Quantum cohomology has its origins in symplectic geometry and algebraic geometry, but is deeply related to differential equations and integrable systems. This text explains in simple terms what is behind the extraordinary success of quantum cohomology, leading to its connections with many existing areas of mathematics as well as its appearance in new areas such as mirror symmetry. Aimed at graduate students in mathematics who want to learn about quantum cohomology in a broad context, and theoretical physicists who are interested in the mathematical setting, the text assumes basic familiarity with differential equations and cohomology.

1. The many faces of cohomology; 2. Quantum cohomology; 3. Quantum differential equations; 4. Linear differential equations in general; 5. The quantum D-module; 6. Abstract quantum cohomology; 7. Integrable systems; 8. Solving integrable systems; 9. Quantum cohomology as an integrable system; 10. Integrable systems and quantum cohomology; References

Martin A. GuestTokyo Metropolitan University