Sophus Lie Days
at Cornell University
February 27 – March 1, 2012
Alex Lubotzky, Hebrew University of Jerusalem
Short presentations of finite groups
Finding nice and compact presentations of various groups has been a subject of much research for more than a century. The Coxeter presentation of the symmetric groups and the Steinberg presentation of groups of Lie type are such. In response to conjectures of Babai and Szemeredi on one hand (motivated by questions in computational group theory) and of Mann on the other hand (motivated by questions on subgroup growth) we show that all non-abelian finite simple groups (with the possible exception of Ree groups) have presentations which are small (bounded number of relations) and short (w.r.t. the length of the relations). This is surprising as the simple abelian groups — the cyclic groups of prime order — do not have such presentations! We will describe the motivations and results, a cohomological application (proving a conjecture of Holt) and some connections with discrete subgroups of Lie groups and topology.
The talk will be suitable for undergradutes with basic knowledge of group theory.
The talk is based on a series of papers with Bob Guralnick, Bill Kantor and Martin Kassabov.
Sieve methods in group theory
The sieve methods are classical methods in number theory. Inspired by the affine sieve method developed by Sarnak, Bourgain, Gamburd and others, as well as by works of Rivin and Kowalsky, we develop in a systemtic way a sieve method for group theory. This method is especially useful for groups with property tau. Hence the recent results of Breuillard-Green-Tao, Pyber-Szabo, Varju and Salehi-Golsefidy are very useful and enables one to apply them to linear groups. We will present the method and some of its applications to linear groups and to the mapping class groups. (Joint work with Chen Meiri.)
Arithmetic groups, Ramanujan graphs, and error correcting codes
While many of the classical codes are cyclic, a long standing conjecture asserts that there are no ‘good’ cyclic codes. In recent years, the interest in symmetric codes has been promoted by Kaufman, Sudan, Wigderson and others (where symmetric means that the acting group can be any group). Answering their main question (contrary to the common expectation), we show that there DO exist good symmetric codes. In fact, our codes satisfy all the “golden standards” of coding theory. Our construction is based on the Ramanujan graphs contructed by Lubotzky-Samuels-Vishne as a special case of Ramanujan complexes. The crucial point is that these graphs are edge transitive and not just vertex transitive as in previous constructions of Ramanujan graphs. All notions will be explained. Joint work with Tali Kaufman. [view poster]
Boris Tsygan, Northwestern University
I will review the definition of the cyclic homology and its roots in Lie theory and geometry. I will proceed to current results on a new definition of the Hochschild and cyclic homology and its relation to a new set of Lie-theoretical and geometric questions.
Noncommutative calculus and formality theory
Noncommutative calculus is a theory that studies the standard algebraic constructions from the calculus on manifolds in terms of the algebra of functions, and in a way that is applicable for any associative algebra, commutative or not. Noncommutative analogs of the basic objects of calculus, such as forms and multivectors, turn out to be the standard constructions of homological algebra of rings, such as the Hochschild and cyclic complexes. There are two basic questions: (a) What are the classical algebraic structures that can be generalized to the noncommutative case? (For example, both forms and multivectors carry graded commutative products; multivectors act on forms; multivectors carry a graded Lie bracket, etc.) (b) If our algebra was the algebra of functions to begin with, are the algebraic structures arising from noncommutative calculus the same as the classical ones? I will review the current state of the theory, as well as its applications to deformation quantization (the Formality Theorem of Kontsevich and its generalizations) and to index theorems.