This proposal focuses on several important problems in representation theory of finite groups and its applications. It ties together different areas of mathematics, such as finite groups and algebraic groups, finite permutation group theory, group cohomology, combinatorics and finite geometry, algebraic geometry, and string theory, with the main unifying ingredient being the representation theory. Many of the problems addressed in the proposal come up naturally -- some long-standing and play a central role -- in the group representation theory, and others are motivated by various important applications. The PI will study several problems along the lines of the local-global principle, including Brauer's height zero conjecture and some conjectures concerning rationality and divisibility properties of complex and Brauer characters of finite groups. The PI will also continue his long-term project to classify cross characteristic representations of finite groups of Lie type of low dimension. He will then apply his results to seek significant progress on a number of applications, including the Kollar-Larsen Problem on linear groups with elements of bounded age (with applications in algebraic geometry and string theory), the Ore Conjecture on commutators in simple groups, a strengthening of Holt's Conjecture on the dimension of the second cohomology group for finite groups and their presentations, and representations of finite quasisimple groups with special properties (with application in the subgroup structure of finite simple groups).
The main area of research in this proposal is the group representation theory. The concept of a group in mathematics grew out of the notion of symmetry. The symmetries of an object in nature or science are encoded by a group, and this group carries a lot of important information about the structure of the object itself. The representation theory allows one to study groups via their actions on vector spaces which model the ways they arise in the real world. It has fascinated mathematicians for more than a century and had many important applications in physics and chemistry, particularly in quantum mechanics and in the theory of elementary particles. Finite groups and their representations have already proved valuable in coding theory and cryptography, and are expected to continue to play an important role in the modern world of computers and digital communications. The investigator's research will lead to important advances in understanding the representation theory of finite groups and help achieve significant progress in a number of its applications.
Project outcomes of the research grant DMS-0901241 The NSF grant DMS-0901241 focuses on several important problems in representation theory of finite groups, and their applications in finite permutation group theory, group cohomology, algebraic geometry, combinatorics and finite geometry. Among the major outcomes of the PI's research within this grant, we mention -- a reduction theorem for the Alperin Weight Conjecture, published in Inventiones Math. 2011. This conjecture, announced in 1986 by J. Alperin, is one of the the deeper and more challenging problems of the representation theory of finite groups; -- an upper bound (in terms of the rank of the group in question) for the dimension of the first cohomology group of finite groups of Lie type in cross-characteristic, published in Annals of Math. 2011; -- solution of the irreducible restriction problem for finite groups of Lie type of type A, published in Amer. J. Math. 2010. This problem is at the core of the Aschbacher-Scott program on maximal subgroups of finite classical groups and has important implications in the primitive permutation group theory; -- rational extensions of a classical theorem of J. G. Thompson (for all primes), published in Advances in Math. 2010; -- proof of Brauer's height zero conjecture for 2-blocks of maximal defect of arbitrary finite groups, published in Crelle's journal 2012. This conjecture, stated by R. Brauer in 1955, is one of the main conjectures in the modular representation theory of finite groups; -- proof of the Ore conjecture (of 1951), which says that every element of any non-abelian finite simple group is a commutator. In spite of its simple formulation, this conjecture had resisted great efforts of many mathematicians over more than half a century, until the PI and his collaborators completed its proof and published it in J. Europ. Math. Soc. 2012; The PI's research within this grant has led to a better understanding of basic properties, in particular linear representations and also permutation representations, of finite groups, especially finite groups of Lie type, long-standing conjectures about them, and their interrelations with other areas of mathematics. It has also led to significant advances in a number of applications in structural and cohomological group theory, combinatorics, Galois theory, arithmetic of curves, algebraic geometry, and string theory. Results obtained within this grant have helped train two graduate students. One of them, Mrs. Schaeffer Fry, has recently defended her Ph. D. Thesis under the PI's supervision and accepted a postdoctoral position at the Michigan State University. The PI's results have been lectured at many national and international conferences, including a plenary address at the 2012 AMS Spring Western Section Meeting (Honolulu, HI March 3 - 4, 2012). Some of these results have also been used in graduate courses taught by the PI at the University of Arizona, and summer schools in Florida and Hanoi.
