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

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. 

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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. 

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Joint Colloquium web site: 

http://www.maths.usyd.edu.au/u/SemConf/JointColloquium/index.html