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A symplectic Lie group is a Lie group with a left-invariant symplectic form. Its Lie algebra structure is that of a quasi-Frobenius Lie algebra. In this note, we identify the groupoid analogue of a symplectic Lie group. We call the…

For $\textrm{SL}(n,\mathbb{R})$ ($n\geq3$), $\textrm{SO}(n+1,n)$ ($n\geq2$), $\textrm{Sp}(2n,\mathbb{R})$ ($n\geq2$) and for the adjoint real split form of the exceptional group $\textrm{G}_2$, we exhibit non-uniform lattices in which we…

For finite dimensional hermitean inner product spaces $V$, over $*$-fields $F$, and in the presence of orthogonal bases providing form elements in the prime subfield of $F$, we show that quantifier free definable relations in the subspace…

We introduce the notion of a symplectic hopfoid, which is a "groupoid-like" object in the category of symplectic manifolds where morphisms are given by canonical relations. Such groupoid-like objects arise when applying a version of the…

Given a finite connected graph $\Lambda$, the space of $SU(2)$ lattice gauge-fields on $\Lambda$, modulo gauge transformations, is a Lagrangian submanifold -- with mild singularities -- of the $SU(2)$ character variety (= phase-space of…

Denote by $Sp(k,l)$ the quaternionic symplectic group of signature $(k,l)$. We study the deformation rigidity of the embedding $Sp(k,l) \times Sp(1) \hookrightarrow H$, where $H$ is either $Sp(k+1,l)$ or $Sp(k,l+1)$, this is done by…

Let $G$ be a Lie group, and let $(M,\omega)$ be a symplectic manifold. If $G$ admits a Hamiltonian action on $(M,\omega)$ with momentum map $\mu$, then $M$, the zero-level set of $\mu$, the orbit space, and the corresponding symplectic…

We show that continuous bounded group cohomology stabilizes along the sequences of real or complex symplectic Lie groups, and deduce that bounded group cohomology stabilizes along sequences of lattices in them, such as…

We are interested in comparing properties of symplectic mapping class groups of symplectic manifolds of dimension four or higher with properties of classical mapping class groups of surfaces. For $n \geq 2$, consider a configuration of…

We construct smooth embeddings of spherical quandles into conjugation quandles of Lie groups, where the ambient Lie groups can be taken to be orthogonal, Spin, or Pin groups. Moreover, in dimensions $1$ and $3$, we compare our embeddings…

There is a natural action of the braid group on the symmetric matrices with units on the diagonal, appearing in various fields as Singularity Theory, Frobenius Manifolds or Isomonodromic deformations of certain classes of linear…

A new class of $A^{(1)}_n$ integrable lattice models is presented. These are interaction-round-a-face models based on fundamental nimrep graphs associated with the $A^{(1)}_n$ conjugate modular invariants, there being a model for each value…

A manifold is said to be $n$-plectic if it is equipped with a closed, nondegenerate $(n+1)$-form. This thesis develops the theory of \emph{relative $n$-plectic structures}, where the classical condition is replaced by a closed,…

The set of isotopy classes of ordered n-component links in the 3-sphere is acted on by the symmetric group via permutation of the components. The intrinsic symmetry group of the link, S(L), is defined to be the set of elements in the…

We show that there is an hierarchy of intersection rigidity properties of sets in a closed symplectic manifold: some sets cannot be displaced by symplectomorphisms from more sets than the others. We also find new examples of rigidity of…

In this paper we prove the connectedness of symplectic ball packings in the complement of a spherical Lagrangian, S^2 or RP^2, in symplectic manifolds that are rational or ruled. Via a symplectic cutting construction this is a natural…

In this paper we define a notion of Witt group for sesquilinear forms in hermitian categories, which in turn provides a notion of Witt group for sesquilinear forms over rings with involution. We also study the extension of scalars for…

We use the local theta correspondences between the quaternionic Hermitian groups and the quaternionic skew-Hermitian groups to understand the distinction problem for the symmetric pair SL(2,E)/SL(1,D), where E is a quadratic field extension…

We consider conformal actions of simple Lie groups on compact Lorentzian manifolds. Mainly motivated by the Lorentzian version of a conjecture of Lichnerowicz, we establish the alternative: Either the group acts isometrically for some…

We classify the connected Lie subgroups of the symplectic group $Sp(2,\mathbb{R})$ whose elements are matrices in block lower triangular form. The classification is up to conjugation within $Sp(2,\mathbb{R})$. Their study is motivated by…