UBC, Vancouver

There are two basic theories of curve counting on Calabi-Yau threefolds. Donaldson-Thomas theory arises by considering curves as subschemes; Gromov-Witten theory arises by considering curves as the image of maps. Both theories can also be formulated for orbifolds. Let X be a dimension three Calabi-Yau orbifold and let

Y --> X be a Calabi-Yau resolution. The Gromov-Witten theories of X and Y are related by the Crepant Resolution Conjecture. The Gromov-Witten and Donaldson-Thomas theories of Y are related by the famous MNOP conjecture. In this talk I will (with some provisos) formulate the remaining equivalences: the crepant resolution conjecture in Donaldson-Thomas theory and the MNOP conjecture for orbifolds. I will discuss examples to illustrate and provide evidence for the conjectures.

A singular perturbation analysis is presented to analyze various PDE models in a
two-dimensional domain that contain localized regions of non-uniform behavior. A
key theme of this talk is to present a unified mathematical approach, based on
an asymptotic analysis involving logarithmic series and certain Green's function
techniques, that can be used to treat a variety of PDE models such as diffusion
or eigenvalue problems in perforated domains or reaction-diffusion models with
spot-type behavior.

/notices/events/abstracts/applied-analysis/tt05/Heywood.pdf