R. P. Thomas

Braid group actions on derived categories of coherent sheaves

This paper gives a construction of braid group actions on the derived category of coherent sheaves on a variety

A study of the Hilbert-Mumford criterion for the stability of projective varieties

X

Curves on K3 surfaces and modular forms

. The motivation for this is Kontsevichs homological mirror conjecture, together with the occurrence of certain braid group actions in symplectic geometry. One of the main results is that when

The 3-fold vertex via stable pairs

\dim X \geq 2

Hodge theory and derived categories of cubic fourfolds

, our braid group actions are always faithful. We describe conjectural mirror symmetries between smoothings and resolutions of singularities that lead us to find examples of braid group actions arising from crepant resolutions of various singularities. Relations with the McKay correspondence and with exceptional sheaves on Fano manifolds are given. Moreover, the case of an elliptic curve is worked out in some detail.

Reduced classes and curve counting on surfaces I: theory

We make a systematic study of the Hilbert-Mumford criterion for different notions of stability for polarised algebraic varieties

MOMENT MAPS, MONODROMY AND MIRROR MANIFOLDS

(X,L)

On the Motivic Stable Pairs Invariants of K3 Surfaces

; in particular for K- and Chow stability. For each type of stability this leads to a concept of slope

13/2 ways of counting curves

\mu

An obstructed bundle on a Calabi-Yau 3-fold

for varieties and their subschemes; if