These are expository lectures reviewing (1) recent developments in two-dimensional YangMills theory and (2) the construction of topological field theory Lagrangians. Topological field theory is discussed from the point of view of infinite-dimensional differential geometry.We emphasize the unifying role of equivariant cohomology both as the underlying principle in the formulation of BRST transformation laws and as a central concept in the geometrical interpretation of topological field theory path integrals.

We describe a topological string theory which reproduces many aspects of the 1/N expansion of SU (N ) Yang-Mills theory in two spacetime dimensions in the zero coupling (A = 0) limit. The string theory is a modified version of topological gravity coupled to a topological sigma model with spacetime as target. The derivation of the string theory relies on a new interpretation of Gross and Taylor's "Ω −1 points." We describe how inclusion of the area, coupling of chiral sectors, and Wilson loop expectation values can be incorporated in the topological string approach.

Novel curative therapies using genetic transfer of normal globin-producing genes into autologous hematopoietic stem cells (HSCs) are in clinical trials for patients with sickle cell disease (SCD). The percentage of transferred globin necessary to cure SCD is currently not known. In the setting of allogeneic nonmyeloablative HSC transplants (HSCTs), stable mixed chimerism is sufficient to reverse the disease. We regularly monitored 67 patients after HSCT. After initially robust engraftment, 3 of these patients experienced declining donor myeloid chimerism (DMC) levels with eventual return of disease. From this we discovered that 20% DMC is necessary to reverse the sickle phenotype. We subsequently developed a mathematical model to test the hypothesis that the percentage of DMC necessary is determined solely by differences between donor and recipient red blood cell (RBC) survival times. In our model, the required 20% DMC can be entirely explained by the large differences between donor and recipient RBC survival times. Our model predicts that the requisite DMC and therefore necessary level of transferred globin is lowest in patients with the highest reticulocyte counts and concomitantly shortened RBC lifespans.

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