Dr. Kevin David Corlette

Personal

Prefix: Dr.
First Name: Kevin
Middle Name: David
Last Name: Corlette

Summary

Kevin Corlette is co-chair of the science advisory committee of the Mathematicial Sciences Research Institute.

Biography

Employment: Professor and Chair of the Department of Mathematics University of Chicago

Kevin Corlette is co-chair of the science advisory committee of the Mathematicial Sciences Research Institute. He is also a teacher extraordinaire (algebra and topology) at the University of Chicago. There is a note atesting to his teaching at http://home.uchicago.edu/~nacecire/corlette.html. His establishment of the generalization Siu's Kodaira-Bochner type identity to the case of a parallel differential form on the domain, was applied to the quarternionic and Cayley plane cases and "revitalized the subject."

RESEARCH

Lie Group Theory, Differential Geometry and Mathematical Physics. "I am particularly interested in Kahler geometry and locally symmetric spaces, as well as systems of partial differential equations with geometric meaning, such as the harmonic map and Yang-Mills equations."

At the 1990 Geometry Festival his talk was "Harmonic maps and geometric superrigidity." Here is the abstract from the 1994 International Congress of Mathematicians.

Harmonic maps, rigidity and Hodge theory

Harmonic maps can, in many cases, be viewed as analogues of harmonic 1-forms. As such, techniques from Hodge theory can often be carried over to this setting. This lecture will discuss some of the consequences of arguments of this type.

A harmonic map is an extremal for an energy functional on the space of maps from a Riemannian manifold $M$ into another Riemannian manifold $N$ (or, more generally, $N$ may be a metric space with suitable properties). In the case of a map $f\colon M\to N$ between manifolds, one defines the energy to be the $L^2$-norm of the differential $df$; the Euler-Lagrange equation satisfied by an extremal takes the form $D^*df=0$, where $D$ is the exterior derivative associated with the pullback of the Levi-Civita connection on $TN$, and $df$ is regarded as a 1-form with values in $f^*TN$. Thus, the condition that $f$ is harmonic is precisely the condition that its differential is a harmonic 1-form with values in $f^*TN$.

One of the basic facts of Hodge theory is that a harmonic 1-form on a compact Kähler manifold is the sum of a holomorphic 1-form and its complex conjugate. The analogous fact for harmonic maps was discovered by Siu: if $f\colon M\to N$ is a harmonic map from a compact Kähler manifold to a Riemannian manifold with nonpositive complex sectional curvature, then $\partial f$ is holomorphic in a suitable sense. Thi is one of the starting points for a nonabelian Hodge theory for compact Kähler manifolds, reported on by Simpson at the previous ICM. Recent work by A. Reznikov has used these ideas to prove a conjecture of Bloch which asserts that the Cheeger-Chern-Simons classes of fact vector bundles over smooth complex projective varieties are torsion.

In another direction, one can consider harmonic 1-forms on a compact locally symmetric space with nonpositive sectional curvature. Under certain hypotheses, Matsushima proved vanishing of such forms; analogoues for forms with values in certain bundles were proved by Weil, Calabi and others. Analogues of these results can be found for harmonic maps from appropriate symmetric spaces into Riemannian manifolds with nonpositive curvature in a suitably strong sense. The author was able to give a proof of superrigidity over the reals for lattices in the groups ${\rm Sp}(n,1)$ and $F^{-20}_4$ using similar ideas in a nonlinear setting, extending results of Margulis to these rank one groups. Gromov and Schoen developed a theory of harmonic maps into singular spaces, and used this to prove superrigidity in the $p$-adic case. As a consequence, they were able to prove arithmeticity of lattices in these groups. More recently, Mok-Siu-Yeung and Jost-Yau showed how to apply harmonic map techniques in the higher rank case as well, recovering many of Margulis' results in that case. These results also give geometric forms of superrigidity, and lead in particular to metric rigidity theorems for locally symmetric spaces. In the higher rank case, this leads to a new proof of Gromov's metric rigidity theorem. In yet another direction, Korevaar and Schoen have developed a vast generalization of the Gromov-Schoen theory of harmonic maps into singular spaces. Their theory allows one to treat harmonic maps into length spaces with nonpositive curvature, a category of spaces which includes infinite-dimensional Riemannian manifolds of nonpositive curvature, ${\bf R}$-buildings, and spaces of Riemannian metrics. In another direction, Simpson and the author have used the theory of harmonic maps into trees to study the question of when a representation of the fundamental group of
a complex quasiprojective variety in ${\rm SL}(2,{\bf C})$ arises as a pullback by a holomorphic map into a Hilbert modular variety.

References: Math Reviews; communications with Dr. Corlette

SELECTED PUBLICATIONS

  1. Corlette, Kevin; Hernández Lamoneda, Luis; Iozzi, Alessandra A vanishing theorem for the tangential de Rham cohomology of a foliation with amenable fundamental groupoidGeom. Dedicata 103 (2004), 205--223.
  2. Corlette, Kevin; Wald, Robert M. Morse theory and infinite families of harmonic maps between spheres. Comm. Math. Phys. 215 (2001), no. 3, 591--608.
  3. Corlette, Kevin; Iozzi, Alessandra. Limit sets of discrete groups of isometries of exotic hyperbolic spaces. Trans. Amer. Math. Soc. 351 (1999), no. 4, 1507--1530.
  4. Jaume Amorós, Marc Burger, Kevin Corlette, Dieter Kotschick, Domingo Toledo. Fundamental Groups of Compact Kähler Manifolds. Mathematical Surveys and Monographs, 44. American Mathematical Society 1996.
  5. Corlette, KevinHarmonic maps, rigidity, and Hodge theory. Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich,1994), 465--471, Birkhäuser, Basel, 1995.
  6. Corlette, Kevin; Zimmer, Robert J. Superrigidity for cocycles and hyperbolic geometry. Internat. J. Math. 5 (1994), no. 3, 273--290.
  7. Kevin Corlette. Nonabelian Hodge theory. Differential geometry: geometry in mathematical physics and related topics(Los Angeles, CA, 1990), 125--144, Proc. Sympos. Pure Math., 54, Part 2, Amer. Math. Soc., Providence, RI, 1993.
  8. Kevin CorletteArchimedean superrigidity and hyperbolic geometry. Ann. of Math. (2) 135 (1992), no. 1, 165-182.
  9. Kevin Corlette. Rigid representations of Kählerian fundamental groups. J. Differential Geom. 33 (1991), no. 1, 239--252.
  10. Corlette, Kevin. Immersions with bounded curvature. Geom. Dedicata 33 (1990), no. 2, 153--161.
  11. Kevin CorletteHausdorff dimensions of limit sets I. Invent. Math. 102 (1990), no. 3, 521--541.
  12. Corlette, Kevin Flat $G$-bundles with canonical metrics. J. Differential Geom. 28 (1988), no. 3, 361--382.
  13. Corlette, KevinGauge theory and representations of Kähler groups. Geometry of group representations (Boulder, CO, 1987), 107--124, Contemp. Math., 74, Amer. Math. Soc., Providence, RI, 1988.

 

At the 1990 Geometry Festival his talk was "Harmonic maps and geometric superrigidity." Here is the abstract from the 1994 International Congress of Mathematicians.

 

Lifetime

Year Born: 1960