Research
My research is in geometric analysis. I am particularly interested in analytic representations of topological invariants, analysis on non-compact or singular spaces, spectral geometry, heat kernels, and Dirac operators.Articles
Index theory on stratified spaces
The index of Dirac operators on incomplete edge spaces4,5 with Jesse Gell-Redman establishes a formula for the index of a spin Dirac operator on a stratified space with a single singular stratum. We excise a tube around the singularity and impose the boundary conditions of Atiyah-Patodi-Singer, then we study the limit as the size of the tube shrinks. It turns out that these domains do not converge, but the index of the Dirac operator on the manifold with boundary is eventually equal to the index of the Dirac operator on the singular space.
( SIGMA )
The index formula for families of Dirac type operators on pseudomanifolds5,6 with Jesse Gell-Redman carries out the heat equation proof of the families index theorem for Dirac-type operators on pseudomanifolds. We assume that the operators satisfy a `Witt assumption', but we allow for pseudodifferential perturbations so this includes any Dirac-type operator with a Fredholm domain. In the process we write out the edge calculus and wedge heat calculus on pseudomanifolds. Our final formula involves the Bismut-Cheeger J-forms instead of their η forms.
Signatures of singular spaces
The signature package on Witt spaces3 with Eric Leichtnam, Rafe Mazzeo, and Paolo Piazza is about the signature operator with C* algebra coefficients on a stratified manifold satisfying the topological `Witt' condition. Although these spaces are singular, Jeff Cheeger showed that the L2 cohomology of certain natural metrics satisfies Poincaré duality and in fact is isomorphic to intersection cohomology. We generalize another of his results, namely that the signature operator is Fredholm, to the context of C* algebra coefficients. We show that its analytic index class is equal to a topologically defined signature, due to Markus Banagl. In contrast to the case of closed manifolds, we deduce homotopy invariance of the topological signature from that of the analytic signature instead of the other way around. We also define higher signatures on Witt spaces and show that the strong Novikov conjecture implies their homotopy invariance, just like on closed manifolds.( Annales de l'ENS )
Hodge theory on Cheeger spaces4 with Eric Leichtnam, Rafe Mazzeo, and Paolo Piazza generalizes Cheeger's ideal boundary conditions to a large class of stratified spaces. We construct L2 de Rham and Hodge cohomology theories and show that they coincide.
( Crelle's Journa l)
The Novikov conjecture on Cheeger spaces4 with Eric Leichtnam, Rafe Mazzeo, and Paolo Piazza establishes that L2 de Rham cohomology is stratified homotopy invariant, even with C* algebra coefficients, and establishes the Novikov conjecture for Cheeger spaces with appropriate fundamental groups.
( Journal of Noncommutative Geometry )
Refined intersection homology on non-Witt spaces4 with Markus Banagl, Eric Leichtnam, Rafe Mazzeo, and Paolo Piazza gives a purely topological description of L2 de Rham cohomology with Cheeger ideal boundary conditions, extending Banagl's self-dual sheaves and refining the intersection homology of Goresky and MacPherson.
( Journal of Topology and Analysis )
On the Hodge theory of stratified spaces5 is a survey of the four papers above with a detailed discussion of the resolution of a stratified space. It also includes a discussion of how Brylinsky's de Rahm description of intersection homology is natural on the resolution of a stratified space. The final section describes the Borel-Serre compactification of a locally symmetric space and its resolution to a manifold with corners with an iterated fibration structure.
( Proceedings of "Hodge Theory and L2-Cohomology" )
Stratified surgery and K-theory invariants of the signature operator5,6 with Paolo Piazza takes advantage of the invariance of the signature of Witt or Cheeger spaces under appropriate cobordisms and homotopies to show that Higson-Roe's mapping surgery to analysis works on singular spaces. We use the surgery sequence of Browder-Quinn and include a proof that the sequence is exact.
