Homological Mirror Symmetry, Spectral Gaps & Applications

The role of monodromy techniques, vanishing cycles, and Hodge theory have historically formed the foundational tools of mirror symmetry. These traditional approaches have novel and deep meaning when revisited through the modern homological perspective. Manifolds are transformed into categories, Hodge structures are generalized into a sophisticated noncommutative language, and monodromic information is encoded in the gaps of Orlov spectra. This proposal consists of two stages. The first stage is the expansion of our theory of gaps in Orlov spectra. We have already had remarkable progress in development of this theory under the support of previous grants. The second stage is the application of our theory to an array of unsolved problems. Among these problems are questions in rationality theory, the theory of algebraic cycles, and symplectic geometry. We stress the following points: This is an international project, combining efforts of the field`s most prominent researchers, each representing one of the world`s leading institutions. Together, these researchers have an enormous history of producing cutting-edge research through a wealth of collaboration. The project is based on established cooperation with physicists in Vienna and long-term partnerhips with MIT, the IHES, KSU and LAGA Moscow, A. Renyi Institute, and the University of Zagreb. This project will greatly enhance scientific life in Vienna and further promote the city as an international center for the study of algebraic geometry and homological mirror symmetry. This project will have tremendous educational impact on students and post-doctoral fellows in Vienna, giving them an opportunity to learn from the best in the field, visit leading mathematical centers, and forge strong long-term research partnerships. This proposal is a natural continuation of previous FWF and ERC grants with clear and massive impact on several subjects - algebraic geometry, symplectic geometry, homological algebra, and string theory.

Mirror symmetry originated in physics as a duality between N = 2 superconformal quantum field theories. In 1990, the CO PI of this proposal, Maxim Kontsevich, interpreted this duality in a consistent, powerful mathematical framework called Homological Mirror Symmetry (HMS). The ideas put forth by Kontsevich have led to dramatic developments in how the mathematical community approaches ideas from theoretical physics, and indeed our conception of space itself. These developments created a frenzy of activity in the mathematical community which has led to a remarkable synergy of diverse mathematical disciplines, notably symplectic geometry, algebraic geometry, and category theory. HMS is now the foundation of a wide range of contemporary mathematical research dedicated to the ideas. The new achievements of this projects are following directions: 1) Proving Homological Mirror Symmetry.2) Developing the theory of Categorical Linear Systems. 3) Connection between dynamical system an derived categories. These directions have an immense impact on some classical questions of algebraic and Symplectic Geometry. The last two directions - developed in the last 3 years - are really ground breaking and open new venues of cutting edge research. We have produced high level postdoc and very well prepared graduate students - A. Noll, F. Haiden, G. Dimitrov. The results we have obtained were recorded in several papers and 3 conferences allowed us to disseminate our new results. The above project has significant and broad output: 1. Deepening the connection with theoretical physics. 2. Establishing unexpected connection between category theory and complexity and dynamical systems. 3. Helping educate new generation of researchers through several. Our work has had a broad educational impact and is related to Physics. All directions mentioned above have helped ideas of combining wall crossing, algebraic cycles and spectra crystallize and give back our due to Physics.