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We study the orbifold Hirzebruch-Riemann-Roch (HRR) theorem for quotient Deligne-Mumford stacks, explore its relation with the representation theory of finite groups, and derive a new orbifold HRR formula via an orbifold Mukai pairing. As a…

We show that the vortex moduli space in non-abelian supersymmetric N=(2,2) gauge theories on the two dimensional plane with adjoint and anti-fundamental matter can be described as an holomorphic submanifold of the instanton moduli space in…

This work is motivated by two central questions in the birational geometry of moduli spaces of curves -- Fulton's conjecture and the effective cone of $\bar M_g$. We study the algebro-geometric aspect of Teichmuller curves parameterizing…

Any homogeneous polynomial $P(x, y, z)$ of degree $d$, being restricted to a unit sphere $S^2$, admits essentially a unique representation of the form $\lambda + \sum_{k = 1}^d [\prod_{j = 1}^k L_{kj}]$, where $L_{kj}$'s are linear forms in…

Any homogeneous polynomial $P(x, y, z)$ of degree $d$, being restricted to a unit sphere $S^2$, admits essentially a unique representation of the form $\lambda_0 + \sum_{k = 1}^d \lambda_k [\prod_{j = 1}^k L_{kj}]$, where $L_{kj}$'s are…

This paper shows that immersed totally geodesic $m$-dimensional suborbifolds of $n$-dimensional arithmetic hyperbolic orbifolds correspond to finite subgroups of the commensurator whenever $m \geqslant \frac{n-1}{2}$. We call such totally…

Coassociative 4-folds are a particular class of 4-dimensional submanifolds which are defined in a 7-dimensional manifold M with a G_2 structure given by a `positive' differential 3-form, sometimes called G_2-form. Assuming that a G_2-form…

We prove a Cauchy-type integral representation for classes of functions holomorphic in four priviledged tuboid domains of the complexified one-sheeted two-dimensional hyperboloid. From a physical viewpoint, this hyperboloid can be used for…

Let $\Gamma \subset \mathbf{PU}(2,1)$ be a lattice which is not co-compact, of finite Bergman-covolume and acting freely on the open unit ball $\mathbf{B} \subset \mathbb{C}^2$. Then the compactification $X = \bar{\Gamma \setminus…

Let $Y(E_n)$ denote the moduli space of pairs $(S,B)$ where $S$ is a del Pezzo surface of degree $9-n$ and $B$ is the labeled (marked) sum of its finitely many lines. When $n=6$, $Y(E_6)$ is the classical moduli space of marked cubic…

Let $M$ be a closed hyperbolic $3$-manifold. A homotopy class $[S]$ of surfaces in $M$ is filling if any representative cuts $M$ into components contractible in $M$. We prove that there exist $\epsilon_0, g_0>0$ such that every homotopy…

Some ball-quotient orbifolds are related by covering maps. We exploit these coverings to find infinite towers of orbifolds uniformized by the complex 2-ball and some orbifolds over K3 surfaces uniformized by the 2-ball. Corresponding…

We shall obtain unobstructed deformations of four geometric structures: Calabi-Yau, HyperK\"ahler, $\G$ and Spin(7) structures in terms of closed differential forms (calibrations). We develop a direct and unified construction of smooth…

We investigate the geometry of the Simpson moduli space M of stable sheaves on P_3 with Hilbert polynomial H(m)=3m+1 and describe explicitly the two smooth, rational components, their 11-dimensional smooth, transversal intersection and the…

Many-body wavefunctions usually lie in high-dimensional Hilbert spaces. However, physically relevant states, i.e, the eigenstates of the Schr\"odinger equation are rare. For many-body systems involving only pairwise interactions, these…

We find the moduli space of multi-solitons in noncommutative scalar field theories at large theta, in arbitrary dimension. The existence of a non-trivial moduli space at leading order in 1/theta is a consequence of a Bogomolnyi bound obeyed…

Moduli spaces of doubly periodic monopoles, also called monopole walls or monowalls, are hyperk\"ahler; thus, when four-dimensional, they are self-dual gravitational instantons. We find all monowalls with lowest number of moduli. Their…

The Eisenstein-Picard modular surface $M$ is the quotient space of the complex hyperbolic plane by the modular group $\rm PU(2,1; \mathbb{Z}[\omega])$. We determine the global topology of $M$ as a 4-orbifold.

We study the metric and topological properties of the space $\mathscr{D}(G)$ of left-invariant hyperbolic pseudometrics on the non-elementary hyperbolic group $G$ that are quasi-isometric to a word metric, up to rough similarity. This space…

We construct some cusped finite-volume hyperbolic $n$-manifolds $M_n$ that fiber algebraically in all the dimensions $5\leq n \leq 8$. That is, there is a surjective homomorphism $\pi_1(M_n) \to \mathbb Z$ with finitely generated kernel.…