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57
Rational functions with real critical points and the B. and M. Shapiro conjecture in real enumerative geometry
, 2001
"... Suppose that 2d − 2 tangent lines to the rational normal curve z ↦ → (1: z:...: z d)inddimensional complex projective space are given. It was known that the number of codimension 2 subspaces intersecting all these lines is always finite; for a generic configuration it is equal to the dth Catalan n ..."
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Cited by 55 (16 self)
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Suppose that 2d − 2 tangent lines to the rational normal curve z ↦ → (1: z:...: z d)inddimensional complex projective space are given. It was known that the number of codimension 2 subspaces intersecting all these lines is always finite; for a generic configuration it is equal to the dth Catalan number. We prove that for real tangent lines, all these codimension 2 subspaces are also real, thus confirming a special case of a general conjecture of B. and M. Shapiro. This is equivalent to the following result: If all critical points of a rational function lie on a circle in the Riemann sphere (for example, on the real line), then the function maps this circle into a circle.
TRIPLES, ALGEBRAS AND COHOMOLOGY
 REPRINTS IN THEORY AND APPLICATIONS OF CATEGORIES
, 2003
"... ..."
A Spectral Sequence For Motivic Cohomology
 Invent. Math
"... this paper is to construct a spectral sequence from the ..."
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Cited by 31 (0 self)
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this paper is to construct a spectral sequence from the
and G.Sardanashvily, Lagrangian supersymmetries depending on derivatives. Global analysis and cohomology
"... Abstract: Lagrangian contact supersymmetries (depending on derivatives of arbitrary order) are treated in very general setting. The cohomology of the variational bicomplex on an arbitrary graded manifold and the iterated cohomology of a generic nilpotent contact supersymmetry are computed. In partic ..."
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Cited by 16 (8 self)
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Abstract: Lagrangian contact supersymmetries (depending on derivatives of arbitrary order) are treated in very general setting. The cohomology of the variational bicomplex on an arbitrary graded manifold and the iterated cohomology of a generic nilpotent contact supersymmetry are computed. In particular, the first variational formula and conservation laws for Lagrangian systems on graded manifolds using contact supersymmetries are obtained. 1
The Category Theoretic Understanding of Universal Algebra: Lawvere Theories and Monads
, 2007
"... Lawvere theories and monads have been the two main category theoretic formulations of universal algebra, Lawvere theories arising in 1963 and the connection with monads being established a few years later. Monads, although mathematically the less direct and less malleable formulation, rapidly gained ..."
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Cited by 14 (0 self)
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Lawvere theories and monads have been the two main category theoretic formulations of universal algebra, Lawvere theories arising in 1963 and the connection with monads being established a few years later. Monads, although mathematically the less direct and less malleable formulation, rapidly gained precedence. A generation later, the definition of monad began to appear extensively in theoretical computer science in order to model computational effects, without reference to universal algebra. But since then, the relevance of universal algebra to computational effects has been recognised, leading to renewed prominence of the notion of Lawvere theory, now in a computational setting. This development has formed a major part of Gordon Plotkin’s mature work, and we study its history here, in particular asking why Lawvere theories were eclipsed by monads in the 1960’s, and how the renewed interest in them in a computer science setting might develop in future.
Global Hochschild (co)homology of singular spaces
, 2006
"... We introduce Hochschild (co)homology of morphisms of schemes or analytic spaces and study its fundamental properties. In analogy with the cotangent complex we introduce the so called (derived) Hochschild complex of a morphism; the Hochschild cohomology and homology groups are then the Ext and Tor ..."
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Cited by 6 (1 self)
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We introduce Hochschild (co)homology of morphisms of schemes or analytic spaces and study its fundamental properties. In analogy with the cotangent complex we introduce the so called (derived) Hochschild complex of a morphism; the Hochschild cohomology and homology groups are then the Ext and Tor groups of that complex. We prove that these objects are well defined, extend the known cases, and have the expected functorial and homological properties such as graded commutativity of Hochschild cohomology and existence of the characteristic homomorphism from Hochschild cohomology to the (graded) centre of the derived category.
(CO)CYCLIC (CO)HOMOLOGY OF BIALGEBROIDS: AN APPROACH VIA (CO)MONADS
, 2008
"... ... there is a (co)simplex Z ∗: = ΠTl ∗+1 X in C. The aim of this paper is to find criteria for para(co)cyclicity of Z ∗. Our construction is built on a distributive law of Tl with a second (co)monad Tr on M, a natural transformation i: ΠTl → ΠTr, and a morphism w: TrX → TlX in M. The (symmetrical) ..."
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Cited by 3 (1 self)
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... there is a (co)simplex Z ∗: = ΠTl ∗+1 X in C. The aim of this paper is to find criteria for para(co)cyclicity of Z ∗. Our construction is built on a distributive law of Tl with a second (co)monad Tr on M, a natural transformation i: ΠTl → ΠTr, and a morphism w: TrX → TlX in M. The (symmetrical) relations i and w need to satisfy are categorical versions of Kaygun’s axioms of a transposition map. Motivation comes from the observation that a (co)ring T over an algebra R determines a distributive law of two (co)monads Tl = T ⊗R (−) and Tr = (−) ⊗R T on the category of Rbimodules. The functor Π can be chosen such that Z n = T b⊗R... b⊗RT b⊗RX is the cyclic Rmodule tensor product. A natural transformation i: T b⊗R(−) → (−)b⊗RT is given by the flip map and a morphism w: X ⊗R T → T ⊗R X is constructed whenever T is a (co)module algebra or coring of an Rbialgebroid. The notion of a stable anti YetterDrinfel’d module over certain bialgebroids, so called ×RHopf algebras, is introduced. In the particular example when T is a module coring of a ×RHopf algebra B and X is a stable anti YetterDrinfel’d Bmodule, the paracyclic object Z ∗ is shown to project to a cyclic structure on T ⊗ R ∗+1 ⊗B X. For a BGalois extension S ⊆ T, a stable anti YetterDrinfel’d Bmodule TS is constructed, such that