Mathematics Colloquia and Seminars

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In 1985, Donaldson, Gromov, and Jones made revolutionary discoveries that led to the three independent theories named after them. Very recent works in representation theory and physics due to Nakajima, Nekrasov, Vafa and others have revealed that sometimes generating functions of the three different theories of three different manifolds are completely identical, after "tensoring" Witten to each theory. This talk sets its starting point at Atiyah's influential paper, "New invariants of 3- and 4-dimensional manifolds" (1988), and motivates the ideas of Gromov-Witten invariants from the consideration of TQFT of dimensions 2, 3 and 4. We review the ideas of, and the inter-relations among, Donaldson, Seiberg-Witten, and Jones-Witten invariants, Floer homology, Chern-Simons gauge theory, and conformal blocks. We then introduce Gromov-Witten invariants, and calculate their generating functions for rather simple (2D) examples using representation theory of symmetric groups, following Okounkov and Pandharipande. Through this calculation a connection to conformal field theory, and hence to Jones-Witten theory, is suggested. We also observe that the generating functions are solutions to a famous integrable system of nonlinear partial differential equations.

Prerequisite: To save time, theory of connections, curvatures, characteristic classes, de Rham theory, symplectic geometry, complex Kaehler geometry, sheaf cohomology, Heegaard splittings of three-manifolds, and representation theory of symmetric grou