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We show that whenever a closed symplectic manifold admits a Hamiltonian diffeomorphism with finitely many simple periodic orbits, the manifold has a spherical homology class of degree two with positive symplectic area and positive integral…

Inspired by constructions of Kova\v{c}evi\'{c}, we introduce the amalgamated free product of circle actions, obtained by blowing up two actions along prescribed orbits and rearranging the inserted intervals. Under natural orbit and index…

We show that for a Hamiltonian action of a compact torus $G$ on a compact, connected symplectic manifold $M$, the $G$-equivariant cohomology is determined by the residual $S^1$ action on the submanifolds of $M$ fixed by codimension-1 tori.…

In this article, we consider fixed spin-1/2 particles interacting through the quantized electromagnetic field in a constant magnetic field. We give approximate evolutions of coherent states. This uses spins-photon classical Hamiltonian…

- We have suggested using the action-angle variables for the study of a (quasi)particle in quantum ring. We have presented the action-angle variables for three two-dimensional singular oscillator systems - We have suggested a procedure of…

We prove several results on symplectic varieties with a Hamiltonian action of a reductive group having invariant Lagrangian subvarieties. Our main result states that the images of the moment maps of a Hamiltonian variety and of the…

We study compact complex manifolds $M$ admitting a conformal holomorphic Riemannian structure invariant under the action of a complex semi-simple Lie group $G$. We prove that if the group $G$ acts transitively and essentially, then $M$ is…

Let $M$ be a compact symplectic manifold on which a compact torus $T$ acts Hamiltonialy with a moment map $\mu$. Suppose there exists a symplectic involution $\theta:M\to M$, such that $\mu\circ\theta=-\mu$. Assuming that 0 is a regular…

For actions with a dense orbit of a connected noncompact simple Lie group $G$, we obtain some global rigidity results when the actions preserve certain geometric structures. In particular, we prove that for a $G$-action to be equivalent to…

We consider the pseudo-Riemannian Lichnerowicz conjecture in the homogeneous setting. In particular, we show that any compact connected pseudo-Riemannian manifold $M$ on which a semisimple group $G$ acts conformally, essentially and…

We study conformal actions of connected nilpotent Lie groups on compact pseudo-Riemannian manifolds. We prove that if a type-(p,q) compact manifold M supports a conformal action of a connected nilpotent group H, then the degree of…

We develop a set of sufficient conditions for guaranteeing that an integrable system with a symmetry group $K$ on a manifold $M$ descends to an integrable system on a dense open subset of the quotient Poisson space $M/K$. The higher…

Let $S$ be an oriented surface of finite type, $\mathcal{MCG}(S)$ its mapping class group, and $\mathcal{T}(S)$ its Teichm\"uller space with the Teichm\"uller metric. Let $H \leq \mathcal{MCG}(S)$ be a finite subgroup and consider the…

Let $X$ be a finitistic space with its rational cohomology isomorphic to that of the wedge sum $P^2(n)\vee S^{3n} $ or $S^{n} \vee S^{2n}\vee S^{3n}$. We study continuous $\mathbb{S}^1$ actions on $X$ and determine the possible fixed point…

Let $M$ be a simple holomorphically symplectic manifold, that is, a simply connected holomorphically symplectic manifold of Kahler type with $h^{2,0}=1$. We prove that the group of holomorphic automorphisms of $M$ acts on the set of faces…

We study the orbit structure and the geometric quantization of a pair of mutually commuting hamiltonian actions on a symplectic manifold. If the pair of actions fulfils a symplectic Howe condition, we show that there is a canonical…

We consider compact manifolds $M$ with a cohomogeneity one action of a compact Lie group $G$ such that the orbit space $M/G$ is a closed interval. For $T$ a maximal torus of $G$, we find necessary and sufficient conditions on the group…

We prove a variant of the Chance-McDuff conjecture for pseudo-rotations: under certain additional conditions, a closed symplectic manifold which admits a Hamiltonian pseudo-rotation must have deformed quantum product and, in particular,…

We discuss how the global geometry and topology of manifolds depend on different group actions of their fundamental groups, and in particular, how properties of a non-trivial compact 4-dimensional cobordism $M$ whose interior has a complete…

For any compact connected Lie group $G$, we study the Hamiltonian sum of two compact Hamiltonian group $G$-manifolds $(X^+,\omega^+,\mu^+)$ and $(X^-,\omega^-,\mu^-)$ with a common codimension 2 Hamiltonian submanifold $Z$ of the opposite…