JESUS ANTONIO
ALVAREZ LOPEZ

In my Ph.D. thesis and later, with a Fulbright/MEC fellowship, I studied the spectral sequence of Riemannian foliations (those with rigid transverse dynamics), showing a behavior similar to the spectral sequence of a fiber bundle. I also collaborated in its use to characterize the existence of a metric so that the leaves have zero mean curvature. Subsequently, my work evolved in various directions, with quite a few collaborators. First, hard Analysis was used to show that the heat flow along the leaves of Riemannian foliations preserves smoothness on the ambient manifold at infinite time, yielding a leafwise Hodge decomposition preserving smoothness on the ambient manifold, a trace formula for simple foliated flows without fixed points, and a description of the spectral sequence in terms of adiabatic limits. All of this was surprising by the use of operators that are only leafwise elliptic. This work attracted much interest, but more generality was needed. After solving many difficulties, we extended the trace formula to simple foliated flows with fixed points (the foliation is not Riemannian). Many new ingredients were used in this achievement, like spaces of conormal distributions, b-calculus, zeta functions, and heat invariants. This project took much of my work in the last 20 years. In the second direction, we extended many properties of Riemannian foliations to equicontinuous foliated spaces. This developed into the study of the coarse geometry of leaves, turbulent relations, graph colorings, and Cantor actions. In the third direction, we extended Witten’s analytic proof of Morse inequalities to stratified spaces, using the Dunkl harmonic oscillator. In the fourth direction, some of the techniques used with Riemannian foliations were applied to symplectic foliations. Recently, I began three new collaborations. One is to show the existence of solutions for some highly nonlinear hypoelliptic PDEs on stratified Lie groups. Another one is to prove that there is a perturbation of the leafwise heat flow for transversely affine foliations keeping transverse smoothness at infinite time. And another one about variants of the twisted Kähler-Ricci flow. In addition to this variety of directions, quite a few relevant results were achieved and published in very prestigious scientific journals, such as Amer. J. Math. 111 (1989) and 123 (2001), Trans. Amer. Math. Soc. 329 (1992), J. Differential Geom. 10 (1993), Ann. Global Anal. Geom. 10 (1992), J. Funct. Anal. 99 (1991), Geom. Funct. Anal. 10 (2000), Compositio Math. 125 (2001).