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It is shown that certain transformations on quiver-dimension vector pairs induce isomorphisms on the corresponding moduli spaces of quiver representations and map a stable dimension vector to a stable dimension vector. This result combined…

We survey old and new results on the existence of moduli spaces of semistable coherent sheaves both in algebraic and in complex geometry.

We construct moduli spaces of representations of quivers over arbitrary schemes and show how moduli spaces of pointed curves of genus zero like the Grothendieck-Knudsen moduli spaces $\overline{M}_{0,n}$ and the Losev-Manin moduli spaces…

We extend the scope of a former paper to vector bundle problems involving more than one vector bundle. As the main application, we obtain the solution of the well-known moduli problems of vector bundles associated with general quivers.

We construct good moduli spaces from moduli of objects in the sense of To\"en-Vaqui\'e. As an application, we construct good moduli spaces for perverse sheaves.

We construct a new compactification of the moduli space of maps from pointed nonsingular projective stable curves to a nonsingular projective variety with prescribed ramification indices at the points. It is shown to be a proper…

This is a note in which we first review symmetries of moduli spaces of stable meromorphic connections on trivial vector bundles over the Riemann sphere, and next discuss symmetries of their integrable deformations as an application. In the…

We prove several results on the structure of solvable quotients of fundamental groups of compact Kahler manifolds (Kahler groups).

Noncommutative phase spaces are generated and classified in the framework of centrally extended anisotropic planar kinematical Lie groups as well as in the framework of noncentrally extended planar absolute time Lie groups. Through these…

We analyse the vector bundle moduli arising from generic heterotic compactifications from the point of view of quiver representations. Phenomena such as stability walls, crossing between chambers of supersymmetry, splitting of non-Abelian…

We discuss the noncommutative generalizations of polynomial algebras which after appropriate completions can be used as coordinate algebras in various noncommutative settings, (noncommutative differential geometry, noncommutative algebraic…

We construct a modular compactification via stable slc pairs for the moduli spaces of K3 surfaces with a nonsymplectic group of automorphisms under the assumption that some combination of the fixed loci of automorphisms defines an effective…

For each positive integer Q there exists a path connected metric compactum X such that the Qth-homotopy group of X is compactly generated but not a topological group (with the quotient topology).

We explore methods for constructing normal forms of indecomposable quiver representations. The first part of the paper develops homological tools for recursively constructing families of indecomposable representations from indecomposables…

Given a noncommutative Hamiltonian space $A$, we prove that the conjecture ``{\it quantization commutes with reduction}'' holds for $A$. We further construct a semidirect product algebra $A \rtimes \mG^A$, and establish a correspondence…

This is a noncommutative-geometric study of the semiclassical dynamics of finite topological D-brane systems. Starting from the formulation in terms of A -infinity categories, I show that such systems can be described by the noncommutative…

A new framework for noncommutative complex geometry on quantum homogeneous spaces is introduced. The main ingredients used are covariant differential calculi and Takeuchi's categorical equivalence for faithfully flat quantum homogeneous…

In this paper, we first show that for an acyclic gentle algebra A, the irreducible components of any moduli space of A-modules are products of projective spaces. Next, we show that the nice geometry of the moduli spaces of modules of an…

Let $RQ$ be the path algebra of a Dynkin quiver $Q$ over a commutative noetherian ring $R$. We show that any homotopically smashing t-structure in the derived category of $RQ$ is compactly generated. We also give a complete description of…

In this paper, for an even dimensional compact manifold with boundary which has the non-product metric near the boundary, we use the noncommutative residue to define a conformal invariant pair. For a 4-dimensional manifold, we compute this…