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We construct the first examples of rationally convex surfaces in the complex plane with hyperbolic complex tangencies. In fact, we give two very different types of rationally convex surfaces: those that admit analytic fillings by…
In this paper, we compare the symplectic (co)homology capacity with the spectral capacity in the relative case. This result establishes a chain of inequalities of relative symplectic capacities, which is an analogue of the non-relative…
We exhibit many examples of closed symplectic manifolds on which there is an autonomous Hamiltonian whose associated flow has no nonconstant periodic orbits (the only previous explicit example in the literature was the torus T^2n (n\geq 2)…
We shall consider nonrestricted representations of $C_l-$ type Lie algebra over an algebraically closed field of characteristic $p\geq7.$ This paper gives some counter examples to important theory relating to the representations of modular…
We present some examples of locally conformal symplectic structures of the first kind on compact nilmanifolds which do not admit Vaisman metrics. One of these examples does not admit locally conformal K\"ahler metrics and all the structures…
This paper deals with the following question: which manifolds can be realized as leaves of codimension-1 symplectic foliations on closed manifolds? We first observe that leaves of symplectic foliations are necessarily strongly geometrically…
In this note we give an example of a group which is locally embeddable into finite groups (in particular it is initially subamenable, sofic and hence hyperlinear) but does not have Kirchberg's factorization property. This group provides…
We survey recent results on the representation theory of symplectic reflection algebras, focusing particularly on connections with symplectic quotient singularities and their resolutions, spaces of representations of quivers, and on…
We describe some examples of non abelian nilpotent Lie algebras which are not algebraic.
In symplectic geometry, symplectic invariants are useful tools in studying symplectic phenomena. Hofer-Zehnder capacity and displacement energy are important symplectic invariants. Usher proved the so-called sharp energy-capacity inequality…
Roughly twenty five years ago Hofer asked: can the cylinder B^2(1) \times \mathbb{R}^{2(n-1)} be symplectically embedded into B^{2(n-1)}(R) \times \mathbb{R}^2 for some R>0? We show that this is the case if R \geq \sqrt{2^{n-1}+2^{n-2}-2}.…
It is well known(cf. Weil, G\'erardin's works) that there are two different Weil representations of a symplectic group over an odd finite field. By a twisted action, we show that one can reorganize them as a representation of a related…
We compute the symplectic reductions for the action of Sp_2n on several copies of C^2n and for all coregular representations of Sl_2. If it exists we give at least one symplectic resolution for each example. In the case Sl_2 acting on…
This is a reaction to the article Symplectic bipotentials, in published form [2] Harakeh M, Ban M, de Saxce G. Symplectic bipotentials. Mathematics and Mechanics of Solids. 2026;0(0) doi:10.1177/10812865251413554, and in preprint form [1]…
We discuss some examples of open manifolds which admit non-isomorphic symplectic structures of Liouville type.
We use positive S^1-equivariant symplectic homology to define a sequence of symplectic capacities c_k for star-shaped domains in R^{2n}. These capacities are conjecturally equal to the Ekeland-Hofer capacities, but they satisfy axioms which…
This paper is devoted to the construction of analogues of higher Ekeland-Hofer symplectic capacities for $P$-symmetric subsets in the standard symplectic space $(\mathbb{R}^{2n},\omega_0)$, which is motivated by Long and Dong's study…
Examples of nonformal simply connected symplectic manifolds are constructed.
We introduce coordinates on the spaces of framed and decorated representations of the fundamental group of a surface with nonempty boundary into the symplectic group $Sp(2n,\mathbf R)$. These coordinates provide a noncommutative…
The First and Second Representation Theorem for sign-indefinite quadratic forms are extended. We include new cases of unbounded forms associated with operators that do not necessarily have a spectral gap around zero. The kernel of the…