We define the notion of a Kirby element of a ribbon category $\mathcal{C}$ (not necessarily semisimple). Kirby elements lead to 3-manifold invariants. We characterize a class of Kirby elements, the algebraic Kirby elements, in terms of the structure maps of a Hopf algebra in $\mathcal{C}$. This class is sufficiently large to recover the quantum invariants of 3-manifolds of Reshetikhin and Turaev, of Hennings, Kauffman and Radford, and of Lyubashenko when these are well defined. The cases of a semisimple ribbon category and of a category of representations are explored in detail.

In this paper, I give a generalized analogue of the string topology results of Chas and Sullivan, and of Cohen and Jones. For a finite simplicial complex $X$ and $k \geq 1$, I construct a spectrum $Maps(S^k, X)^{S(X)}$, which is obtained by taking a generalization of the Spivak bundle on $X$ (which however is not a stable sphere bundle unless $X$ is a Poincaré space), pulling back to $Maps(S^k, X)$ and quotienting out the section at infinity. I show that the corresponding chain complex is naturally homotopy equivalent to an algebra over the $(k + 1)$-dimensional unframed little disk operad $\mathcal{C}_{k + 1}$. I also prove a conjecture of Kontsevich, which states that the Quillen cohomology of a based $\mathcal{C}_k$-algebra (in the category of chain complexes) is equivalent to a shift of its Hochschild cohomology, as well as prove that the operad $C_{\ast}\mathcal{C}_k$ is Koszul-dual to itself up to a shift in the derived category. This gives one a natural notion of (derived) Koszul dual $C_{\ast}\mathcal{C}_k$-algebras. I show that the cochain complex of $X$ and the chain complex of $\Omega^k X$ are Koszul dual to each other as $C_{\ast}\mathcal{C}_k$-algebras, and that the chain complex of $Maps(S^k, X)^{S(X)}$ is naturally equivalent to their (equivalent) Hochschild cohomology in the category of $C_{\ast}\mathcal{C}_k$-algebras.

We characterise the 1-unconditional subsets $(\mathrm{e}_{rc})_{(r,c) \in I}$ of the set of elementary matrices in the Schatten–von-Neumann class $\mathrm{S}^p$. The set of couples $I$ must be the set of edges of a bipartite graph without cycles of even length $4 \lel \le p$ if $p$ is an even integer, and without cycles at all if $p$ is a positive real number that is not an even integer. In the latter case, $I$ is even a Varopoulos set of V-interpolation of constant 1. We also study the metric unconditional approximation property for the space $\mathrm{S}^p_I$ spanned by $(\mathrm{e}_{rc})_{(r,c) \in I}$ in $\mathrm{S}^p$.

Let $1 \le q \le p <\infty$ and let $\mathcal{C}$ be the class of all bounded convex domains $\Omega$ in $\mathbb{R}^n$. Following the approach in `An optimal Poincaré inequality in $L^1$ for convex domains', by G. Acosta and R. G. Durán (Proc. Amer. Math. Soc. 132 (2003) 195–202), we show that the best constant $C$ in the weighted Poincaré inequality

$$ \| f - f_{av} \|_{L^q_w (\Omega)} \le C w(\Omega)^{\frac{1}{q} - \frac{1}{p}} \mbox{diam}(\Omega) \| \nabla f \|_{L^p_w(\Omega)} $$

for all $\Omega \in \mathcal{C}$, all Lipschitz continuous functions $f$ on $\Omega$, and all weights $w$ which are any positive power of a non-negative concave function on $\Omega$ is the same as the best constant for the corresponding one-dimensional situation, where $\mathcal{C}$ reduces to the class of bounded intervals. Using facts from `Sharp conditions for weighted 1-dimensional Poincaré inequalities', by S.-K. Chua and R. L. Wheeden (Indiana Math. J. 49 (2000) 143–175), we estimate the best constant. In the case $q = 1$ and $1 <\infty$, our estimate is between the best constant and twice the best constant. Furthermore, when $p = q = 1$ or $p = q = 2$, the estimate is sharp. Finally, in the case where the domains in $\mathbb{R}^n$ are further restricted to be parallelepipeds, we obtain a slightly different form of Poincaré's inequality which is better adapted to directional derivatives and the sidelengths of the parallelepipeds. We also show that this estimate is sharp for a fixed rectangle.

A generalization of classical theorems on the existence of sections of real, complex and quaternionic Stiefel manifolds over spheres is proved. We obtain a complete list of Lie group homomorphisms $\rho : G \to G_n$, where $G_n$ is one of the groups $SO(n)$, $SU(n)$ or $Sp(n)$ and $G$ is one of the groups $SO(k)$, $SU(k)$ or $Sp(k)$, which reduce the structure group $G_n$ in the fibre bundle $G_n \to G_{n + 1} \to G_{n + 1} / G_n$.

We provide an analogue of Gundy's decomposition for $L_1$-bounded non-commutative martingales. An important difference from the classical case is that for any $L_1$-bounded non-commutative martingale, the decomposition consists of four martingales. This is strongly related with the row/column nature of non-commutative Hardy spaces of martingales. As applications, we obtain simpler proofs of the weak type $(1,1)$ boundedness for non-commutative martingale transforms and the non-commutative analogue of Burkholder's weak type inequality for square functions. A sequence $(x_n)_{n \ge 1}$ in a normed space $\mathrm{X}$ is called 2-co-lacunary if there exists a bounded linear map from the closed linear span of $(x_n)_{n \ge 1}$ to $l_2$ taking each $x_n$ to the $n$th vector basis of $l_2$. We prove (using our decomposition) that any relatively weakly compact martingale difference sequence in $L_1 (\mathcal{M}, \tau)$ whose sequence of norms is bounded away from zero is 2-co-lacunary, generalizing a result of Aldous and Fremlin to non-commutative $L_1$-spaces.

We prove explicit upper bounds for the torsional rigidity of extrinsic domains of minimal submanifolds $P^m$ in ambient Riemannian manifolds $N^n$ with a pole $p$. The upper bounds are given in terms of the torsional rigidities of corresponding Schwarz symmetrizations of the domains in warped product model spaces. Our main results are obtained via previously established isoperimetric inequalities, which are here extended to hold for this more general setting based on warped product comparison spaces. We also characterize the geometry of those situations in which the upper bounds for the torsional rigidity are actually attained and give conditions under which the geometric average of the stochastic mean exit time for Brownian motion at infinity is finite.