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Renaud Gauthier

Publications and source records attributed to Renaud Gauthier.

27 records · Page 2Linked to original sources
arXiv2014

Algebraic Stochastic Calculus

We develop the foundations of Algebraic Stochastic Calculus, with an aim to replacing what is typically referred to as Stochastic Calculus by a purely categorical version thereof. We first give a sheaf theoretic reinterpretation of Probability Theory. We regard probability spaces (X, F, P) as Grothendieck sites (F, J_P) on which Brownian motions are defined via sheaves in symmetric monoidal infinity-categories. Due to the complex nature of such a formalism we are naturally led to considering a purely categorical, time independent formalism in which stochastic differential equations are replaced by studying problems in deformation theory.

math.AG↗
arXiv2013

Infinity Links L, infinity-4-Manifolds M_L and Kirby Categories

We construct what we call a Kirby category, a monoidal category whose morphisms are smooth 4-manifolds, projecting down to another monoidal category whose morphisms are orientable 3-manifolds, the projection being induced by the boundary map on manifolds. We construct a higher categorical generalization of such concepts and introduce the notion of ribbon $\infty$-categories, a generalization of braided monoidal $\infty$-categories (\cite{Lu1}), which gives rise to the concepts of $\infty$-links, $\infty$-4-manifolds as well as the more general notion of walled $\infty$-4-manifolds if one focuses attention on $\infty$-4-manifolds built from gluing thickened sheets on ribbons. These fall into a larger class of constrained $\infty$-4-manifolds whose classical 4-dimensional counterparts are constrained 4-manifolds on which we consider physical theories. We regard pairs of constrained 4-manifolds and Lagrangians densities depicting physical theories defined on such spaces as morphism objects in an enhanced Kirby category, whose objects are regarded as events. We define a universal category $Λ$ of all events that we relate to the $\infty$-category of ribbon $\infty$-categories and conclude in part that Lagrangian field theories can be superseded by using $\infty$-categories.

math.GT↗
arXiv2013

Higher Tannaka and Beyond

We consider the origins of Higher Tannaka duality, as well as it consequences. In a first time we review the work of J. Wallbridge on that subject, which shows in particular that Hopf algebras are essential to generating Tannakian infinity-categories. While doing so we provide an analysis of what it means for Higher Tannaka to be a reconstruction program for stacks. Putting Wallbridge's work in perspective paves the way for causal models in Higher Category Theory. We introduce two interesting concepts that we deem to be instrumental within this framework; that of blow-ups of categories as well as that of fractal infinity-categories.

math.AG↗
arXiv2012

Relaxed Cech Cohomology, Emeralds over Topological Spaces and the Kontsevich Integral

We introduce families of decorations of a same topological space, as well as a family of sheaves over such decorated spaces. Making those families a directed system leads to the concept of emerald over a space. For the configuration space X_N of N points in the plane, connecting points of the plane with chords is a decoration and the sheaf of log differentials over such spaces forms an emerald. We introduce a relaxed form of Cech cohomology whereby intersections are defined up to equivalence. Two disjoint open sets of X_N whose respective points are connected by a chord is one instance of intersection up to equivalence. One paradigm example of such a formalism is provided by the Kontsevich integral.

math.GT↗
arXiv2012

A Stringy Generalization of the Kontsevich Integral

We introduce a "minimal" Kontsevich integral that generates the original Kontsevich integral while at the same time producing ribbons whose boundaries are the braids on which the minimal Kontsevich integral is evaluated. We generalize the definition of the Kontsevich integral to that of graphs in R^3 and study the behavior of such expressions as different graphs are brought together, thus leading to a 2-dimensional generalization of the Kontsevich integral.

math.GT↗
arXiv2010

On the Foundations of the LMO Invariant

We generalize the definition of the framed Kontsevich integral initially presented by T.Q.T.Le and J.Murakami. We define an isotopy invariant $\widetilde{Z}_f$ that behaves well under band sum moves.

math.GT↗
arXiv2010

On the LMO Invariant, the Wheeling Theorem, and the Aarhus Integral

In a previous paper, we generalized the definition of the framed Kontsevich integral initially presented by Le and Murakami. We also defined an isotopy invariant $\widetilde{Z}_f$ that is well-behaved under band sum moves. Using this invariant we study the construction of the LMO invariant, the Wheeling Theorem, and the Aarhus integral.

math.GT↗
arXiv2010

The Kontsevich Integral in Book Notation

We introduce a matrix representation of a chord on a tangle which leads us to representing tangle chord diagrams as stacks of matrices that we call books. We show that band sum moves, Reidemeister moves as well as orientation changes are implemented on \widetilde{Z}_f - a framed link invariant constructed from the Kontsevich integral that's well-behaved under band sum moves - by matrix congruences. We prove that being given the bare framed Kontsevich integral Z_f(L) in book notation for some unknown link L, we can determine what the link L is, as well as the projection of Z_f(L) in the original completed algebra of chord diagrams.

math.GT↗