Noncommutative versions of some classical birational transformations

HyperKähler Manifolds and Birational Transformations

The minimal model program has greatly enhanced our knowledge on birational geometry of varieties of dimension 3 and higher. About the same time, the last two decades have also witnessed increasing interests in HyperKähler manifolds, a particular class of Calabi-Yau manifolds. One interest in this area, which we hope to treat in the future, is to investigate the behavior of the SYZ mirror conjec...

Non-Euclidean Versions of Some Classical Triangle Inequalities

In this paper we recall with short proofs of some classical triangle inequalities, and prove corresponding non-Euclidean, i.e., spherical and hyperbolic versions of these inequalities. Among them are the well known Euler’s inequality, Rouché’s inequality (also called “the fundamental triangle inequality”), Finsler–Hadwiger’s inequality, isoperimetric inequality and others.

Some classical function theory theorems and their modern versions

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On Certain Groups of Birational Contact Transformations*

On birational monomial transformations of plane

We study birational monomial transformations of the form φ(x : y : z) = (ε1x1y1z1 : ε2x2y2z2 : xα3yβ3zγ3), where ε1,ε2 ∈ {−1,1}. These transformations form a group. We describe this group in terms of generators and relations and, for every such transformation φ, we prove a formula, which represents the transformationφ as a product of generators of the group. To prove this formula, we use birati...