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Homological Algebra of Mirror Symmetry
 in Proceedings of the International Congress of Mathematicians
, 1994
"... Mirror Symmetry was discovered several years ago in string theory as a duality between families of 3dimensional CalabiYau manifolds (more precisely, complex algebraic manifolds possessing holomorphic volume elements without zeroes). The name comes from the symmetry among Hodge numbers. For dual Ca ..."
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Cited by 346 (2 self)
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Mirror Symmetry was discovered several years ago in string theory as a duality between families of 3dimensional CalabiYau manifolds (more precisely, complex algebraic manifolds possessing holomorphic volume elements without zeroes). The name comes from the symmetry among Hodge numbers. For dual CalabiYau manifolds V, W of dimension n (not necessarily equal to 3) one has dim H p (V, Ω q) = dim H n−p (W, Ω q). Physicists conjectured that conformal field theories associated with mirror varieties are equivalent. Mathematically, MS is considered now as a relation between numbers of rational curves on such a manifold and Taylor coefficients of periods of Hodge structures considered as functions on the moduli space of complex structures on a mirror manifold. Recently it has been realized that one can make predictions for numbers of curves of positive genera and also on CalabiYau manifolds of arbitrary dimensions. We will not describe here the complicated history of the subject and will not mention many beautiful contsructions, examples and conjectures motivated
Introduction to SH Lie algebras for physicists,” Int
 J. Theor. Phys
, 1993
"... Much of point particle physics can be described in terms of Lie algebras and ..."
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Cited by 129 (14 self)
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Much of point particle physics can be described in terms of Lie algebras and
Cluster algebras II: Finite type classification, Invent
 Department of Mathematics, Northeastern University
"... 1.2. Basic definitions 3 1.3. Finite type classification 5 ..."
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Cited by 105 (16 self)
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1.2. Basic definitions 3 1.3. Finite type classification 5
Tensor constructions of open string theories I
 Foundations,” Nucl. Phys. B
, 1997
"... The possible tensor constructions of open string theories are analyzed from first principles. To this end the algebraic framework of open string field theory is clarified, including the role of the homotopy associative A ∞ algebra, the odd symplectic structure, cyclicity, star conjugation, and twist ..."
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Cited by 88 (7 self)
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The possible tensor constructions of open string theories are analyzed from first principles. To this end the algebraic framework of open string field theory is clarified, including the role of the homotopy associative A ∞ algebra, the odd symplectic structure, cyclicity, star conjugation, and twist. It is also shown that two string theories are offshell equivalent if the corresponding homotopy associative algebras are homotopy equivalent in a strict sense. It is demonstrated that a homotopy associative star algebra with a compatible even bilinear form can be attached to an open string theory. If this algebra does not have a spacetime interpretation, positivity and the existence of a conserved ghost number require that its cohomology is at degree zero, and that it has the structure of a direct sum of full matrix algebras. The resulting string theory is shown to be physically equivalent to a string theory with a familiar open string gauge group.
Categorical mirror symmetry: the elliptic curve
 Adv. Theor. Math. Phys
, 1998
"... We describe an isomorphism of categories conjectured by Kontsevich. If M and ˜ M are mirror pairs then the conjectural equivalence is between the derived category of coherent sheaves on M and a suitable version of Fukaya’s category of Lagrangian submanifolds on ˜ M. We prove this equivalence when M ..."
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Cited by 81 (10 self)
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We describe an isomorphism of categories conjectured by Kontsevich. If M and ˜ M are mirror pairs then the conjectural equivalence is between the derived category of coherent sheaves on M and a suitable version of Fukaya’s category of Lagrangian submanifolds on ˜ M. We prove this equivalence when M is an elliptic curve and ˜ M is its dual curve, exhibiting the dictionary in detail.
DG quotients of DG categories
 J. Algebra
"... Abstract. Keller introduced a notion of quotient of a differential graded category modulo a full differential graded subcategory which agrees with Verdier’s notion of quotient of a triangulated category modulo a triangulated subcategory. This work is an attempt to further develop his theory. ..."
