Pierre Albin
NSF Postdoctoral fellow, MIT
Office 4-182; (617) 258-6895
Email: pierre(at)math.mit.edu
My CV
PhD, Mathematics, Stanford, 2005
BSc, Applied Mathematics, I.T.A.M. (Mexico), 2000
This April there will be a conference at MIT in honor of Richard Melrose's 60th birthday,
Singularities @ MIT.
Research
My research is in geometric analysis, specifically
index theory
and spectral geometry on non-compact and singular spaces.
Articles
1. Renormalizing curvature integrals on Poincaré-Einstein Manifolds 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)
2. A renormalized index theorem for some complete asymptotically regular metrics:
the Gauss-Bonnet theorem 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. Fredholm realizations of elliptic symbols on manifolds with boundary with
Richard Melrose approaches index theory on asymptotically hyperbolic manifolds differently.
Whereas the previous two papers dealt with finding 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)
4. Families index for manifolds with hyperbolic cusp singularities 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)
5. A local families index formula for d-bar operators on punctured Riemann surfaces
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)
6. Relative Chern character, boundaries and index formulae 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.
7. Some index formulae on the moduli space of stable parabolic vector bundles
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.
8. Fredholm realizations of elliptic symbols on manifolds with boundary II: fibered boundary 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")
Teaching
I'm not teaching this semester.
If you'd like to see lecture notes from my course `Introduction to analysis on non-compact manifolds', click here
You can also see an OpenCourseWare webpage about an undergraduate analysis course I taught with Enno Lenzmann,
here