George Dimitrov

In our previous paper, viewing $D^b(K(l))$ as a noncommutative curve, where $K(l)$ is the Kronecker quiver with $l$-arrows, we introduced categorical invariants via counting of noncommutative curves. Roughly, these invariants are sets of subcategories in a given category and their quotients. The noncommutative curve-counting invariants are obtained...

The second author and Katzarkov introduced categorical invariants based on counting of full triangulated subcategories in a given triangulated category $\mathcal T$ and they demonstrated different choices of additional properties of the subcategories being counted, in particular - an approach to make non-commutative counting in $\mathcal T$ dependa...

The second author and Katzarkov introduced categorical invariants based on counting of full triangulated subcategories in a given triangulated category T and they demonstrated different choices of additional properties of the subcategories being counted, in particular-an approach to make non-commutative counting in T dependable on a stability condi...

In this paper we introduce new categorical notions and give many examples. In an earlier paper we proved that the Bridgeland stability space on the derived category of representations of K(l), the l-Kronecker quiver, is biholomorphic to \({{\mathbb {C}}} \times {\mathcal {H}}\) for \(l\ge 3\). In the present paper we define a new notion of norm, wh...

This note contains a written form of a talk given by the first author at the conference on Mirror Symmetry and Related Topics, Miami, January 28-February 2, 2019. Details and related remarks are added. In our previous papers we introduced categorical invariants, which are, roughly, sets of triangulated subcategories in a given triangulated category...

In our previous paper, viewing $D^b(K(l))$ as a non-commutative curve, we observed that it is reasonable to count non-commutative curves in certain triangulated categories, where $K(l)$ is the Kronecker quiver with $l$-arrows. We gave a general definition, which specializes to the non-commutative curve-counting invariants. Roughly it defines the se...

We introduce a remarkable subset "the stem" of the set of positive roots of a reduced root system. The stem determines several interesting decompositions of the corresponding reductive Lie algebra. It gives also a nice simple three dimensional subalgebra and a "Cayley transform". In the present paper we apply the above devices to give a complete cl...

We prove that for $l\geq 3$ the space of Brdgeland stability conditions on the $l$-Kronecker quiver is ${\mathbb C} \times \mathcal H$ as a complex manifold.

We introduce several notions and give examples. We prove that ${\rm Stab}(D^b(K(l)))\cong {\mathbb C}\times \mathcal H$ for $l\geq 3$, where $K(l)$ is $l$-Kronecker quiver. This is an example of SOD, where ${\rm Stab}( \langle \mathcal T_1,\mathcal T_2\rangle )\not \cong{\rm Stab}(\mathcal T_1)\times {\rm Stab}(\mathcal T_2)$. This example suggest...

Using results in a previous paper "Non-semistable exceptional objects in hereditary categories", we focus here on studying the topology of the space of Bridgeland stability conditions on $D^b(Rep_k(Q ))$, where $Q$ is the acyclic triangular quiver (the underlying graph is the extended Dynkin diagram $\widetilde{\mathbb A}_2$). In particular, we pro...

This paper is a continuation of our previous paper, where we studied non-semistable exceptional objects in hereditary categories and introduced the notion of regularity preserving category. To obtain examples of such categories we introduced certain conditions on the Ext-nontrivial couples and found some examples where these conditions are satisfie...

For a given stability condition $\sigma$ on a triangulated category we define a $\sigma$-exceptional collection as an Ext-exceptional collection, whose elements are $\sigma$-semistable with phases contained in an open interval of length one. If there exists a full $\sigma$-exceptional collection, then $\sigma$ is generated by this collection in a p...

We study questions motivated by results in the classical theory of dynamical systems in the context of triangulated and A-infinity 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 categor...

We obtain a complete classification of hypercomplex manifolds, on which a compact group of automorphisms acts transitively. The description of the spaces as well as the proofs of our results use only the structure theory of reductive groups, in particular the notion of "stem" of a reduced root system, introduced in the first paper of this series.