The course will be read on 15, 16, and 18 February at 18:10, and on 17 February at 16:20.
Lecturer: Nikolay Andreevich Tyurin, doctor of physical and mathematical sciences, Professor of the Russian Academy of Sciences, Joint Institute for Nuclear Research (Dubna), Higher School of Economics (Moscow).
Abstract: After the successful use of the anti-self-duality equation to construct new invariants of smooth structures on 4-dimensional manifolds, which led to the appearance of Donaldson polynomials, the natural idea arose to use other equations from physics for the same purposes. As a result, Ed. Witten and N. Seiberg in 1994 proposed a system of equations with an Abelian gauge group, depending on the choice of the metric on the underlying 4-manifold. The moduli varieties of solutions of such systems modulo gauge equivalence for the general metric are finite-dimensional, smooth, and compact (which qualitatively distinguishes this theory from the theory of instantons); as a result, new invariants of smooth structures can be defined. With the help of these invariants, called the Seiberg - Witten invariants, it turned out to be possible to re-prove some of the results obtained by S. Donaldson, and the calculations were greatly simplified. Subsequently, K. Taubes noted that the Seibreg - Witten invariants are nontrivial not only for a complex surface (like the Donaldson invariants), but also for a symplectic manifold of dimension 4. In Taubes's calculations, an essential role was played by almost complex structures that always exist on a symplectic manifold. However, this important result turned out to be only intermediate: in the next paper, Taubes showed that the Seiberg - Witten invariants of a symplectic manifold coincide with the Gromov invariants. From that moment on, a new and popular direction of research emerged, designated by the short abbreviation GM.