Recent papers and preprints

Legendrian skein algebras and Hall algebras – We compare two associative algebras which encode the “quantum topology” of Legendrian curves in contact threefolds of product type S×R. The first is the skein algebra of graded Legendrian links and the second is the Hall algebra of the Fukaya category of S. We construct a natural homomorphism from the former to the latter, which we show is an isomorphism if S is a disk with marked points and injective if S is the annulus. arxiv:1908.10358

An extension of the Siegel space of complex abelian varieties and conjectures on stability structures – We study semi-algebraic domains associated with symplectic tori and conjecturally identified with spaces of stability conditions on the Fukaya categories of these tori. Our motivation is to test which results from the theory of flat surfaces could hold for more general spaces of stability conditions. The main results concern systolic bounds and volume of the moduli space. Published in manuscripta mathematica. journal – arxiv:1808.06364

Iterated logarithms and gradient flows (with P. Pandit, L. Katzarkov, M. Kontsevich) – We consider applications of the theory of balanced weight filtrations and iterated logarithms, initiated in arXiv:1706.01073, to PDEs. The main result is a complete description of the asymptotics of the Yang–Mills flow on the space of metrics on a holomorphic bundle over a Riemann surface. A key ingredient in the argument is a monotonicity property of the flow which holds in arbitrary dimension. The A-side analog is a modified curve shortening flow for which we provide a heuristic calculation in support of a detailed conjectural picture. arxiv:1802.04123

Semistability, modular lattices, and iterated logarithms (with P. Pandit, L. Katzarkov, M. Kontsevich) – We provide a complete description of the asymptotics of the gradient flow on the space of metrics on any semistable quiver representation. This involves a recursive construction of approximate solutions and the appearance of iterated logarithms and a limiting filtration of the representation. The filtration turns out to have an algebraic definition which makes sense in any finite length modular lattice. This is part of a larger project by the authors to study iterated logarithms in the asymptotics of gradient flows, both in finite and infinite dimensional settings. arxiv:1706.01073

Flat surfaces and stability structures (with L. Katzarkov, M. Kontsevich) – We identify spaces of half-translation surfaces, equivalently complex curves with quadratic differential, with spaces of stability structures on Fukaya-type categories of punctured surfaces. This is achieved by new methods involving the complete classification of objects in these categories, which are defined in an elementary way. We also introduce a number of tools to deal with surfaces of infinite area, where structures similar to those in cluster algebra appear. Published in Publ. Math IHES. Springer – arxiv:1409.8611

Dynamical systems and categories (with G. Dimitrov, L. Katzarkov, M. Kontsevich) – We study questions motivated by results in the classical theory of dynamical systems in the context of triangulated and A∞-categories. First, entropy is defined for exact endofunctors and computed in a variety of examples. In particular, the classical entropy of a pseudo-Anosov map is recovered from the induced functor on the Fukaya category. Second, the density of the set of phases of a Bridgeland stability condition is studied and a complete answer is given in the case of bounded derived categories of quivers. Certain exceptional pairs in triangulated categories, which we call Kronecker pairs, are used to construct stability conditions with density of phases. Some open questions and further directions are outlined as well. Published in Contemporary Math. arxiv:1307.8418

An Orbit Construction of Phantoms, Orlov Spectra, and Knörrer Periodicity (with D. Favero, L. Katzarkov) – A phantom category is a split-closed triangulated category with trivial Grothendieck group K_0. We give examples of such categories coming from matrix factorizations. Published in Lecture Notes of the Unione Matematica Italiana. Springer