The project covers a wide range of aspects in algebra such as algebraic geometry, algebraic groups and motivic cohomology. The investigator proposes to use theory of algebraic cycles in geometry to study properties of splitting varieties of algebraic objects. The first topic the investigator proposes to study is the canonical dimension of an algebraic object such as algebraic variety, quadratic form, central simple algebra, etc. The canonical dimension measures complexity of the class of splitting fields of an algebraic object. In particular, the investigator proposes to compute the canonical dimension of projective homogeneous varieties and simple algebraic groups. The investigator also plans jointly with A. Suslin to compute the motivic cohomology of Severi-Brauer varieties and generalize this computation to the case of generic splitting varieties of arbitrary symbols. These varieties were used in the proof of the Milnor and the Bloch-Kato Conjecture.
The main objective of mathematics is to provide an approximation to the picture of the physical world. This project develops methods from topology that studies continuous transformations of structures called topological spaces and algebraic geometry concerning geometric objects coming from graphing polynomial equations and called algebraic varieties. This project is devoted to the study of certain fundamental problems of motivic cohomology theory - a relatively new and very quickly developing branch of algebraic geometry. The new areas of algebra that will hopefully evolve from the work on the project will create new research opportunities for mentoring graduate students and junior faculty and provide material for graduate courses.
The project covered a wide range of aspects in algebra such as algebraic geometry, algebraic groups and motivic cohomology. The first topic of the PI project concerned the essential and canonical dimension of an algebraic object such as algebraic variety, quadratic form, central simple algebra, etc. The essential and canonical dimension of algebraic objects have been studied. These discrete invariants measure complexity of algebraic objects. The problem of computation of essential and canonical dimension required diverse techniques in the area of algebra: algebraic geometry, algebraic K-theory, equivariant algebraic K-theory, algebraic stacks and gerbes, central simple algebras and Severi-Brauer varieties. It was proved (in a joint work with N. Karpenko) that the essential dimension of a p-group coincides with the minimum of the dimension of a faithful representation of the group. In a joint paper with A. Suslin, the PI computed the motivic cohomology of the class of splitting varieties of a symbol in algebraic Milnor K-theory modulo a prime integer. These varieties were used in the proof of the Milnor and the Bloch-Kato Conjecture. The main tool in the proof is the use of Quillen operations that allow one to connect various groups of motivic cohomology. The proposed computation has various applications in K-theory and Galois cohomology theory. The book ``The Algebraic and Geometric Theory of Quadratic Forms" by R.Elman, N.Karpenko and the PI has been published (AMS, Colloquium Publications Series). It is monograph on recent developments in the algebraic theory of quadratic forms. This book is a comprehensive study of the algebraic theory of quadratic forms, from classical theory to recent developments, including results and proofs that have never been published. The book is written from the viewpoint of algebraic geometry and includes the theory of quadratic forms over fields of characteristic two, with proofs that are characteristic independent whenever possible. The following graduate students successfully completed Ph.D. theses on the topics of the proposal: M. Blunk, S. Baek, D.H. Nguyen, A. Ruozzi, W. Wong, B. Mathews, S. Blinstein. The work on the project was reported in various international meetings including : Programm ``Geometric Applications of Homotopy Theory" at the Fields Institute, Toronto, 2007, Abel Symposium ``Algebraic Topology" in Oslo, August 2007, Conference ``Algebraic Geometry, K-theory, and Motives", St. Petersburg, Russia, June 2010, Conference ``Ramification in Algebra and Geometry at Emory, Emory University, Atlanta, May 2011, ``Deformation theory, patching, quadratic forms, and the Brauer group", American Institute of Mathematics, Palo Alto, California ,January 2011. The results of the project were presented in the following serious of lectures intended mostly for graduate students, postdocs and young researches: Mini-course: ``Essential dimension and canonical dimension", University of Lens, June 2018, ``Motives and Milnor conjecture", Summer School, Institut de Math'ematiques de Jussieu, Paris, June 2011. Mini-course: ``Essential dimension in algebra", Summer school ``Algebra and Geometry", Pedagogical University, Yaroslavl, July 2012. The PI was a co-organizer of the following conferences: ``Quadratic Forms and Linear Algebraic Groups", May 2009, Oberwolfach, Germany. ``Linear Algebraic Groups and Related Structures", September 2009, Banff International Research Station, Canada. ``International Algebraic Conference dedicated to the 70th birthday of Anatoly Yakovlev", St. Petersburg, Russia, June 2010. ``Algebraic Geometry, K-theory, and Motives, Conference dedicated to the 60th birthday of Andrei Suslin", St. Petersburg, Russia, June 2010. ``K-Theory and Motives", On the occasion of the 60th birthday of Andrei Suslin, University of California at Los Angeles, March 2011.
