We explore the conformal geometric structures of a pair of second-order partial-differential equations. In particular, we investigate the conditions under which this geometry is conformal to the vacuum Einstein equations of general relativity. Furthermore, we introduce a new version of the conformal Einstein equations, which are used in the analysis of the conformal geometry of the PDE’s.

We study scaling limits and conformal invariance of critical site percolation on triangular lattice. We show that some percolation-related quantities are harmonic conformal invariants, and calculate their values in the scaling limit. As a particular case we obtain conformal invariance of the crossing probabilities and Cardy’s formula. Then we prove existence, uniqueness, and conformal invarianc...

The standard conformal compactification of Euclidean space is the round sphere. We use conformal geodesics to give an elementary proof that this is the only possible conformal compactification.

The string theory is a fast moving subject, both physics wise and in the respect of mathematics. In order to keep up with the discipline it is important to move with new ideas which are being stressed. Here I wish to give extracts from new papers of ideas which I have recently found interesting. There are six papers which are involved: I ."Strings formulated directly in 4 dimensions " A. N...

A conformal transformation is a diffeomorphism which preserves angles; the differential at each point is the composition of a rotation and a dilation. In its original sense, conformal geometry is the study of those geometric properties preserved under transformations of this type. This subject is deeply intertwined with complex analysis for the simple reason that any holomorphic function f(z) o...

We discuss a numerical implementation of conformal welding for finitely connected regions using the geodesic zipper algorithm and Koebe’s iterative method for computing conformal maps to regions bounded by circles. We also show that a conformal map from a finitely connected region to a region bounded by circles can be written as a composition of finitely many conformal maps of simply connected ...

We study scaling limits and conformal invariance of critical site percolation on triangular lattice. We show that some percolation-related quantities are harmonic conformal invariants, and calculate their values in the scaling limit. As a particular case we obtain conformal invariance of the crossing probabilities and Cardy’s formula. Then we prove existence, uniqueness, and conformal invarianc...

Some years ago Koutras presented a method of constructing a conformal Killing tensor from a pair of orthogonal conformal Killing vectors. When the vector associated with the conformal Killing tensor is a gradient, a Killing tensor (in general irreducible) can then be constructed. In this paper it is shown that the severe restriction of orthogonality is unnecessary and thus it is possible that m...