Abstract:

Variations of stability conditions, and geometric methods relating different (semi)stable loci in moduli spaces, are powerful techniques in enumerative geometry and related areas. In particular, one can sometimes obtain "wall-crossing" formulas for enumerative invariants of such loci, in any cohomology theory of Borel-Moore type, via various so-called "master spaces". I will explain some remarkable algebraic structures and properties that appear in the general study of such wall-crossing formulas, e.g. for elliptic genus. The main case of interest, and also the most sophisticated, is Joyce's recent discovery that that, for a wide class of linear categories like categories of coherent sheaves, cohomological wall-crossing formulas are controlled by a vertex algebra. This may be refined to equivariant K-theory, and leads to rich interactions with geometric representation theory. I will conclude by explaining some joint work, with Nikolas Kuhn and Felix Thimm, on applications of all these ideas to enumerative problems of 3-Calabi-Yau type, notably various flavors of Donaldson-Thomas theory.