Global motivic homotopy theory

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This research proposal is in the areas of mathematics known as algebraic geometry and homotopy theory. Algebraic geometry studies algebraic varieties which are of principal importance. First they are relatively easy to understand since they are just defined by polynomial equations, next they usually give a rather accurate approximation to other shapes, most importantly they do appear naturally in quite a lot of subjects in theoretical physics, coding theory and computer sciences. That is why algebraic geometry - the theory of algebraic varieties is so important for the development and applications of mathematics. Homotopy theory is a considerably newer area of mathematics, being an important branch of algebraic topology, the modern development of what is popularly known as "rubber-sheet geometry", that is, the study of the properties of curves, surfaces and objects of higher dimension which are preserved under operations such as bending and stretching; in homotopy theory one allows additional modifications by "continuous deformation". Since its creation homotopy theory has become an essential component of modern mathematics. Homotopy theory has numerous applications both in and out of mathematics, including theoretical physics and computer sciences.Motivic homotopy theory is a blend of algebraic geometry and homotopy theory. Its primary object is to study algebraic varieties from a homotopy theoretic viewpoint. Many of the basic ideas and techniques in this subject originate in algebraic topology. Motivic homotopy theory led to such striking applications as the solution of the Milnor conjecture and the Bloch-Kato conjecture, in algebraic geometry. Besides these quite spectacular applications, the fact that one can use the ideas and techniques of homotopy theory to solve problems in algebraic geometry has attracted mathematicians from both fields and has led to a wealth of new constructions and applications.The principal aim of this project is to develop global motivic homotopy theory, investigate motivic equivariant spectra and a range of associated cohomology theories of algebraic varieties. Its study will shed light on some classical problems in motivic homotopy theory. Also, we want to apply methods of global motivic homotopy theory to classical global algebraic topology. We believe that these investigations will have important computational advantages. The homotopy-theoretic and geometric outlook that we develop will also be useful in other areas of mathematics such as algebraic topology, non-commutative geometry and mathematical physics.Effective developments of the project objectives require methods of algebraic geometry, motivic homotopy theory, equivariant topology and representation theory.