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DEFORMATION THEORY OF REPRESENTATIONS OF PROP(ERAD)S I
"... Abstract. In this paper and its followup [MV08], we study the deformation theory of morphisms of properads and props thereby extending Quillen’s deformation theory for commutative rings to a nonlinear framework. The associated chain complex is endowed with an L∞algebra structure. Its MaurerCarta ..."
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Cited by 9 (4 self)
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Abstract. In this paper and its followup [MV08], we study the deformation theory of morphisms of properads and props thereby extending Quillen’s deformation theory for commutative rings to a nonlinear framework. The associated chain complex is endowed with an L∞algebra structure. Its MaurerCartan elements correspond to deformed structures, which allows us to give a geometric interpretation of these results.
Infinite magmatic bialgebras
 Adv. Appl. Math. (2007
"... Abstract. An infinite magmatic bialgebra is a vector space endowed with an nary operation, and an nary cooperation, for each n, verifying some compatibility relations. We prove a rigidity theorem, analogue to the HopfBorel theorem for commutative bialgebras: any connected infinite magmatic bialge ..."
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Cited by 2 (1 self)
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Abstract. An infinite magmatic bialgebra is a vector space endowed with an nary operation, and an nary cooperation, for each n, verifying some compatibility relations. We prove a rigidity theorem, analogue to the HopfBorel theorem for commutative bialgebras: any connected infinite magmatic bialgebra is of the form Mag ∞ (Prim H), where Mag ∞ (V) is the free infinite magmatic algebra over the vector space V. 1.
DEFORMATION THEORY OF REPRESENTATIONS OF PROP(ERAD)S II
"... Abstract. This paper is the followup of [MV08]. ..."
Comparing definitions of weak higher categories, I
, 2009
"... The theory of operads, defined through categories of labeled graphs, is generalized to suit definitions of higher categories with arbitrary basic shapes. Constructions of cubical, globular and opetopic weak higher categories are obtained as examples. 1 ..."
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The theory of operads, defined through categories of labeled graphs, is generalized to suit definitions of higher categories with arbitrary basic shapes. Constructions of cubical, globular and opetopic weak higher categories are obtained as examples. 1
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"... Feynman graphs, and nerve theorem for compact symmetric multicategories (extended abstract) ..."
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Feynman graphs, and nerve theorem for compact symmetric multicategories (extended abstract)
MOTIVATION AND BACKGROUND
, 904
"... Abstract. The main observable quantities in Quantum Field Theory, correlation functions, are expressed by the celebrated Feynman path integrals which are not well defined mathematical objects. Perturbation formalism interprets such an integral as a formal series of finite– dimensional but divergent ..."
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Abstract. The main observable quantities in Quantum Field Theory, correlation functions, are expressed by the celebrated Feynman path integrals which are not well defined mathematical objects. Perturbation formalism interprets such an integral as a formal series of finite– dimensional but divergent integrals, indexed by Feynman graphs, the list of which is determined by the Lagrangian of the theory. Renormalization is a prescription that allows one to systematically “subtract infinities ” from these divergent terms producing an asymptotic series for quantum correlation functions. On the other hand, graphs treated as “flowcharts”, also form a combinatorial skeleton of the abstract computation theory and various operadic formalisms in abstract algebra. In this role of descriptions of various (classes of) computable functions, such as recursive functions, functions computable by a Turing machine with oracles etc., graphs can be used to replace standard formalisms having linguistic flavor, such as Church’s λ–calculus and various programming languages. The functions in question are generally not everywhere defined due to potentially infinite loops and/or necessity to search in an infinite haystack for a needle which is not there. In this paper I argue that such infinities in classical computation theory can be addressed in the same way as Feynman divergences, and that meaningful versions of renormalization in this context can be devised. Connections with quantum computation are also touched upon.