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Stability structures, motivic Donaldson-Thomas invariants and cluster transformations
We define new invariants of 3d Calabi-Yau categories endowed with a stability structure. Intuitively, they count the number of semistable objects with fixed class in the K-theory of the categoryExpand
Homological mirror symmetry and torus fibrations
In this paper we discuss two major conjectures in Mirror Symmetry: Strominger-Yau-Zaslow conjecture about torus fibrations, and the homological mirror conjecture (about an equivalence of the FukayaExpand
Cohomological Hall algebra, exponential Hodge structures and motivic Donaldson-Thomas invariants
We define a new type of Hall algebras associated e.g. with quivers with polynomial potentials. The main difference with the conventional definition is that we use cohomology of the stack ofExpand
Affine Structures and Non-Archimedean Analytic Spaces
In this paper we propose a way to construct an analytic space over a non-archimedean field, starting with a real manifold with an affine structure which has integral monodromy. Our construction isExpand
Deformations of algebras over operads and Deligne's conjecture
In present paper we develop the deformation theory of operads and algebras over operads. Free resolutions (constructed via Boardman-Vogt approach) are used in order to describe formal moduli spacesExpand
Notes on A∞-Algebras, A∞-Categories and Non-Commutative Geometry
We develop a geometric approach to A-infinity algebras and A-infinity categories based on the notion of formal scheme in the category of graded vector spaces. The geometric approach clarifies severalExpand
Quantum affine algebras and their representations
Abstract Some questions on the representation theory of quantum affine algebras are discussed from the categorical point of view.
Algebras of functions on quantum groups
Introduction Poisson Lie groups Quantized universal enveloping algebras Quantized algebras of functions Quantum Weyl group and the universal quantum $R$-matrix Bibliography.
Algebras of functions on compact quantum groups, Schubert cells and quantum tori
AbstractThe structures of Poisson Lie groups on a simple compact group are parametrized by pairs (a, u), wherea∈R, $$u \in \Lambda ^2 \mathfrak{h}_R$$ , and $$\mathfrak{h}_R$$ is a real CartanExpand
Notes on A-infinity algebras, A-infinity categories and non-commutative geometry. I
We develop geometric approach to A-infinity algebras and A-infinity categories based on the notion of formal scheme in the category of graded vector spaces. Geometric approach clarifies severalExpand
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