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Cited by 79 (0 self)
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Abstract. Keller introduced a notion of quotient of a differential graded category modulo a full differential graded subcategory which agrees with Verdier’s notion of quotient of a triangulated category modulo a triangulated subcategory. This work is an attempt to further develop his theory.
Geometry of the Space of Phylogenetic Trees
 Adv. in Appl. Math
, 1999
"... ields to graphically represent various types of hierarchical relationships, including evolutionary relationships between species, divergent patterns between subpopulations and evolutionary relationships between genes. These trees are generally rooted and semilabeled, i.e., they descend from a singl ..."
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Cited by 77 (1 self)
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ields to graphically represent various types of hierarchical relationships, including evolutionary relationships between species, divergent patterns between subpopulations and evolutionary relationships between genes. These trees are generally rooted and semilabeled, i.e., they descend from a single node called the root, bifurcate at lower nodes and end at terminal nodes, called tips or leaves; the leaves are labeled by the names of the species, subpopulations or genes being studied. In biological studies the latter are called operational taxonomic units (OTU's). Traditionally, trees were inferred form morphological similarities among the OTU's. To build an evolutionary species tree, or phylogenetic tree, two species which shared the most characteristics were classified as `siblings' and assumed to share a common ancestor which is not the ancestor of any other species. Such `siblings' are said to be homologous, and it is this basic homo
Oriented OpenClosed String Theory Revisited
"... String theory on Dbrane backgrounds is openclosed string theory. Given the relevance of this fact, we give details and elaborate upon our earlier construction of oriented openclosed string field theory. In order to incorporate explicitly closed strings, the classical sector of this theory is open ..."
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Cited by 74 (6 self)
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String theory on Dbrane backgrounds is openclosed string theory. Given the relevance of this fact, we give details and elaborate upon our earlier construction of oriented openclosed string field theory. In order to incorporate explicitly closed strings, the classical sector of this theory is open strings with a homotopy associative A ∞ algebraic structure. We build a suitable BatalinVilkovisky algebra on moduli spaces of bordered Riemann surfaces, the construction of which involves a few subtleties arising from the open string punctures and cyclicity conditions. All vertices coupling open and closed strings through disks are described explicitly. Subalgebras of the algebra of surfaces with boundaries are used to discuss symmetries of classical open string theory induced by the closed string sector, and to write classical open string field theory on general closed string backgrounds. We give a preliminary analysis of the ghostdilaton theorem.
Higher dimensional algebra III: ncategories and the algebra of opetopes
, 1997
"... We give a definition of weak ncategories based on the theory of operads. We work with operads having an arbitrary set S of types, or ‘Soperads’, and given such an operad O, we denote its set of operations by elt(O). Then for any Soperad O there is an elt(O)operad O + whose algebras are Soperads ..."
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Cited by 74 (6 self)
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We give a definition of weak ncategories based on the theory of operads. We work with operads having an arbitrary set S of types, or ‘Soperads’, and given such an operad O, we denote its set of operations by elt(O). Then for any Soperad O there is an elt(O)operad O + whose algebras are Soperads over O. Letting I be the initial operad with a oneelement set of types, and defining I 0+ = I, I (i+1)+ = (I i+) +, we call the operations of I (n−1)+ the ‘ndimensional opetopes’. Opetopes form a category, and presheaves on this category are called ‘opetopic sets’. A weak ncategory is defined as an opetopic set with certain properties, in a manner reminiscent of Street’s simplicial approach to weak ωcategories. In a similar manner, starting from an arbitrary operad O instead of I, we define ‘ncoherent Oalgebras’, which are n times categorified analogs of algebras of O. Examples include ‘monoidal ncategories’, ‘stable ncategories’, ‘virtual nfunctors ’ and ‘representable nprestacks’. We also describe how ncoherent Oalgebra objects may be defined in any (n + 1)coherent Oalgebra.