Characterizing algebraic groups via maximal tori

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Understanding symmetry, and how and why it arises in nature, is important in both Mathematics and Physics. Recall that the mathematical objects that measure symmetry are called ``groups'' and their study is known as ``group theory''. Classical examples of groups are those of rotations or translations (continuous groups) and the symmetry of a square or a snowflake (discrete groups). The mid 20th century saw the birth of Algebraic Groups, objects that capture the spirit of continuous groups (the so-called Lie groups) and discrete groups, but that are much more universal. Over the course of subsequent decades the theory of algebraic groups has been used to give a unified treatment of several key areas of algebra and number theory, including the theories of quadratic forms, central simple algebras, algebras with involution and some non-associative algebras.******The research program centers on understanding the very nature of algebraic groups themselves, and their applications to several areas of Mathematics. The main goal of the project is to achieve important results in characterizing algebraic groups via their maximal tori and ramification locus. Recall that any algebraic group can be thought of as a ``union of its simple subobjects", called maximal tori. A natural question appears immediately: ******What can one say about two algebraic groups given that they have the same maximal tori?******In other words, using analogy with ``children puzzles", we can rephrase it as follows: if we destroy all connections and relations between maximal tori in a given group G and take their disjoint union, one can ask how to glue these tori together in order to reconstruct G itself. Also, one can ask in how many ways we can glue a family of given tori in order to construct a new group. ******The problem of characterizing absolutely almost simple algebraic groups having the same maximal tori is rooted in the classical results on the maximal subfields and the splitting fields of division algebras and it has recently received a good deal of attention in algebra and geometry. This was due in part to newly discovered connections with geometric problems involving isospectral and length-commensurable Riemannian manifolds and locally symmetric spaces, but in fact questions of this kind are relevant also for other areas. ******We intend to attack this problem by the studying the ramification behavior of algebraic groups. We expect that for a given group G defined over a finitely generated field there are only finitely many groups which have the same ramification properties as G . To obtain this result we are going to study different forms of local-global principles for torsors. Recall that torsors are tools that help us to construct groups out of some local data. Any success in understanding local-global behavior of torsors would lead us to solutions of many open long-standing conjectures in the theory of algebraic groups and geometry.***********