This NSF grant supported the PI's research in the field of geometric analysis. This is the study of problems of geometric nature (i.e. how a certain high-dimensional space is curved), using the tools of analysis and differential equations. The fundamental laws of physics are described, in the language of mathematics, by differential equations. Understanding the behavior of solutions to differential equations is the key to unraveling the mystery of the geometry and structure of the universe.

One of the main themes of the research which has been carried out is the geometry of Calabi-Yau spaces, which have become of central importance to the study of realistic string theory models. In a nutshell, string theorists believe that our physical world is not four-dimensional (three spatial directions plus time) but rather ten-dimensional. The remaining six dimensions are extremely small, so that we don't normally perceive them, but are crucial for understanding elementary particles. These six dimensions together form a (tiny) geometric space, which is a Calabi-Yau space. Physically, these spaces capture essential features of particle physics and cosmology, and understanding their geometry will push string  theorists closer to the goal of making contact with observations.

Of special interest is the study of degenerations of Calabi-Yau spaces, which is a key ingredient in the theory of mirror symmetry (a mysterious duality between families of Calabi-Yau spaces). Understanding how Calabi-Yau spaces can degenerate leads not only to a broadening of our mathematical knownledge but also to applications in high energy physics. The findings of the research that the PI has done have greatly clarified the possible degenerations of Calabi-Yau manifolds, and have shed some light on the mechanisms that underlie mirror symmetry.

Another main strand of the research that has been performed is the study of complex manifolds, which are geometric spaces which are described using the complex numbers. Complex manifolds are ubiquitous objects in mathematics, and have wideranging applications in physics and engineering. The types of questions that the PI studied are how to find the "best" shape of a given complex manifold, and how these shapes vary when we change the underlying space. Perhaps the most spectacular result obtained under the support of this grant is a proof of a 1984 conjecture of Gauduchon on constructing such best shapes on all complex manifolds.

The research funded by this grant has resulted in several publications in leading international mathematical journals, dissemination of the research findings at international conferences, workshops and seminars at Universitites, training of graduate students through informal reading seminars and graduate courses, mentoring of postdocs, and collaboration with mathematicians at institutions in the US and abroad. Furthermore, the PI has co-organized two conferences and three workshops at his institution, two thematic periods, two workshops at other institutions and one Special Session at a Sectional AMS Meeting in Chicago. A sizable portion of the speakers for these events was chosen among graduate students, early career mathematicians, and underrepresented groups.

Last Modified: 09/03/2016
Modified by: Valentino Tosatti