Pierre Albin

Assistant professor at the University of Illinois at Urbana-Champaign
Office: Illini Hall 237
Email: palbin(at)illinois.edu

Teaching

Fall 2011 I am teaching Math 423: Elementary Differential Geometry
Spring 2012 I will teach Math 524: Linear Analysis on Manifolds

CV

Foundation Sciences Mathématiques de Paris Postdoctoral Fellow at Jussieu; Mentor: Eric Leichtnam.
IAS and Courant, 2009-2010, Joint Institutes Postdoctoral Fellow; Mentors: Jeff Cheeger and Peter Sarnak
MIT, 2005-2009, Moore instructor and NSF Postdoctoral fellow; Mentor: Richard Melrose
Stanford, 2005, PhD, Mathematics; Advisor: Rafe Mazzeo
I.T.A.M. (Mexico), 2000, Licenciatura en Matemáticas Aplicadas

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

Lie group actions

Equivariant cohomology and resolution ² with Richard Melrose extends to general group actions the simple statement: the equivariant cohomology of a space is the cohomology of the space of orbits. This is literally true only for free actions; we show that otherwise the equivariant cohomology can be computed by a de Rham-like complex on a compactification of the regular part of the orbit space. We also extend the `delocalized' cohomology of Baum, Brylinski, and MacPherson from Abelian group actions to arbitrary compact group actions.

Ricci flow on non-compact surfaces

Ricci flow and the determinant of the Laplacian on non-compact surfaces ^2,3 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.

Symmetric signature

The signature package on Witt spaces, I. Index classes ² with Eric Leichtnam, Rafe Mazzeo, and Paolo Piazza is about the signature operator with C^* algebra coefficients on a stratified manifold. 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.

The signature package on Witt spaces, II. Higher signatures ³ with Eric Leichtnam, Rafe Mazzeo, and Paolo Piazza is again about the signature operator with C^* algebra coefficients on a stratified manifold. We show that the analytic index class defined in the first paper 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.

Smooth K-theory and explicit index formulae

Fredholm realizations of elliptic symbols on manifolds with boundary ² 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 formulae ² 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 boundary ² 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")

Manifolds with hyperbolic cusps

Families index for manifolds with hyperbolic cusp singularities ² 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 surfaces ² 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 bundles ² 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.

Asymptotically hyperbolic manifolds

Renormalizing curvature integrals on Poincaré-Einstein Manifolds ¹ 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 theorem ¹ 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). Together with Rafe Mazzeo, I'm working on understanding the topological content of the general renormalized index theorem.
(Advances in Mathematics)

³ Research funded by the NSF through grant DMS-0635607002. ² Research partially funded by the NSF through a postdoctoral fellowship. ¹ Research partially funded by the NSF through grant DMS-0204730 of Rafe Mazzeo.

Lecture Notes

In Spring 2008, I taught `Introduction to analysis on non-compact manifolds,' if you'd like to see the lecture notes click here

In January 2011 I co-organized a conference at Luminy in honor of Rafe Mazzeo's 50^th birthday, Analyse Géométrique.
In April 2009 I co-organized a conference at MIT in honor of Richard Melrose's 60^th birthday, Singularities @ MIT.