Analytic torsion on manifolds with wedge singularities
A Cheeger-Müller theorem for manifolds with wedge singularities6 with Frédéric Rochon and David Sher establishes a Cheeger-Müller theorem for spaces with a wedge singularity.Analytic torsion on manifolds with hyperbolic cusps
Resolvent, heat kernel and torsion under degeneration to fibered cusps4,5 with Frédéric Rochon and David Sher establishes a Cheeger-Müller theorem on a class of non-compact manifolds that includes most locally symmetric spaces of rank one. We establish two topological interpretations of the analytic torsion, one in terms of a compactification to a manifold with boundary and the other in terms of a singular compactification to a stratified space. We work with unimodular representations of the fundamental group whose associated cohomology satisfies an `acyclic at infinity' condition.
( Memoirs of the AMS )
Analytic torsion and R-torsion of Witt representations on manifolds with cusps4,5 with Frédéric Rochon and David Sher specializes our Cheeger-Müller theorem from fibered cusps to non-compact manifolds whose ends are asymptotically like hyperbolic cusps. We work with unimodular representations of the fundamental group whose associated cohomology satisfies the well-known `Witt condition'.
( Duke )
Inverse problems
Inverse Boundary Problems for Systems in Two Dimensions3 with Colin Guillarmou, Leo Tzou, and Gunther Uhlmann considers a Dirac operator or Laplacian on a surface with boundary. We show that if you know the Cauchy data on the boundary you can recover the operator, up to gauge transformations fixing the boundary.( Annales Henri Poincaré )
Compactness of relatively isospectral sets of surfaces via conformal surgeries4 with Clara Aldana and Frédéric Rochon looks at the famous question `Can one hear the shape of a drum?', but for non-compact drums. We propose a new version of isospectral that generalizes those in the literature, and we show that isophonic drums form a compact set.
( Journal of Geometric Analysis )
Nonlinear Schrödinger equation
Nonlinear quasimodes near elliptic periodic geodesics3 with Hans Christianson, Jeremy Marzuola, and Laurent Thomann shows that you can construct an almost-eigenfunction of the NLS that localize near a periodic geodesic.( Physica D: Nonlinear Phenomena )
Lie group actions
Resolution of smooth group actions2,3 with Richard Melrose constructs a smooth model for the orbit space of a group action. If the action is free, the orbit space is a smooth manifold. We show that the orbit space of a general action is a smooth manifold with corners. An application to equivariant cohomology is currently under revision.( Proceedings of "Spectral Theory and Geometric Analysis" )
Compactification of SL(2)6 with Panagiotis Dimakis and Richard Melrose introduces a `hd-compactification' of SL(2) and gives an geometric approach to the theorems of A Second Adjoint Theorem for SL(2,R) by Tyrone Crisp and Nigel Higson
Ricci flow on non-compact surfaces
Ricci flow and the determinant of the Laplacian on non-compact surfaces2,3 with Clara Aldana and Frédéric Rochon considers non-compact surfaces of finite topology whose metrics either decay like a hyperbolic cusp or expand like a hyperbolic funnel. We use renormalized integrals to define the determinant of the Laplacian and then show the analogue of a famous theorem of Osgood, Phillips, and Sarnak: the maximum value of the determinant occurs at constant curvature metrics. Our tool is a Polyakov formula and the Ricci flow. We prove long time convergence extending the result of Ji, Mazzeo, and Šešum to the case of infinite area.( Communications in PDE )
Smooth K-theory and explicit index formulae
Fredholm realizations of elliptic symbols on manifolds with boundary2 with Richard Melrose approaches index theory on asymptotically hyperbolic manifolds differently. Whereas before I had looked for indices of non-Fredholm Dirac-type operators, this paper answers the question: How restrictive is Fredholmness on the principal symbol? We compute some `smooth K-theory' groups and show that the answer is the same for asymptotically hyperbolic manifolds, asymptotically Euclidean manifolds, and manifolds with boundary. Namely, the Atiyah-Bott obstruction must vanish.( Crelle's Journal )
Relative Chern character, boundaries and index formulae2 with Richard Melrose returns to index theory on asymptotically hyperbolic manifolds. Previously we had described the index as a map in K-theory, now we wanted an explicit formula for the Chern character of the index bundle. We were able to write down a formula for general Fredholm pseudodifferential operators, involving only the model operators in the interior and at the boundary, by eschewing the usual description of relative cohomology and adapting a formula of Boris Fedosov. An appendix includes an improvement over the renormalized trace of Richard Melrose and Victor Nistor in that the resulting trace-defect formula has only half as many terms.
