About

Pierre Albin was born in California and grew up in Houston and Mexico City. He obtained his undergraduate degree in applied mathematics and worked in financial risk management in Mexico before pursuing his PhD in mathematics at Stanford University, which he completed in 2005 under the supervision of Rafe Mazzeo. Following his doctoral studies, he conducted postdoctoral research at MIT, NYU/IAS, and in Paris before joining the University of Illinois at Urbana-Champaign in 2011. His research is centered on geometric analysis, with particular interests in analytic representations of topological invariants, analysis on non-compact or singular spaces, spectral geometry, heat kernels, and Dirac operators.

Research topics

• Mathematical analysis
• Pure mathematics
• Mathematics
• Geometry
• Anatomy

Selected publications

• Smooth atlas stratified spaces, K-Homology Orientations, and Gysin maps

  ArXiv.org · 2025-05-20

  preprintOpen access1st authorCorresponding

  We introduce smooth atlas stratified spaces. We show that this class is closed under cartesian products; consequently, it is possible to define fiber bundles of smooth atlas stratified spaces. We describe the resolution of such a space to a manifold with fibered corners and use this result in order to prove that the class of smooth atlas stratified spaces coincides with that of Thom-Mather stratified spaces. We then consider Witt pseudomanifolds (such as singular complex algebraic varieties) where it is well-known that a bordism invariant signature is available and equal to the Fredholm index of a realization of the signature operator. To each oriented fiber bundle of stratified spaces, with Witt fibers, we assign a class in bivariant KK-theory (with 2 inverted). Kasparov multiplication by this element defines a Gysin map in analytic K-homology and one of our main results is that this Gysin map preserves the analytic signature class of Witt spaces. We prove in fact a more general result: functoriality for fiber bundles in the sense that if one fiber bundle is the composition of two others then the KK-class of the former is the Kasparov product of the classes of the latter. We also discuss this result for other Dirac-type operators satisfying an analytic Witt condition, for example the spin-Dirac operator on a fibration of psc-Witt spin pseudomanifolds. We next define the analytic Gysin map associated to an oriented normally non-singular inclusion of Witt spaces and prove that it also preserves the signature class. Finally, we relate the analytic signature class of a Witt space with the topological Siegel-Sullivan orientation. Specifically we show that if one applies the inverse of the second Adams operation to the Sullivan orientation and complexifies then one obtains our analytic signature class under the natural identification between analytic and topological K-homology.

  PublisherOA PDFDOI

• The index formula for families of Dirac type operators on pseudomanifolds

  Journal of Differential Geometry · 2023 · 19 citations

  1st authorCorresponding
  • Mathematics
  • Pure mathematics
  • Mathematical analysis

  We study families of Dirac-type operators, with compatible perturbations, associated to wedge metrics on stratified spaces. We define a closed domain and, under an assumption of invertible boundary families, prove that the operators are self-adjoint and Fredholm with compact resolvents and trace-class heat kernels. We establish a formula for the Chern character of their index.

  PublisherDOI

• A Cheeger–Müller theorem for manifolds withwedge singularities

  Analysis & PDE · 2022-06-10 · 6 citations

  articleOpen access1st authorCorresponding

  We study the spectrum and heat kernel of the Hodge Laplacian with coefficients in a flat bundle on a closed manifold degenerating to a manifold with wedge singularities. Provided the Hodge Laplacians in the fibers of the wedge have an appropriate spectral gap, we give uniform constructions of the resolvent and heat kernel on suitable manifolds with corners. When the wedge manifold and the base of the wedge are odd dimensional, this is used to obtain a Cheeger-M\"uller theorem relating analytic torsion with the Reidemeister torsion of the natural compactification by a manifold with boundary.

  PublisherOA PDFDOI

• Stratified surgery and K-theory invariants of the signature operator

  Annales Scientifiques de l École Normale Supérieure · 2022-01-01 · 4 citations

  articleOpen access1st authorCorresponding

  In the works of Higson-Roe the fundamental role of the signature as a homotopy and bordism invariant for oriented manifolds is the starting point for an investigation of the relationships between analytic and topological invariants of smooth orientable manifolds. The signature and related K-theory invariants, primary and secondary, are used to define a natural transformation between the (Browder-Novikov-Sullivan-Wall) surgery exact sequence and a long exact sequence of C*-algebra K-theory groups.In recent years the primary signature invariants have been extended from closed oriented manifolds to a class of stratified spaces known as L-spaces or Cheeger spaces. In this paper we showt hat secondary invariants, such as the rho-class, also extend from closed manifolds to Cheeger spaces. We give a rigorous account of a surgery exact sequence for stratified spaces originally introduced by Browder-Quinn and obtain a natural transformation analogous to that of Higson-Roe. We also discuss geometric applications.

