Related papers: The Tangle Hypothesis: Dimension 1

We construct an infinite family of homology theories of framed links in thickened surfaces, as well as a homology theory whose graded Euler characteristic is exactly the Kauffman bracket of the link in the surface. Both theories are based…

We define a symmetric tensor enhancement $\mathrm{E}\mathbb{F}$ with full duals of the 3-category $\mathbb{F}$ of fusion categories in which every Reshetikhin--Turaev theory has a fully local realization. Our $\mathrm{E}\mathbb{F}$ is a…

We give a generalization of the Reshetikhin-Turaev functor for tangles to get a combinatorial formula for the universal Vassiliev-Kontsevich invariant of framed oriented links which is coincident with the Kontsevich integral. The universal…

In the present paper we study a bordism theory related to pairs $(M,\, \xi),$ where $M$ is a closed smooth oriented manifold with a stably trivial normal bundle and $\xi$ is a virtual $\SU$-bundle of virtual dimension 1 over $M$. The main…

Let $\Lambda$ be a finite-dimensional basic algebra over an algebraically closed field $k$. The well-known Drozd's theorem asserts, that $\Lambda$ is either tame or wild. The Crawley-Boevey's Theorem states that for a given tame algebra…

Given a proper toric variety and a line bundle on it, we describe the morphism on singular cohomology given by the cup product with the Chern class of that line bundle in terms of the data of the associated fan. Using that, we relate the…

The celebrated Drozd's theorem asserts that a finite-dimensional basic algebra $\Lambda$ over an algebraically closed field $k$ is either tame or wild, whereas the Crawley-Boevey's theorem states that given a tame algebra $\Lambda$ and a…

In this paper, we develop a general framework of geometric functorial field theories, meaning that all bordisms in question are endowed with geometric structures. We take particular care to establish a notion of smooth variation of such…

We consider Legendrian links and tangles in $J^1S^1$ and $J^1[0,1]$ equipped with Morse complex families over a field $\mathbb{F}$ and classify them up to Legendrian cobordism. When the coefficient field is $\mathbb{F}_2$ this provides a…

We give a presentation of the $n$-dimensional oriented cobordism category $\text{Cob}_n$ with generators corresponding to diffeomorphisms and surgeries along framed spheres, and a complete set of relations. Hence, given a functor $F$ from…

With a view towards applications in the theory of infinite-dimensional representations of finite-dimensional Lie supergroups, we introduce a new category of supermanifolds. In this category, supermanifolds of `maps' and `fields' (fibre…

We establish the equivalence between $U(1)$ Chern-Simons and Reshetikhin-Turaev TQFTs associated with finite quadratic modules. For gauge group $U(1)$ and even level $k$, we prove that the corresponding Chern-Simons TQFT is naturally…

We introduce the category of singular 2-dimensional cobordisms and show that it admits a completely algebraic description as the free symmetric monoidal category on a twin Frobenius algebra, by providing a description of this category in…

A K(pi,1)-foliation is one for which the universal covers of all leaves are contractible (thus all leaves are K(pi,1)'s for some pi). In the first part of the paper we show that the tangential Lusternik--Schnirelmann category cat F of a…

The first and shorter part of this thesis deals with the structural assumption of invertibility in a Lie groupoid. When this assumption is dropped, we obtain the notion of a Lie category: a small category, endowed with a compatible…

Two new classes of metrizable vector bundles have been presented in the papers [1] and [4]. The Lie algebroid generalized tangent bundle of a dual vector bundle is presented. This Lie algebroid is a new example of metrizable vector bundle.…

Consider a three dimensional partially hyperbolic diffeomorphism. It is proven that under some rigid hypothesis on the tangent bundle dynamics, the map is (modulo finite covers and iterates) either an Anosov diffeomorphism, a skew-product…

We construct extended TQFTs associated to Rozansky--Witten models with target manifolds $T^*\mathbb{C}^n$. The starting point of the construction is the 3-category whose objects are such Rozansky--Witten models, and whose morphisms are…

We present a proof of the fact that in a symmetric monoidal category over a field of characteristic zero, objects with an invertible exterior power are rigid. As an application we prove two recent conjectures on dimensions in symmetric…

We show that string theory on a compact negatively curved manifold, preserving a U(1)^{b_1} winding symmetry, grows at least b_1 new effective dimensions as the space shrinks. The winding currents yield a "D-dual" description of a Riemann…