Abstract:
The key result of superstring theory will be formulated as an existence theorem. A definition of a model of superstring theory will be given by using properties of solutions of the two-dimensional wave equation and representation theory. The proof of the theorem is outlined that there exist at least 5 models of superstring theory, called type I, type IIA, type IIB, type HO and type HE string theories.
The spacetime (a Lorentzian manifold together with a timelike vector field) should be 10-dimensional and the gauge group SO(32) or $E_8\times E_8$.
All necessary notions will be defined in the talk.
Remarks:
It would be nice to have one unified theory. What is known about it? These 5 models of superstring theory are related by S-duality, which has an analogue in the Langlands program, and by T-duality, which is based on the mirror symmetry of cohomology of Calabi-Yau manifolds. There is a conjecture that each of the five types string theories become special cases of so called M-theory. Amazing AdS/CFT-correspondence is a holography relationship between superstring model of the type IIB on AdS_$5\times S^5$ and the conformal-invariant $N = 4$ supersymmetric Yang-Mills theory in the four-dimensional Minkowski spacetime.
Some mathematical topics in superstring theory: hypothesis of the absence of divergences in higher orders of perturbation theory; moduli of super Riemann surfaces; nonperturbative string field theory and a sum over topologies; quantum gravity and classification of Riemannian manifolds; cobordisms;
$(\infty,n)$-category; holography.
Some other problems: black hole information paradox and firewalls; did the universe exist before the Big Bang; hypothesis oа quantum fluctuations of the number field.

References

M. B. Green, J. H. Schwarz, E. Witten, Superstring theory, v. 1, 2, Cambridge University Press, 1987 ; M.B. Grin, Dzh. Shvarts, E. Vitten, Teoriya superstrun, v. 1, 2, 1990

Hisham Sati, Urs Schreiber (eds.), Mathematical foundations of quantum field theory and perturbative string theory, Proceedings of Symposia in Pure Mathematics, 83, Amer. Math. Soc., Providence, RI, 2011, viii+354 pp.

I.V. Volovich, “Number theory as the ultimate physical theory”, P-Adic Numbers, Ultrametric Anal. Appl., 2:1 (2010), 77–87

I.V. Volovich, “From $p$-adic strings to étale strings”, Proc. Steklov Inst. Math., 203 (1995), 37–42

I.Ya. Aref'eva, I.V. Volovich, “Quantization of the Riemann zeta-function and cosmology”, Int. J. Geom. Methods Mod. Phys., 4:5 (2007), 881–895, arXiv: hep-th/0701284