Mirror symmetry (MS) was discovered several years ago in string theory as a duality between families of 3-dimensional Calabi-Yau manifolds (more precisely, complex algebraic manifolds possessing holomorphic volume elements without zeros). The name comes from the symmetry among Hodge numbers. For dual Calabi-Yau manifolds V, W of dimension n (not necessarily equal to 3) one has 

$$\dim {H^p}(V,{\Omega ^q}) = \dim {H^{n - p}}(W,{\Omega ^q}).$$

.

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Homological Algebra of Mirror Symmetry

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 category ('number of BPS states with given charge' in physics language). Formally, our motivic DT-invariants are elements of quantum tori over a version of the Grothendieck ring of varieties over the ground field. Via the quasi-classical limit 'as the motive of affine line approaches to 1' we obtain numerical DT-invariants which are closely related to those introduced by Behrend. We study some properties of both motivic and numerical DT-invariants including the wall-crossing formulas and integrality. We discuss the relationship with the mathematical works (in the non-triangulated case) of Joyce, Bridgeland and Toledano-Laredo, as well as with works of physicists on Seiberg-Witten model (string junctions), classification of N=2 supersymmetric theories (Cecotti-Vafa) and structure of the moduli space of vector multiplets. Relating the theory of 3d Calabi-Yau categories with distinguished set of generators (called cluster collection) with the theory of quivers with potential we found the connection with cluster transformations and cluster varieties (both classical and quantum).

Stability structures, motivic Donaldson-Thomas invariants and cluster transformations

This paper introduces the notion of a stability condition on a triangulated category. The motivation comes from the study of Dirichlet branes in string theory, and especially from M.R. Douglas's notion of $\Pi$-stability. From a mathematical point of view, the most interesting feature of the definition is that the set of stability conditions $\Stab(\T)$ on a fixed category $\T$ has a natural topology, thus defining a new invariant of triangulated categories. After setting up the necessary definitions I prove a deformation result which shows that the space $\Stab(\T)$ with its natural topology is a manifold, possibly infinite-dimensional.

https://annals.math.princeton.edu/wp-content/uploads/annals-v166-n2-p01.pdf

Stability conditions on triangulated categories

The classical McKay correspondence relates representations of a finite subgroup
G ⊂ SL(2,C) to the cohomology of the well-known minimal resolution of the
Kleinian singularity C2/G. Gonzalez-Sprinberg and Verdier [10] interpreted the
McKay correspondence as an isomorphism on K theory, observing that the representation
ring of G is equal to the G-equivariant K theory of C2. More precisely,
they identify a basis of the K theory of the resolution consisting of the classes of
certain tautological sheaves associated to the irreducible representations of G.

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The McKay correspondence as an equivalence of derived categories

This paper gives a construction of braid group actions on the derived category of coherent sheaves on a variety $X$. The motivation for this is M. Kontsevich's homological mirror conjecture, together with the occurrence of certain braid group actions in symplectic geometry. One of the main results is that when dim $X\geq 2$, our braid group actions are always faithful. We describe conjectural mirror symmetries between smoothings and resolutions of singularities which lead us to find examples of braid group actions arising from crepant resolutions of various singularities. Relations with the McKay correspondence and with exceptional sheaves on Fano manifolds are given. Moreover, the case of an elliptic curve is worked out in some detail.

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Braid group actions on derived categories of coherent sheaves

We give a sufficient condition for an Ext-finite triangulated category to be saturated. Saturatedness means that every contravariant cohomological functor of finite type to vector spaces is representable. The condition consists in the existence of a strong generator. We prove that the bounded derived categories of coherent sheaves on smooth proper commutative and noncommutative varieties have strong generators, and are hence saturated. In contrast, the similar category for a smooth compact analytic surface with no curves is not saturated. 2000 Math. Subj. Class. Primary 18E30.

Generators and representability of functors in commutative and noncommutative geometry

A criterion for a functor between derived categories of coherent sheaves to be full and faithful is given. A semiorthogonal decomposition for the derived category of coherent sheaves on the intersection of two even dimensional quadrics is obtained. The behaviour of derived categories with respect to birational transformations is investigated. A theorem about reconstruction of a variety from the derived category of coherent sheaves is proved.

https://arxiv.org/pdf/alg-geom/9506012v1.pdf

Semiorthogonal decomposition for algebraic varieties

It is proved that a triangulated category generated by a strong exceptional collection is equivalent to the derived category of modules over an algebra of homomorphisms of this collection. For the category of coherent sheaves on a Fano variety, the functor of tightening to a canonical class is described by means of mutations of an exceptional collection generating the category. The connection between mutability of strong exceptional collections and the Koszul property is studied. It is proved that in the geometric situation mutations of exceptional sheaves consist of present sheaves and the corresponding algebra of homomorphisms is Koszul and selfconsistent. Bibliography: 18 titles.

https://iopscience.iop.org/article/10.1070/IM1990v034n01ABEH000583/pdf

Representation of associative algebras and coherent sheaves

A solution is given to the problem of describing a triangulated category generated by a finite number of objects. It requires the notion of "enhancement" of a triangulated category, by means of the complexes RHom.

http://iopscience.iop.org/article/10.1070/SM1991v070n01ABEH001253/pdf

Enhanced triangulated categories

We consider smooth algebraic varieties with ample either canonical or anticanonical sheaf. We prove that such a variety is uniquely determined by its derived category of coherent sheaves. We also calculate the group of exact autoequivalences for these categories. The technics of ample sequences in Abelian categories is used.

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Alexey Bondal

Papers

Generators and representability of functors in commutative and noncommutative geometry

Semiorthogonal decomposition for algebraic varieties

Representation of associative algebras and coherent sheaves

Enhanced triangulated categories

Reconstruction of a variety from the derived category and groups of autoequivalences