Since the mid 1990’s, the leading candidate for a unified theory of all fundamental physical interactions has been M Theory.
A full formulation of M Theory is still not available, and it is only understood through its limits in certain regimes, which are either one of five 10-dimensional string theories, or 11-dimensional supergravity. The equations for these theories are mathematically interesting in themselves, as they reflect, either directly or indirectly, the presence of supersymmetry. We discuss recent progresses and open problems about two of these theories, namely supersymmetric compactifications of the heterotic string and of 11-dimensional supergravity. This is based on joint work of the speaker with Sebastien Picard and Xiangwen Zhang, and with Teng Fei and Bin Guo.
Differential equations deal with the same matters as children do: pictures in the plane. If a picture related to a differential equation remains (topologically) the same after the equation is slightly perturbed, this equation is structurally stable. If it is not, abrupt changes of the corresponding picture may occur under a small perturbation. These abrupt changes are the subject of the bifurcation theory. This talk gives a survey of the first three years of development of a new branch of the bifurcation theory: global bifurcations on the two sphere. Bifurcations in generic one-parameter families were classified; the answer appeared to be quite unexpected. An important and non-trivial question ”who bifurcates?” was answered. Natalya Goncharuk and the speaker defined a set called large bifurcation support; bifurcations that occur in a small neighborhood of this set determine the global bifurcations on the two-sphere. This result is a starting point for systematic classification of global bifurcations in two-parameter families. New examples of structurally unstable three-parameter families will be demonstrated. These are joint results of the speaker and his collaborators: N. Goncharuk, D. Filimonov, Yu. Kudryashov, N. Solodovnikov, I. Schurov and others. The talk will be addressed to a broad audience.
Non-self-adjoint operators appear in many settings, from kinetic theory
and quantum mechanics to linearizations of equations of mathematical
physics. The spectral analysis of such operators, while often notoriously
difficult, reveals a wealth of new phenomena, compared with their
self-adjoint counterparts. Spectra for non-self-adjoint operators display
fascinating features, such as lattices of eigenvalues for operators of
Kramers-Fokker-Planck type, say, and eigenvalues for operators with
analytic coefficients in dimension one, concentrated to unions of curves
in the complex spectral plane. In this talk, after a general introduction,
we shall discuss spectra for non-self-adjoint perturbations of
self-adjoint operators in dimension two, under the assumption that the
classical flow of the unperturbed part is completely integrable.
The role played by the flow-invariant Lagrangian tori of the completely
integrable system, both Diophantine and rational, in the spectral analysis
of the non-self-adjoint operators will be described. In particular, we
shall discuss the spectral contributions of rational tori, leading to
eigenvalues having the form of the "legs in a spectral centipede". This
talk is based on joint work with Johannes Sj\"ostrand.