A promising way to gain more insights into how people read and understand a map is by analyzing their eye gaze while working with a map. However, eye gaze logging on interactive maps is challenging due to of various reasons. One of the biggest challenges comes from the interaction with a map that can drastically change the visual representation of the depicted content, making it difficult to de...

It is shown that the signature of a manifold with a symplectic circle action, having only isolated fixed points, equals the alternating sum of the Novikov numbers corresponding to the cohomology class of the generalized moment map. The same is true for more general fixed point sets 1. Theorem. Let M be a symplectic manifold with a symplectic circle action having only isolated fixed points. Then...

K-polystability is, on the one hand, conjecturally equivalent to existence of certain canonical K\"ahler metrics polarised varieties, and, other gives correct notion form moduli. We introduce a stability for families K-polystable extending classical slope bundle, viewed as family varieties via associated projectivisation. conjecture that this is condition forming moduli fibrations. Our main res...

Let (G, K) be one of the following classical irreducible Hermitian symmetric pairs of noncompact type: (SU(p, q), S (U(p) × U(q))), (Sp(n,R),U(n)), or (SO(2n),U(n)). Let GC and KC be complexifications of G and K, respectively, and let P be a maximal parabolic subgroup of GC whose Levi subgroup is KC. Let V be the holomorphic part of the complexifiaction of the tangent space at the origin of G/K...

Let G be a reductive complex Lie group with Lie algebra g. We call a subgroup H ⊂ G cramped if there is an integer b(G,H) such that each finite-dimensional representation of G has a non-trivial invariant subspace of dimension less than b(G,H). We show that a subgroup is cramped if and only if the moment map T ∗(K/L) → k∗ is surjective, where K and L are compact forms of G and H. We will use thi...

This paper gives methods for understanding invariants of symplectic quotients. The symplectic quotients that we consider are compact symplectic manifolds (or more generally orbifolds), which arise as the symplectic quotients of a symplectic manifold by a compact torus. A companion paper [23] examines symplectic quotients by a nonabelian group, showing how to reduce to the maximal torus. Through...

Let G be a compact Lie group, and let LG denote the corresponding loop group. Let (X,ω) be a weakly symplectic Banach manifold. Consider a Hamiltonian action of LG on (X,ω), and assume that the moment map μ : X −→ Lg∗ is proper. We consider the function |μ|2 : X −→ R, and use a version of Morse theory to show that the inclusion map j : μ(0) −→ X induces a surjection j∗ : H∗ G(X) −→ H∗ G(μ−1(0))...