Phase change and new Hodge-type categorical invariants

This proposal constitutes a three-year project lead by L. Katzarkov and D. Favero which will be carried out at the University of Vienna to study phase change. Our recent findings place this field at a critical stage of discovery. Indeed, many new theorems have been proven and proliferated by the primary investigators of this proposal, dramatically increasing interest in the subject. These new theorems have broadly opened the field, allowing one to use computations of Orlov spectra to answer classical questions in algebraic geometry about rationality and algebraic cycles (notably the study of the Hodge conjecture and Griffiths groups). A tenant of our understanding is that the implementation of phase change will result in a vast overhaul of homological projective duality, providing, not only, a colossal generalization of previous mathematical work and an immense number of new examples, but also an amazingly fresh perspective. This surprising and exciting mathematical discovery is massive in scope and will require new expertise, computational power, and the combined efforts of many researchers to discover its secrets. As a result, we are requesting a total of EUR 449.982 to cover costs for personnel and travel. This grant is benefited by official contracts with F. Bogomolov, A. Kuznetsov, and D. Orlov in Moscow and the Institute for Mirror Symmetry at Kansas State University (KSU) based on our collaboration with Y. Soibelman and I. Zharkov and will adhere to the Open Access Policy of the FWF. Our strategic plan to better this field is as follows. In the first year D. Favero, M. Ballard and L. Katzarkov will further develop the mathematical phenomenon inspired by phase changes in physics, and apply the results to the study of Griffiths groups and the Hodge conjecture. In the second year, having developed these phase change techniques, we will apply our results to the theory of Orlov spectra and gaps. The project leaders and the post-doctoral fellows will benefit immensely from the expertise of visitors and co-investigators, D. Orlov, T. Pantev, M. Kontsevich., C. Simpson, and Y. Soibelman, who will each be visiting Vienna for two three-month periods. Furthermore, we will hold a conference at the Erwin Schrödinger Institute (ESI) dedicated to Bogomolov`s work on rationality and two additional workshops with counterparts in Moscow. In the third year, we will concentrate on developing the connection between spectra and wall crossing in the moduli spaces of stability conditions. During this period, our contributions to Viennese and world scientific life will culminate with an educational semester at the ESI dedicated to homological mirror symmetry, matrix factorizations, and phase change. This educational semester will provide great opportunities, for both new researchers and experts in the field to learn and share their knowledge. It will also serve to disseminate the body of work produced by our team of researchers during the course of this grant. Aside from workshops and the educational semester, two other conferences are planned, the proceedings will be published in the International Press and PLMS. Let us emphasize first that 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. Second, that this research will have clear and massive repercussions on several subjects -algebraic geometry, symplectic geometry, homological algebra, and string theory. Third, that this project will greatly enhance scientific life and 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.

The ideas of Homological Mirror Symmetry (HMS), put forth by Kontsevich, have led to dramatic developments in how the mathematical community approaches ideas from theoretical physics. 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.This project aims at applying the new achievements in HMS to geometric applications.1) Proving Homological Mirror Symmetry and establishing a connection with phase change and VGIT.2) Developing the theory of Categorical Linear Systems.3) Connection between dynamical system and derived categories.4) Developing the notion of categorical Kaehler metric.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, 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.