Related papers: Simpson-Mochizuki Correspondence for $\lambda$-Fla…
We construct a relative compactification of Dolbeault moduli spaces of Higgs bundles for reductive algebraic groups on families of projective manifolds that is compatible with the Hitchin morphism.
The moduli spaces of stable quasimaps unify various moduli appearing in the study of Gromov-Witten Theory. This note is a survey article on the moduli of stable quasimaps, based on joint papers with Ciocan-Fontanine and Maulik as well as…
The main result of this paper is the construction of a family of superintegrable Hamiltonian systems on moduli spaces of flat connections on a principle $G$-bundle on a surface. The moduli space is a Poisson variety with Atiyah-Bott Poisson…
We study the moduli spaces which classify smooth surfaces along with a complex line bundle. There are homological stability and Madsen--Weiss type results for these spaces (mostly due to Cohen and Madsen), and we discuss the cohomological…
We use the theory of logarithmic line bundles to construct compactifications of spaces of roots of a line bundle on a family of curves, generalising work of a number of authors. This runs via a study of the torsion in the tropical and…
We use geometric invariant theory and the language of quivers to study compactifications of moduli spaces of linear dynamical systems. A general approach to this problem is presented and applied to two well known cases: We show how both…
Geometric structures on manifolds became popular when Thurston used them in his work on the geometrization conjecture. They were studied by many people and they play an important role in higher Teichm\"uller theory. Geometric structures on…
Let $X$ be a complex irreducible smooth projective curve, and let ${\mathbb L}$ be an algebraic line bundle on $X$ with a nonzero section $\sigma_0$. Let $\mathcal{M}$ denote the moduli space of stable Hitchin pairs $(E,\, \theta)$, where…
We show that on any closed contact manifold of dimension greater than 1 a contact structure with vanishing contact homology can be constructed. The basic idea for the construction comes from Giroux. We use a special open book decomposition…
In this paper we develop a Kobayashi-Hitchin type correspondence between solutions of the extended Bogomolny equations on $\Sigma\times \RP$ with Nahm pole singularity at $\Sigma \times \{0\}$ and the Hitchin component of the stable…
We consider the moduli space $\mathscr{N}$ of stable vector bundles of degree $0$ over a compact Riemann surface and the affine bundle $\mathscr{A}\to\mathscr{N}$ of flat connections. Following the similarity between the Teichm\"{u}ller…
For the Riemannian manifold $M^{n}$ two special connections on the sum of the tangent bundle $TM^{n}$ and the trivial one-dimensional bundle are constructed. These connections are flat if and only if the space $M^{n}$ has a constant…
We survey recent progress in the study of moduli of vector bundles on higher-dimensional base manifolds. In particular, we discuss an algebro-geometric construction of an analogue for the Donaldson-Uhlenbeck compactification and explain how…
We study how a general steady configuration of finitely-many point vortices, with Newtonian interaction or generalized surface quasi-geostrophic interactions, can be desingularized into a steady configuration of vortex patches. The…
We give an explicit finite-dimensional model for the derived moduli stack of flat connections on $\mathbb{C}^k$ with logarithmic singularities along a weighted homogeneous Saito free divisor. We investigate in detail the case of plane…
We propose a novel approach to contact Hamiltonian mechanics which, in contrast to the one dominating in the literature, serves also for non-trivial contact structures. In this approach Hamiltonians are no longer functions on the contact…
Let $X$ be a compact Riemann surface of genus $g \geq 3$ and $S$ a finite subset of $X$. Let $\xi$ be fixed a holomorphic line bundle over $X$ of degree $d$. Let $\mathcal{M}_{pc}(r, d, \alpha)$ (respectively, $\mathcal{M}_{pc}(r, \alpha,…
Harmonic maps that minimise the Dirichlet energy in their homotopy classes are known as lumps. Lump solutions on real projective space are explicitly given by rational maps subject to a certain symmetry requirement. This has consequences…
In this paper, we show that the family of moduli spaces of $\balpha'$-stable $(\bt, \blambda)$-parabolic $\phi$-connections of rank 2 over $\BP^1$ with 4-regular singular points and the fixed determinant bundle of degree -1 is isomorphic to…
The moduli space of holomorphic fiber bundles ${\cal M}_n(\Si)$ over a compact Riemann surface $\Si$ is considered. A formula for the regularised determinant and an other for the symplectic form at trivial bundle are proposed.