( Journal of Topology and Analysis )
Fredholm realizations of elliptic symbols on manifolds with boundary II: fibered boundary2 with Richard Melrose uses an indirect approach to compute the smooth K-theory of pseudodifferential operators associated with complete metrics with asymptotic `edges'. A direct approach to these groups would use constructions similar to those occuring in C*-algebra K-theory, but these constructions can not be done smoothly within this calculus (essentially because of a lack of commutativity `at infinity'). We show that a particular degeneration of the geometry at infinity takes these operators to a better behaved calculus whose smooth K-theory groups were computed by Richard Melrose and Frederic Rochon. That computation, together with the results of our previous paper, allow us to compute the groups we are interested in.
( Proceedings of "Motives, Quantum Field Theory, and Pseudodifferential Operators" )
Index theory on manifolds with hyperbolic cusps
Families index for manifolds with hyperbolic cusp singularities2 with Frédéric Rochon improves an index theorem of Vaillant for Dirac-type operators on manifolds with fibered hyperbolic cusps. We improve the theorem by extending it to families and allowing perturbations by smoothing operators. The latter extension is useful because Fredholm perturbations of Dirac-type operators can sometimes be used to generate smooth K-theory groups, in which case solving the index problem for these operators gives a solution of the index problem for all Fredholm operators.( International Mathematics Research Notices )
A local families index formula for d-bar operators on punctured Riemann surfaces2 with Frédéric Rochon specializes our families index theorem to natural families of d-bar operators on the Teichmüller space of Riemann surfaces of a fixed genus and number of cusps. After identifying the terms in the formula, we recover a formula of Leon Takhtajan and Peter Zograf. for the curvature of the associated determinant line bundle. This article also shows that the determinant defined by renormalized zeta-functions is essentially the same as the determinant defined by the Selberg zeta function, when the latter makes sense.
( Communications in Mathematical Physics )
Some index formulae on the moduli space of stable parabolic vector bundles2 with Frédéric Rochon specializes our families index theorem to natural families of d-bar operators on the moduli space of stable parabolic vector bundles. We identify the terms in the formula for the universal parabolic bundle and for its bundle of endomorphisms. In the latter case, our formula implies one of Leon Takhtajan and Peter Zograf. for the curvature of the associated determinant line bundle. We include a discussion of the short-time expansion of the renormalized trace of the heat kernel for manifolds with fibered hyperbolic cusps and explain how the renormalization produces unexpected log-terms.
( Journal of the Australian Mathematical Society )
Asymptotically hyperbolic manifolds
Renormalizing curvature integrals on Poincaré-Einstein Manifolds1 constitutes the first half of my thesis. It compares different ways of renormalizing integrals and shows that, in the usual circumstances, they give the same answer. It shows that scalar Riemannian invariants have well-defined renormalized integrals in this context, and it extends the Gauss-Bonnet formula to these manifolds via renormalization.( Advances in Mathematics )
A renormalized index theorem for some complete asymptotically regular metrics: the Gauss-Bonnet theorem1 constitutes the second half of my thesis. It shows that the Gauss-Bonnet formula is a particular case of a renormalized Atiyah-Singer index theorem. The full theorem requires a precise description of the heat kernel including the `even-ness' of its expansion at the boundary (at infinity).
( Advances in Mathematics )
Poincare-Lovelock metrics on conformally compact manifolds6 shows that conformally compact metrics that satisfy any Lovelock equation have the same asymptotics as those satisfying an Einstein equation. The Lovelock equations are generalizations of the Einstein equation in dimensions greater than four. In some sense they are as natural as the Einstein equation.