Projective Invariants of Projective Structures and Applications

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Global topology of the Hitchin system (2011, methodology): “Using Geometric Invariant Theory [Mu, Ne2] one can construct N d; the moduli space of isomorphism classes of stable rank n degree d vector bundles (equivalently GLn-bundles) on C.”
Looking out for stable syzygy bundles (2003, background): “The degrees determine the Chern classes of the syzygy bundle and therefore the question is equivalent to the following.”
A functorial construction of moduli of sheaves (2006, background): “Castelnuovo-Mumford regularity.”
Moduli spaces for principal bundles in arbitrary characteristic (2005, unknown): “Based on his development of Geometric Invariant Theory, David Mumford proposed the notion of a (semi)stable vector bundle on a Riemann surface [25].”
Stability Conditions on An-Singularities (2010, background): “It is a fine mixture of the theory of t-structures [3] and the slope stability [36], which allows us to represent any object in a triangulated category as a successive mapping cone of semistable objects in a unique way.”
Slope filtrations (2008, background): “Then the quotient μ = deg rk , where deg andrk have their usual meaning, is a slope function on C (MumfordTakemoto [77][102] (18)).”