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I show the existence of a new type of vortex solution which is non-static but stationary and carries angular momentum. This {\it spinning vortex} can be embedded in models with trivial vacuum topology like a model with $SU(2)_{global}\times…
We study moduli stabilisation in four-dimensional $N=1$ supergravity theories which originate from compactifications of the heterotic string on certain manifolds with $SU(3)$ structure. These theories have a non-trivial superpotential…
We revisit the topic of the existence and azimuthal modulational stability of solitary vortices (alias vortex solitons) in the two-dimensional (2D) cubic-quintic nonlinear Schr{\"o}dinger equation. We develop a semi-analytical approach,…
We prove homological stability for both general linear groups of modules over a ring with finite stable rank and unitary groups of quadratic modules over a ring with finite unitary stable rank. In particular, we do not assume the modules…
We formulate a mathematical setup for computational neural networks using noncommutative algebras and near-rings, in motivation of quantum automata. We study the moduli space of the corresponding framed quiver representations, and find…
This work deals with twinlike models that support topological structures such as kinks, vortices and monopoles. We investigate the equations of motion and develop the first order framework to show how to build distinct models with the same…
We study the elliptic zastava spaces, their versions (twisted, Coulomb, Mirkovic local spaces, reduced) and relations with monowalls moduli spaces and Feigin-Odesskii moduli spaces of $G$-bundles with parabolic structure on an elliptic…
In this paper, we construct smooth travelling counter-rotating vortex pairs with circular supports for the generalized surface quasi-geostrophic equation. These vortex pairs are analogues of the Lamb dipoles for the two-dimensional…
The quantum Euclidean spheres, $S_q^{N-1}$, are (noncommutative) homogeneous spaces of quantum orthogonal groups, $\SO_q(N)$. The *-algebra $A(S^{N-1}_q)$ of polynomial functions on each of these is given by generators and relations which…
We study moduli spaces of certain sextic curves with a singularity of multiplicity 3 from both perspectives of Deligne-Mostow theory and periods of K3 surfaces. In both ways we can describe the moduli spaces via arithmetic quotients of…
In order to extend the geometrization of Yangian $R$-matrices from Lie algebras $gl(n)$ to superalgebras $gl(M|N)$, we introduce new quiver-related varieties which are associated with representations of $gl(M|N)$. In order to define them…
For an elliptic surface $q:Y \to \Sigma$, with prescribed singular fibres, Stefan Bauer proved directly via algebraic geometry that the stable bundles over $Y$, whose chern classes are pull backs from $\Sigma$, correspond to the stable…
Vortices symmetric with respect to simultaneous parity and time reversing transformations are considered on the square lattice in the framework of the discrete nonlinear Schr\"{o}dinger equation. The existence and stability of vortex…
We describe two geometrically meaningful compactifications of the moduli space of elliptic K3 surfaces via stable slc pairs, for two different choices of a polarizing divisor, and show that their normalizations are two different toroidal…
We compute the homotopy type of the moduli space of flat, unitary connections over aspherical surfaces, after stabilizing with respect to the rank of the underlying bundle. Over the orientable surface M^g, we show that this space has the…
We provide evidence for conjectural dualities between nonrelativistic Chern-Simons-matter theories and theories of (fractional, nonAbelian) quantum Hall fluids in $2+1$ dimensions. At low temperatures, the dynamics of nonrelativistic…
We prove that the moduli space of gauge equivalence classes of symplectic vortices with uniformly bounded energy in a compact Hamiltonian manifold admits a Gromov compactification by polystable vortices. This extends results of Mundet i…
We analyse the vector bundle moduli arising from generic heterotic compactifications from the point of view of quiver representations. Phenomena such as stability walls, crossing between chambers of supersymmetry, splitting of non-Abelian…
Semitopological Vortices (Q-Rings) are identified to be classical soliton configurations whose stability is attributed to both topological and nontopological charges. We discuss some recent work on the simplest possible realization of such…
This paper concerns the investigation of the stability properties of relative equilibria which are rigidly rotating vortex configurations sometimes called vortex crystals, in the N-vortex problem. Such a configurations can be characterized…