NSA JOINT PROJECT PROPOSAL ( WITH GORDANA ) CLUSTER THEORY AND RELATED TOPICS Bibliography for Kiyoshi Igusa

A. Summary of items most closely related to the proposal: (1) In [7] (The generalized Grassmann invariant ...), the PI introduces pictures, derives their basic properties, and uses them to construct an element of order 16 in K 3 (Z). (2) In [40] (Links, pictures and the homology of nilpotent groups), the PI, with Kent Orr, analyze picture for torsion free nilpotent groups and construct special pictures which they called " atomic pictures " and showed that these labels the cells in a CW-complex which is a K(π, 1) for the nilpotent group. They use this construction to prove the " k-slice conjecture " for Milnor µ-link invari-ants. This conjecture states that a link is k-slice if and only if the µ-invariants of length ≤ 2k vanish. (3) In [41] (Cluster complexes via semi-invariants), the PIs, with Kent Orr and Jerzy Weyman, introduce presentation spaces: these are Hom(P, Q) where P, Q are projective modules. They use semi-invariants defined on presentation spaces to extend the domains of semi-invariants to negative dimension vectors. They use these to show that the stability conditions for semi-invariants determine the cluster complex for a hereditary algebra over an algebraically closed field. (4) In [42] (Modulated semi-invariants and nilpotent groups), the PIs, with Kent Orr and Jerzy Weyman, extend the results of [41] to modulated quivers. This paper also proves the crucial fact that, up to sign, the c-vectors of a cluster are equal to the weights of the virtual semi-invariants associated to the cluster. (5) In [43] (Picture groups of finite type ...), the PIs, with Kent Orr and Jerzy Weyman, define picture groups of finite type and compute the cohomology of the picture group of type A n with straight orientation. This paper relies on two other papers [33], [58] to prove that the " picture space " is a K(π, 1) and, therefore, the cohomology of this space is the cohomology of the group. (6) In [33] (The category of noncrossing partitions), the PI gives a combinatorial construction of the cluster morphism category of type A n with straight orientation and proves that the classifying space is locally CAT(0) and therefore a K(π, 1). (7) In [54] (Continuous Frobenius categories) the PIs use representations of the circle S 1 to construct various Frobenius categories whose stable categories are the continuous cluster categories. This is the first paper in the " continuous categories " …