  PublisherOA PDFDOI

• Construction of nonlinear quasimodes near elliptic periodic orbits

  UNC Libraries · 2021-08-28

  articleOpen access

  We consider the nonlinear Schroedinger equation on a compact manifold near an elliptic periodic geodesic. Using a geometric optics construction, we construct quasimodes to a nonlinear stationary problem which are highly localized near the periodic geodesic. We show the nonlinear Schroedinger evolution of such a quasimode remains localized near the geodesic, at least for short times.

  PublisherDOI

• Resolvent, heat kernel, and torsion under degeneration to fibered cusps

  Memoirs of the American Mathematical Society · 2021 · 13 citations

  1st authorCorresponding
  • Mathematics
  • Pure mathematics
  • Mathematical analysis

  Manifolds with fibered cusps are a class of complete non-compact Riemannian manifolds including many examples of locally symmetric spaces of rank one. We study the spectrum of the Hodge Laplacian with coefficients in a flat bundle on a closed manifold undergoing degeneration to a manifold with fibered cusps. We obtain precise asymptotics for the resolvent, the heat kernel, and the determinant of the Laplacian. Using these asymptotics we obtain a topological description of the analytic torsion on a manifold with fibered cusps in terms of the R-torsion of the underlying manifold with boundary.

  PublisherOA PDFDOI

• Poincaré-Lovelock metrics on conformally compact manifolds

  Advances in Mathematics · 2020-03-16 · 1 citations

  preprintOpen access1st authorCorresponding
  PublisherOA PDFDOI

• Sub-Riemannian Limit of the Differential Form Heat Kernels of Contact Manifolds

  International Mathematics Research Notices · 2020-09-15 · 5 citations

  articleOpen access1st author

  Abstract We study the behavior of the heat kernel of the Hodge Laplacian on a contact manifold endowed with a family of Riemannian metrics that blow-up the directions transverse to the contact distribution. We apply this to analyze the behavior of global spectral invariants such as the $\eta $-invariant and the determinant of the Laplacian. In particular, we prove that contact versions of the relative $\eta $-invariant and the relative analytic torsion are equal to their Riemannian analogues and hence topological.

  PublisherOA PDFDOI

• Compactification of semi-simple Lie groups

  arXiv (Cornell University) · 2019-10-07

  preprintOpen access1st authorCorresponding

  We discuss the `hd-compactification' of a semi-simple Lie group to a manifold with corners; it is the real analog of the wonderful compactification of deConcini and Procesi. There is a 1-1 correspondence between the boundary faces of the compactification and conjugacy classes of parabolic subgroups with the boundary face fibering over two copies of the corresponding flag variety with fiber modeled on the (compactification of the) reductive part. On the hd-compactification Harish-Chandra's Schwartz space is identified with a space of conormal functions of rapid-logarithmic decay relative to square-integrable functions.

  PublisherOA PDFDOI

• Sub-Riemannian limit of the differential form heat kernels of contact manifolds

  arXiv (Cornell University) · 2019-12-05 · 2 citations

  preprintOpen access1st authorCorresponding

  We study the behavior of the heat kernel of the Hodge Laplacian on a contact manifold endowed with a family of Riemannian metrics that blow-up the directions transverse to the contact distribution. We apply this to analyze the behavior of global spectral invariants such as the eta-invariant and the determinant of the Laplacian. In particular we prove that contact versions of the relative eta-invariant and the relative analytic torsion are equal to their Riemannian analogues and hence topological.

  PublisherOA PDFDOI

Recent grants

• Index Theory on Singular Spaces

  NSF · $148k · 2011–2015

• Index Theory on Singular Spaces

  NSF · $200k · 2017–2023

• PostDoctoral Research Fellowship

  NSF · $108k · 2006–2010

Frequent coauthors

• Frédéric Rochon

  25 shared

• Éric Leichtnam

  Sorbonne Université

  22 shared

• Rafe Mazzeo

  19 shared

• Hans Christianson

  18 shared

• Laurent Thomann

  Institut Élie Cartan de Lorraine

  18 shared

• Jeremy L. Marzuola

  18 shared

• David A. Sher

  DePaul University

  17 shared

• Paolo Piazza

  Sapienza University of Rome

  16 shared