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The realization problem asks: When does an algebraic complex arise, up to homotopy, from a geometric complex? In the case of 2- dimensional algebraic complexes, this is equivalent to the D2 problem, which asks when homological methods can…

For a manifold $M$ with an integral closed 3-form $\omega$, we construct a $PU(H)$-bundle and a Lie groupoid over its total space, together with a curving in the sense of gerbes. If the form is non-degenerate, we furthermore give a natural…

Let $M_{l,m}$ be the total space of the $S^3$-bundle over $S^4$ classified by the element $l\sigma+m\rho\in{\pi_4(SO(4))}$, $l,m\in\mathbb Z$. In this paper we study the homotopy theory of gauge groups of principal $G$-bundles over…

Based on [1], we study the complexity of horizontality in each twistor space $\hat{E}_{\varepsilon}$ associated with an oriented vector bundle $E$ of rank $4$ with a positive-definite metric over the $2$-torus $T^2$, and obtain…

We introduce a notion of Poincar\'e duality for pairs of $\infty$-categories, extending Poincar\'e-Lefschetz duality for pairs of spaces. This categorical extension yields an efficient book-keeping device that affords, among other things, a…

Wall's D(2) problem asks if a cohomologically 2-dimensional geometric 3-complex is necessarily homotopy equivalent to a geometric 2-complex. We solve part of the problem when the fundamental group is dihedral of order $2^n$, and offer a…

Let $G$ be a simply-connected simple compact Lie group and let $M$ be an orientable smooth closed 4-manifold. In this paper we calculate the homotopy type of the suspension of $M$ and the homotopy types of the gauge groups of principal…

A natural explicit condition is given ensuring that an action of the multiplicative monoid of non-negative reals on a manifold F comes from homotheties of a vector bundle structure on F, or, equivalently, from an Euler vector field. This is…

A double disk bundle is any smooth closed manifold obtained as the union of the total spaces of two disk bundles, glued together along their common boundary. The Double Soul Conjecture asserts that a closed simply connected manifold…

Let M be a simply connected Riemannian symmetric space, with at most one flat direction. We show that every Riemannian (or unitary) vector bundle with parallel curvature over M is an associated vector bundle of a canonical principal bundle,…

We present several examples of T^8/P orbifolds with $P \subset SU(4)$. We compute their Hodge numbers and consider turning on discrete torsion. We then study supersymmetric compactifications of type II, heterotic, and type I strings on…

We consider the isomonodromy problems for flat $G$-bundles over punctured elliptic curves $\Sigma_\tau$ with regular singularities of connections at marked points. The bundles are classified by their characteristic classes. These classes…

This paper is devoted to the study of the uniformization of the moduli space of pairs (X, E) consisting of an algebraic curve and a vector bundle on it. For this goal, we study the moduli space of 5-tuples (X, x, z, E, \phi), consisting of…

The structure space S(M) of a closed topological m-manifold M classifies bundles whose fibers are closed m-manifolds equipped with a homotopy equivalence to M. We construct a highly connected map from S(M) to a concoction of algebraic…

On a smooth complex projective variety $X$ of dimension $n$, consider an ample vector bundle $\mathcal{E}$ of rank $r \leq n-2$ and an ample line bundle $H$. A numerical character $m_2=m_2(X,\mathcal{E},H)$ of the triplet…

We describe relations between hyperbolic geometry and codimension two knots or, more exactly, between varieties of conjugacy classes of discrete faithful representations of the fundamental groups of hyperbolic n-manifolds M into…

We construct manifestly superconformal field theories in six dimensions which contain a non-Abelian tensor multiplet. In particular, we show how principal 3-bundles over a suitable twistor space encode solutions to these self-dual tensor…

We introduce the concept of Spin^G-structure in a SO-bundle, where $G\subset U(V)$ is a compact Lie group containing $-id_V$. We study and classify $Spin^G(4)$-structures on 4-manifolds, we introduce the G-Monopole equations associated with…

We describe explicitly the moduli spaces $M^{pst}_g(S,E)$ of polystable holomorphic structures $E$ with $\det E\cong K$ on a rank 2 vector bundle $E$ with $c_1(E)=c_1(K)$ and $c_2(E)=0$ for all minimal class VII surfaces $S$ with $b_2(S)=1$…

Let $M$ be a closed, oriented, simply connected 6-manifold. After localization away from 2, we give a homotopy decomposition of $\Sigma M$ in terms of spheres, Moore spaces and other recognizable spaces. As applications we calculate…