Sheaves of Categories and their applications

During the last three years the Vienna research group of L. Katzarkov has made several interesting connections between category theory and geometry. These connections have led to several new notions that are geometric in nature and contain a deep categorical meaning. These include the notion of Stability Hodge Structure and the notion of a generalized Landau-Ginzburg (LG) model that does not admit a description in terms of a potential. This proposal outlines a program whose main goals are to 1. Develop the general theory of coefficient systems for Fukaya categories: this is the theory of sheaves of categories of the title. 2. Apply this theory to explore the symplectic geometry of the generalized LG-models mentioned in (2) above. 3. Develop the theory of Stability Hodge Structures, and study geometric examples derived from the theory of sheaves of categories. 4. Use the theory of sheaves of categories to categorify the recent approach to stable rationality developed by Voisin, Colliott-Thélène, Pirutka and Totaro, and to explore several applications to long-standing questions in classical algebraic geometry. This project will be carried out at the University of Vienna in collaboration with C. Simpson (Nice) and M. Kontsevich (IHES). In order to introduce young people to the field and to disseminate our results, we plan to run an ESI program on classical and categorical approaches to stable rationality in Spring 2017. The program has already been approved. The collaboration of Pandit and Katzarkov with Simpson and Kontsevich is well established. The synergy between these researchers will contribute to further establishing Vienna as a center of modern category theory.

The concept of counting is among the oldest in Mathematics and it has a central role. The notion of category was introduced in 20 century and it is no less important. Grothendieck, Kontsevich et al. used the latter to invent a new type of geometry, so called non-commutative algebraic geometry. Motivated by String theory Kontsevich formulated a remarkable equivalence between categories called Homological mirror symmetry (HMS). HMS in turn is a source of insight for both Physics and Mathematics. The project gives a new way of looking at Homological mirror symmetry as a correspondence between so called perverse sheaves of categories and holomorphic data of stability conditions to holomorphic families of categories and perverse sheaves of stability conditions. During 2018 the PI was changed from Ludmil Katzarkov to George Dimitrov. The PI and the ex-PI take a new approach by replacing stability conditions with a large class of new categorical invariants: non-commutative counting invariants and categorical versions of curve complexes. This is a program initiated by the new PI, in joint works with the ex-PI. Guiding idea of this program is a novel application of the concept of counting in a categorical context. The non-commutative counting invariants are, roughly, sets of subcategories in certain categories and their quotients. These sets carry additional structures: partial order and a directed graph. Computing these invariants so far demonstrates connections with combinatorics, number theory, classical geometry. For example, non-commutative counting in some primary cases has a geometric combinatorial parallel - counting of maps between polygons. Performing non-commutative counting in a category related to the complex projective plane the PI obtained finiteness and showed that completing the problem of counting here is equivalent to solving an old open problem due to Markov: are there two different solutions (x1, y1, z1) N^3, (x2, y2, z2) N^3 of the Diophantine equation x^2 + y^2 + z^2 = 3xyz with max{x1, y1, z1} = max{x2, y2, z2} . In summary, within this project the concept of non-commutative counting opens a new chapter in non-commutative algebraic geometry and new connections to classical fields.