Type: Seminar

Modified: Wed 27 Feb 2013 0022

Distribution: World

Expiry: 27 Mar 2013

CalTitle1: SYD- UNSW Joint Colloquium: Scott -- The Legacy of the Wall and Guralnick Conjectures

Auth: zhangou@60.225.181.121 (zhouz) in SMS-WASM

Speaker: Prof. Leonard Scott (Virginia) http://pi.math.virginia.edu/~lls2l/ Time: Friday, March 8, 2:30--3:30PM Room: Chemistry Lecture Theatre 2, the University of Sydney Lunch plan: we meet near Level 2 entrance to Carslaw Building around 1PM. The lunch would be at Law Annex Cafe with reservation at 1:10PM. ----------------------------------------------- Title: The Legacy of the Wall and Guralnick Conjectures Abstract: In 1961 G.E. Wall conjectured that the number of maximal subgroups of a finite group is less than the order of the group. The conjecture holds for all finite solvable groups (proved by Wall himself in his original paper) and holds for almost all finite simple groups, possibly all of them (proved by Liebeck, Pyber and Shalev in 2007). it is now known to be false in general, at least as originally stated, with infinitely many negative composite group examples found through a combination of computational and theoretical techniques. (I cite in particular computer calculations of Frank Luebeck, as partly inspired and later confirmed by calculations of my undergraduate student, Tim Sprowl, with theoretical input from myself and Bob Guralnick.) Somewhat surprisingly, the Wall conjecture, through a related 1986 conjecture of Bob Guralnick, has had a tremendous impact on the development of cohomology theory of finite and algebraic groups, with many positive results proved regarding first 1-cohomology with irreducible coefficients and, most recently, higher degree cohomology. The negative examples that have been now found arise from combining deep considerations in algebraic group theory (such as rational and generic cohomology theory, cohomology theory related to the Lusztig conjecture, and the Lusztig conjecture itself) with computer calculations of Kazhdan-Lusztig polynomials. I will discuss all of these things, and, as time permits, a new algorithm my student and I have developed for calculating individual Kazhdan-Lusztig polynomials (which might have been guessed ahead of time to be especially interesting), or more accurately, individual Kazhdan-Lusztig basis elements in a Hecke algebra, with minimal input from lower degree calculations. ----------------------------------------------- Joint Colloquium web site: http://www.maths.usyd.edu.au/u/SemConf/JointColloquium/index.html