Related papers: GKM Theory for Manifolds of Isospectral Matrices i…

For a strongly connected directed graph $\Lambda$, it is shown that the quantum automorphism group of $\Lambda$ acts isometrically in the sense of D. Goswami et al. on the spectral triple $(C^{\ast}(\Lambda), L^{2}(\Lambda^{\infty},M),D)$…

The aim of this note is to provide a comprehensive treatment of the homotopy theory of $\Gamma$-$G$-spaces for $G$ a finite group. We introduce two level and stable model structures on $\Gamma$-$G$-spaces and exhibit Quillen adjunctions to…

We prove a {\Gamma}-equivariant version of the algebraic index theorem, where {\Gamma} is a discrete group of automorphisms of a formal deformation of a symplectic manifold. The particular cases of this result are the algebraic version of…

We introduce the notion of Q-filtrable varieties: projective varieties with a torus action and a finite number of fixed points, such that the cells of the associated Bialynicki-Birula decomposition are all rationally smooth. Our main…

An algebraic GKM manifold is a non-singular algebraic variety equipped with an algebraic action of an algebraic torus, with only finitely many torus fixed points and finitely many 1-dimensional orbits. In this expository article, we use…

An approach, based on the Smith Normal Form, is introduced to study the spectra of symmetric matrices with a given graph. The approach serves well to explain how the path cover number (resp. diameter of a tree T) is related to the maximum…

We study equivariant Gromov-Witten invariants and quantum cohomology in GKM theory. Building on the localization formula, we prove that the resulting expression is independent of the choice of compatible connection, and provide an…

An equivariant map queer Lie superalgebra is the Lie superalgebra of regular maps from an algebraic variety (or scheme) $X$ to a queer Lie superalgebra $\mathfrak{q}$ that are equivariant with respect to the action of a finite group…

GKM theory is a powerful tool in equivariant topology and geometry that can be used to generalize classical ideas from (quasi)toric manifolds to more general torus actions. After an introduction to the topic this survey focuses on recent…

Given a graph $\Gamma$, one may conside the set $X$ of its vertices as a metric space by assuming that all edges have length one. We consider two versions of homology theory of $\Gamma$ and their $K$-theory counterparts -- the $K$-theory of…

Let $G$ and $G'$ be simple Lie groups of equal real rank and real rank at least $2$. Let $\Gamma <G$ and $\Lambda < G'$ be non-uniform lattices. We prove a theorem that often implies that any quasi-isometric embedding of $\Gamma$ into…

We formulate the notion of an isomorphism of GKM graphs. We then show that two GKM graphs have isomorphic graph equivariant cohomology algebras if and only if the graphs are isomorphic.

In this paper we study non-negatively curved and rationally elliptic GKM$_4$ manifolds and orbifolds. We show that their rational cohomology rings are isomorphic to the rational cohomology of certain model orbifolds. These models are…

Let T be a torus of dimension at least k and M a T-manifold. M is a GKM_k-manifold if the action is equivariantly formal, has only isolated fixed points, and any k weights of the isotropy representation in the fixed points are linearly…

In previous work we equipped quiver Grassmannians for nilpotent representations of the equioriented cycle with an action of an algebraic torus. We show here that the equivariant cohomology ring is acted upon by a product of symmetric groups…

Let $G/K$ be a simply connected compact irreducible symmetric space of real rank one. For each $K$-type $\tau$ we compare the notions of $\tau$-representation equivalence with $\tau$-isospectrality. We exhibit infinitely many $K$-types…

It is shown that for a given infinite graph $G$ on countably many vertices, and a compact, infinite set of real numbers $\Lambda$ there is a real symmetric matrix $A$ whose graph is $G$ and its spectrum is $\Lambda$. Moreover, the set of…

It follows from the GKM description of equivariant cohomology that the GKM graph of a GKM manifold has free equivariant graph cohomology, and satisfies a Poincar\'e duality condition. We prove that these conditions are sufficient for an…

Let $(M,\omega)$ be a compact symplectic manifold with a Hamiltonian GKM action of a compact torus. We formulate a positive condition on the space; this condition is satisfied if the underlying symplectic manifold is monotone. The main…

We study the space $X_h$ of Hermitian matrices having staircase form and the given simple spectrum. There is a natural action of a compact torus on this space. Using generalized Toda flow, we show that $X_h$ is a smooth manifold and its…