The research topics of this project are motivated by questions in geometry and topology of low dimensional manifolds and mathematical physics. Firstly, the P.I. plans to study the deformations of calibrated submanifolds in G_2 and Spin(7) manifolds. In joint work with S. Akbulut, the P.I. studied complex associative and complex Cayley submanifolds and showed that the moduli space of these submanifolds is smooth without any obstructions to such deformations. She also introduced the mathematical definitions of mirror Calabi-Yau and G_2 manifolds. This project aims to follow these footsteps and to obtain a compactification of these moduli spaces and to study the mirror dualities in Calabi-Yau and G_2 manifolds. These will lead to the construction of new counting invariants and provide a better understanding of the mirror symmetry phenomenon. In the second part, the P.I. plans to continue her work in noncompact G_2 manifolds. In joint work with D. Joyce, the P.I. studied deformations of asymptotically cylindrical coassociative submanifolds and their Topological Quantum Field Theories. Also, in recent joint work with C. Robles, she studied the Cartan-Kahler Theory for associative and Cayley embeddings into G_2 and Spin(7) manifolds. She plans to use these results to construct new examples of G_2 and Spin(7) manifolds. Similarly, special Lagrangian submanifolds of Calabi-Yau manifolds are expected to give analogous Topological Quantum Field Theories. Understanding the special Lagrangian moduli spaces inside degenerating Calabi-Yau manifolds will provide a rigorous framework for the Floer homology program. Therefore, the P.I. plans to study the moduli spaces of asymp. cylindrical special Lagrangian submanifolds with boundary.
The long range goal of studying the geometry and topology of manifolds with special holonomy is to bring a broader mathematical understanding of M-theory in physics. Despite its highly conceptual nature, M-theory has proven to be of great interest to the U.S. public, yielding popular literature and television specials that describe the essence of this potential ``theory of everything''. The P.I. believes that this makes M-theory an excellent subject with which to catch the interest of students who might decide to study math and science,and as a female researcher in this exciting area she feels a particular responsibility to encourage young women to begin and to continue their studies of mathematics. She is currently supervising two Ph.D students and is serving as the faculty advisor of the Society of Undergraduate Mathematics Students. She also organizes the geometry seminars and the colloquium talks at UR. Additionally, she is the co-organizer of the joint Cornell-UR geometry seminars. She will continue to encourage both undergraduate and graduate students and to collaborate with them in this research field which is expected to have a long-lasting impact on both mathematics and theoretical physics.
