Exact Results in Quantum Theory

I will present the relationship between a wide class of amplitudes intopological string theory and quiver representation theory. For agiven string theory model, we find its dual description in terms ofeffective spacetime theory, or supersymmetric quantum mechanics, whoseinteractions are succinctly encoded in a quiver graph. This relation,in various more specific situations, leads to very interestingmathematical and physics statements. For example, for topologicalstring configurations related to knots, the quiver point of viewenables to prove some old conjectures about knot invariants and leadsto new knot invariants. On the other hand, we show that amplitudes intopological strings on toric Calabi-Yau manifolds without compactfour-cycles, which take the form of q-hypergeometric functions, can bealso encoded in some quiver. This leads to interesting new statementsabout hypergeometric functions and their q-analogs. In this talk, Iwill summarize these relations and discuss prospects for future work.