By now there is a long list of questions in analysis and algebra which are known to be undecided in the standard set theory (Zermelo-Fraenkel). In particular, no standard methods accepted and used by mathematicians can provide a proof deciding such questions. Yet, a definitive answer is often desirable. I will discuss some axioms that settle most of these open questions, provide useful extensions of standard set theory, and are intersting on their own. These axioms rely on the existence of sets that are significantly "larger" than any sets mainstream mathematics works with. 

In this talk, I will present some rigidity problems and theorems from analysis, partial differential equations and differential geometry. For examples, the uniqueness theorem of holomorphic functions upper rigidity of harmonic mapping. In particular, I will present some rigidity theorem for proper holomorphic mapping and smooth solutions of some degenerate elliptic partial differential equations. 

Yau's conjecture states that the volume of the nodal set of
Laplace eigenfunctions on a compact Riemannian manifold is comparable to
the square root of the corresponding eigenvalue. Donnelly and Fefferrman
proved Yau's conjecture for real analytic metrics but the conjecture stays
widely open for smooth metrics specially in dimensions n>2. Recently
Sogge-Zelditch and Colding-Minicozzi have established new lower bounds for
the volume of the nodal sets. In this talk we give a new proof of
Colding-Minicozzi's result using a different method. This is a joint work
with Christopher Sogge and Zuoqin Wang.

Penrose tilings and substitution sequences, spectral properties of operators in Hilbert space and dynamical systems, fractals and convolutions of singular measures - we will see how all these topics meet in the study of mathematical quasicrystals.