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In this paper, we give both positive and negative answers to Gromov's compactness question regarding positive scalar curvature metrics on noncompact manifolds. First we construct examples that give a negative answer to Gromov's compactness…

Let $S$ be a surface of finite type which is not a sphere with at most four punctures, a torus with at most two punctures, or a closed surface of genus two. Let $\mathcal{MF}$ be the space of equivalence classes of measured foliations of…

Let $M$ be a smooth manifold. When $\Gamma$ is a group acting on the manifold $M$ by diffeomorphisms one can define the $\Gamma$-co-invariant cohomology of $M$ to be the cohomology of the differential complex…

The Lichnerowicz formula yields an index theoretic obstruction to positive scalar curvature metrics on closed spin manifolds. The most general form of this obstruction is due to Rosenberg and takes values in the $K$-theory of the group…

For a grading-restricted vertex superalgebra $V$ and an automorphism $g$ of $V$, we give a linearly independent set of generators of the universal lower-bounded generalized $g$-twisted $V$-module $\widehat{M}^{[g]}_{B}$ constructed by the…

The decomposition $\Gamma=BH$ of a group $\Gamma$ into a subset $B$ and a subgroup $H$ of $\Gamma$ induces, under general conditions, a group-like structure for $B$, known as a gyrogroup. The famous concrete realization of a gyrogroup,…

This is preprint HAL-00429963 (2009). I describe a combinatorial construction of the cohomology classes in compactified moduli spaces of curves $\widehat{Z}_{I}\in H^{*}(\bar{\mathcal{M}}_{g,n})$ starting from the following data: an odd…

The pro-isomorphic zeta function of a finitely generated nilpotent group $\Gamma$ is a Dirichlet generating function that enumerates finite-index subgroups whose profinite completion is isomorphic to that of $\Gamma$. Such zeta functions…

We begin a study of torsion theories for representations of an important class of associative algebras over a field which includes all finite W-algebras of type A, in particular the universal enveloping algebra of gl(n) (or sl(n)) for all…

We study semigroups of convex monotone operators on spaces of continuous functions and their behaviour with respect to $\Gamma$-convergence. In contrast to the linear theory, the domain of the generator is, in general, not invariant under…

We prove that II$_1$ factors $M$ have a unique (up to unitary conjugacy) cross-product type decomposition around ``core subfactors'' $N \subset M$ satisfying the property HT and a certain ``torsion freeness'' condition. In particular, this…

Let M be a Riemann surface with boundary $\partial M$ and genus greater than zero. Let $\Gamma$ be the mapping class group of M fixing $\partial M$. The group $\Gamma$ acts on ${\mathcal M}_{\mathcal C} = \Hom_{\mathcal…

Let $\Gamma$ be a finite dimensional Lie group and consider the smooth double loop group, i.e. the Fr\'echet Lie group of smooth maps from the 2-torus to $\Gamma$. For a finite dimensional Hilbert space V, let H denote the Hilbert space of…

We prove that many spaces of positive scalar curvature metrics have the homotopy type of infinite loop spaces. Our result in particular applies to the path component of the round metric inside $\mathcal{R}^+ (S^d)$ if $d \geq 6$. To achieve…

For a convex cocompact subgroup $G<Mod(S)$, and points $x,y \in Teich(S)$ we obtain asymptotic formulas as $R\to \infty$ of $|B_{R}(x)\cap Gy|$ as well as the number of conjugacy classes of pseudo-Anosov elements in $G$ of dilatation at…

$\Gamma$-structures are weak forms of multiplications on closed oriented manifolds. As shown by Hopf the rational cohomology algebras of manifolds admitting $\Gamma$-structures are free over odd degree generators. We prove that this…

In part II we constructed the lower bound, in the spirit of $\Gamma$- $\liminf$ for some general classes of singular perturbation problems, with or without the prescribed differential constraint, taking the form E_\e(v):=\int_\Omega…

We study the higher genus equivariant Gromov-Witten theory of the Hilbert scheme of n points of the plane. Since the equivariant quantum cohomology is semisimple, the higher genus theory is determined by an R-matrix via the Givental-Teleman…

Let $\Gamma$ be a finite group acting on a Lie group $G$. We consider a class of group extensions $1 \to G \to \hat{G} \to \Gamma \to 1$ defined by this action and a $2$-cocycle of $\Gamma$ with values in the centre of $G$. We establish and…

The cohomology $H^*(\Gamma, E) $ of a torsion-free arithmetic subgroup $\Gamma$ of the special linear $\mathbb{Q}$-group $\mathsf{G} = SL_n$ may be interpreted in terms of the automorphic spectrum of $\Gamma$. Within this framework, there…