Algebraic Transformation Groups

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In a general sense, this program proposes to investigate symmetries of certain mathematical objects. ******Symmetries are at the heart of mathematical investigation. These can be symmetries of geometric objects (triangles, squares, but also significantly more intricate things, in mathematical language referred to as manifolds, or their algebraic generalizations, varieties). ***Such symmetries are usually described by structures called groups. This builds on the elementary but fundamental insight that if an object of whatever nature has two symmetries, then usually applying one of these symmetries after the other again results in a symmetry (for example, a square can be rotated by 90 degrees. Performing two such rotations consequently results in another symmetry, namely rotation by 180 degrees). This generalizes to much more complicated objects. ******The purpose of this program is, in a broad sense, to understand the symmetries of complicated algebraic varieties, and use these to describe their singularities. Here, a "singularity" in geometric terms is roughly speaking an "exception" in the general shape of the thing; for example, in the case of a triangle, the three corners would be considered singularities, as they look significantly different from the rest of the triangle. It is not easy to define precisely what "singularity" means without resorting to technical jargon, but understanding such singularities has widespread mathematical applications. ******The second part of the project deals with an abstract concept, called "derivations." Derivations (not all, but the most important ones) arise as approximations of the kind of symmetries alluded to above. For reasonable classes of symmetries, there exists a class of derivations that describes these symmetries "infinitesimally" (for examples a circle admits as symmetries arbitrary rotations; an infinitesimal approximation of such a rotation would the a (very "infinitesimally" short) translation along a straight tangent of the circle). To understand the original symmetries, one should first understand their approximations in the form of derivations. ******Understanding symmetries and related objects (Transformation Groups and Lie Algebras) has wide spread applications both within Mathematics, but also in Physics and other sciences. For example the theory of transformation groups can help in analysing phylogenetic trees (describing hereditary relationships between species) in Biology. **