A Morse function on a manifold decomposes the manifold into infinitely many hypersurfaces (level sets). The typical level set is smooth but a few of them have singularities. A Dirac operator on the manifold induces Dirac-type operators on these level sets. The investigator intends to prove that the index of the original operator can be recovered from invariants capturing infinitesimal behavior of this family of induced operators along finitely many of these level sets. More precisely the index will be a sum of two types of invariants: soft and hard invariants. The soft invariants are described by infinitesimal spectral flows near finitely many smooth level sets, while the hard invariants are described by the Kashiwara-Wall indices of triplets of infinite dimensional lagrangian spaces canonically determined by the singular level sets.
The Dirac type equations are generalizations of the famous Maxwell's equations of the electromagnetism. A solution of such an equation can be viewed as a sort of stationary electromagnetic wave on the Universe under investigation, known as the background manifold. The number of solutions of a Dirac equation is highly dependent on the shape of the manifold (or Universe). Morse theory is a technique of investigating the shape of a manifold by decomposing it into certain elementary pieces. The investigator intends to explain how to compute the number of solutions of a Dirac equation by studying the behavior of this equation on these elementary pieces of space.
The investigations in this project have to do with manifolds which are higher dimensional generalization of the more intuitive concept of `curved surface' such as the surface of a donut, or the surface of a planet with all its montains and valleys. Morse theory, named after the American mathematician Marston Morse, is one method of investigating manifolds. In non technical terms, this technique assumes the existence of a temperature distribution and extracts global information about the manifold from the patterns formed by the temperature fronts. The front associated to a fixed value of the temperature T is the locus of all points on the manifold where the temperature is equal to that fixed value T. There is thus one front for each value of the temeprature. As we vary the teperature, the front changes. Morse observed that most of the time the changes in the front as the temperature increases are mild but, r at certain values of the temperature, the front changes suddenly and dramatically. Such values of the temperature are called critical and the corresponding fronts are called critical fronts. The critical fronts display singularities which are, roughly speaking sharp points on the front; think of the vertex of a cone. Such points are called critical points of the temperature distribution. Morse theory is about extracting information about the original manifold by investigating these critical points. The shape of the original manifold is difficult to quantify but mathematicians have associated to manifolds numerical quantities called invariants that allow us to distinguish between various shapes. I have achieved three types of outcomes. 1. Many invariants of a manifold are obtained by counting the solutions of certain partial differential equations on that manifold. The corresponding number is clled the index of the equation. By analyzing the geometry of the critical points of a temperature distribution on a two-dimensional manifold I was able to compute the index of certain equations on that manifold called Dolbeault equations. The answers reveals beautiful and deep conections to qunatities that appear in quantum mechanics, more precisely the Maslov index. This was joint work with my former student Daniel Cibotaru, now professor in Fortaleza Brazil. 2. I have addressed the following question: how does a random distritribution of temperature look like. One result I proved is a universality result which rougly speaking says that, random temperature distributions of temeprature on different manifolds have on average the same numbe of critical points and critical values. This result rougly says that the statistics of raandom temperature distribution is blind to the shape of the manifold. However, I discovered more refined statistical invariants very sensitive to the shape of the original manifold. In fact, I proved that they are so sensitive that they can be used to probabilitically reconstruct the original shape. 3. The third outcome has its origine in the doctoral dissertation of my former student Brandon Rowekamp now assistant professor at Minnesota State University at Mankato. Together we used Morse theoretic techniques to solve a problem in computer imaging: recover a black-and-white picture given a pixelated version of it. We were both pleasantly suprised by the power of geometric ideas in solving this problem. I am personally very excited about this line of research which has forced me to go into unexpected and rather diverse parts of mathematics. The work on pixelations has given me an my former student the opportunity to glimpse into parts as science less frequented by theoretical mathematicians such as computer imaging. We began to gain some understanding on the type of questions of interest to researchers in this rather applied field and hopefully we were able to convey to computer scientists the power and effectiveness of seemingly very abstract mathematical theories.
