Keywords defining my area of research
The area of my research is geometry, to be more precise, differential geometry and its applications in theoretical physics . Geometry is one of the most ancient mathematical disciplines, which arose at the dawn of mankind. Why did I choose differential geometry? First of all, probably because I really liked this subject at school. The second reason is that geometry is very closely related to our lives. Everything that is around us (including ourselves) is located in space and the geometry of this space directly affects all the processes taking place around us. Another reason for this choice, perhaps the most important, is that the University of Tartu has a historically strong scientific school in differential geometry. A short description of the development of this scientific school can be found under the button "Differential geometry in Tartu".
What is modern differential geometry?
From the school course in geometry, we know that geometry is theorems about the equality of triangles, the Pythagorean theorem, the lengths of some curves (circles), the area of the simplest figures and the volumes of some geometric bodies. A huge impetus to the development of geometry was given by Descartes' method of coordinates. This method was truly revolutionary, it put into the hands of geometers a powerful tool of algebraic methods, that is, it translated geometry into the language of algebra with its equations and their solutions. The next stage in this development of geometry is associated with the emergence of the differential calculus, the application of which to geometry led to the emergence of differential geometry. Currently, the field of differential geometry is a synthesis of many mathematical disciplines. I would like to especially note the widespread use of algebraic structures and methods in modern differential geometry. This is quite understandable, since the concept of symmetry plays an important role in geometry, and from the algebraic point of view, symmetries form a group . This is precisely the starting point from which an extremely important area of modern differential geometry grows and it is called the theory of Lie groups, Lie algebras and their representations. Note that this theory plays an equally important role in the modern theory of elementary particles. More ...