Related papers: A multiplicative analogue of complex symplectic im…
We consider the (direct sum over all $n$ of the) $K$-theory of the semi-nilpotent commuting variety of $\mathfrak{gl}_n$, and describe its convolution algebra structure in two ways: the first as an explicit shuffle algebra (i.e. a…
The Marle-Guillemin-Sternberg (MGS) form is local model for a neighborhood of an orbit of a Hamiltonian Lie group action on a symplectic manifold. One of the main features of the MGS form is that it puts simultaneously in normal form the…
We study a class of algebras with non-Lie commutation relations whose symplectic leaves are surfaces of revolution: a cylinder or a torus. Over each of such surfaces we introduce a family of complex structures and Hilbert spaces of…
A theorem of Delzant states that any symplectic manifold $(M,\om)$ of dimension $2n$, equipped with an effective Hamiltonian action of the standard $n$-torus $\T^n = \R^{n}/2\pi\Z^n$, is a smooth projective toric variety completely…
Motivated by the symmetric version of matrix multiplication we study the plethysm $S^k(\mathfrak{sl}_n)$ of the adjoint representation $\mathfrak{sl}_n$ of the Lie group $SL_n$. In particular, we describe the decomposition of this…
In this article we generalize a theorem by Palais on the rigidity of compact group actions to cotangent lifts. We use this result to prove rigidity for integrable systems on symplectic manifolds including sytems with degenerate…
Smooth threefold flops with irreducible centres are classified by the length invariant, which takes values 1, 2, 3, 4, 5 or 6. This classification by Katz and Morrison identifies 6 possible partial resolutions of Kleinian singularities that…
The purpose of this paper is to establish several new results about the Hodge theory of Lagrangian fibrations on (not necessarily compact) holomorphic symplectic manifolds. Let $M$ be a holomorphic symplectic manifold of dimension $2n$ that…
Symmetric product orbifolds, i.e. permutation orbifolds of the full symmetric group S_{n} are considered by applying the general techniques of permutation orbifolds. Generating functions for various quantities, e.g. the torus partition…
Let X be a normal affine T-variety of complexity at most one over a perfect field k, where T stands for the split algebraic torus. Our main result is a classification of additive group actions on X that are normalized by the T-action. This…
On one side, from the properties of Floer cohomology, invariant associated to a symplectic manifold, we define and study a notion of symplectic hyperbolicity and a symplectic capacity measuring it. On the other side, the usual notions of…
For closed and oriented hyperbolic surfaces, a formula of Witten establishes an equality between two volume forms on the space of representations of the surface in a semisimple Lie group. One of the forms is a Reidemeister torsion, the…
We quantise the Euclidean torus universe via a combinatorial quantisation formalism based on its formulation as a Chern-Simons gauge theory and on the representation theory of the Drinfel'd double DSU(2). The resulting quantum algebra of…
We study fixed-point loci of Nakajima varieties under symplectomorphisms and their anti-symplectic cousins, which are compositions of a diagram automorphism, a reflection functor and a transpose defined by certain bilinear forms. These…
We analyze the hamiltonian quantization of Chern-Simons theory associated to the universal covering of the Lorentz group SO(3,1). The algebra of observables is generated by finite dimensional spin networks drawn on a punctured topological…
We prove that any coadjoint orbit with real eigenvalues of a complex semisimple Lie group, equipped with the real part of the canonical holomorphic symplectic form, is symplectomorphic to the cotangent bundle of a (partial) flag manifold.…
Let $p$ be a fixed odd prime number, $\mu$ be a Hida family over the Iwasawa algebra of one variable, $\rho_{\mu}$ its Galois representation, $\mathbb{Q}_\infty/\mathbb{Q}$ the $p$-cyclotomic tower and $S$ the variable of the cyclotomic…
We call complex quasifold of dimension k a space that is locally isomorphic to the quotient of an open subset of the space C^k by the holomorphic action of a discrete group; the analogue of a complex torus in this setting is called a…
We classify filtered quantizations of conical symplectic singularities and use this to show that all filtered quantizations of symplectic quotient singularities are spherical Symplectic reflection algebras of Etingof and Ginzburg. We…
Given a Hamiltonian torus action on a symplectic manifold, Teleman and Fukaya have proposed that the Fukaya category of each symplectic quotient should be equivalent to an equivariant Fukaya category of the original manifold. We lay out new…