operators plus a locally computable term from the resolvent trace asymptotics.
- Multiparameter resolvent trace expansion for elliptic boundary problems, submitted (2013) extends the new observation of a polyhomogeneous multiparameter resolvent trace expansion for scalar operators from above, to certain elliptic boundary value problems with potentially many new applications and generalizations.
Discrete Quantum Field Theory
The mathematical framework of quantum field theory was introduced by Michael Atiyah und Graeme Segal and defines quantum field theory as a functor from the category of cobordisms into the category of vector spaces. The framework was originally developed to describe conformal quantum field theory, but encompasses further theories including the standard model.
The functorial attribution of a quantum field theory is formally represented by a path integral. A mathematically rigorous definition of the path integral poses a central problem of the modern mathematical physics. Two fundamental approaches to a rigorous mathematical description of the path integral are given by the constructive as well as the semi-classical (perturbative) quantum field theories.
In the constructive approach one studies a finite-dimensional approximation of the path integral, with a proof of limit existence. In the semi-classical approach the path integral is regarded as a formal asymptotic expansion in its parameter $\hbar$. The coefficients of that formal sum are described in terms of Feynman diagrams.
- Combinatorial Quantum Field Theory and Gluing Formula for Determinants with Nicolai Reshetikhin (2013) We provide a careful construction of the discrete Gaussian quantum field theory. In particular we define the combinatorial Dirichlet-to-Neumann operator and establish a gluing formula for determinants of discrete Laplacians in the quantum field theoretic framework. We relate the gluing formula to the corresponding Mayer-Vietoris formula by Burghelea, Friedlander and Kappeler for zeta-determinants of analytic Laplacians, using the approximation theory of Dodziuk. Our argument motivates existence of gluing formulas as a consequence of a gluing principle on the discrete level. We hope to provide a first step towards asymptotic expansions of determinants for discrete Laplacians in more general configurations