Kentaro Nagao

Donaldson–Thomas theory and cluster algebras

We provide a transformation formula of non-commutative Donaldson-Thomas invariants under a composition of mutations. Consequently, we get a description of a composition of cluster transformations in terms of quiver Grassmannians. As an application, we give an alternative proof of Fomin-Zelevinskys conjectures on

HYPERBOLIC 3-MANIFOLDS AND CLUSTER ALGEBRAS

F

K-THEORY OF QUIVER VARIETIES, q-FOCK SPACE AND NONSYMMETRIC MACDONALD POLYNOMIALS

-polynomials and

Motivic Donaldson–Thomas invariants of small crepant resolutions

g

Counting Invariant of Perverse Coherent Sheaves and its Wall-crossing

-vectors.

DERIVED CATEGORIES OF SMALl TORIC CALABI–YAU 3-FOLDS AND CURVE COUNTING INVARIANTS

We advocate the use of cluster algebras and their

Wall-crossing of the motivic Donaldson-Thomas invariants

y

Quiver varieties and Frenkel–Kac construction

-variables in the study of hyperbolic 3-manifolds. We study hyperbolic structures on the mapping tori of pseudo-Anosov mapping classes of punctured surfaces, and show that cluster

Motivic Donaldson-Thomas invariants of the conifold and the refined topological vertex

y

The Non-commutative Topological Vertex and Wall Crossing Phenomena

-variables naturally give the solutions of the edge-gluing conditions of ideal tetrahedra. We also comment on the completeness of hyperbolic structures.