Group Theory Today
Group theory is the part of mathematics which deals with symmetries of objects and structures. For this reason it is very aesthatic and useful in mathematics,physics,chemistry,visual arts and in creating funny,annoyig toys like the magic cube. One of the biggest breakthroughs in group theory is the classification of finite simple groups (CFSG). The proof is twenty thousand pages long(!!!) so mathematicians can believe in it or not but they don't have the chance to check the proof. Some people think that it finishes the whole subject but they are at least three-fold wrong. 1.) Even though finite simple groups are "decribed" one can ask very hard questions about them (similarly as about prime numbers which are also "described") 2.) Finite groups are put together from simple groups in a highly non-trivial way. 3.) There are infinite groups which have nothing to do with CFSG.
The purpose of this course is to give an idea about what group theorists are up to these days. I will use the groups SL(n,Z) to demonstrate many directions. Some of the keywords are: profinite groups, Kazhdan's property-T, expander graphs, subgroup growth, arithmetic groups, lattices in Lie groups, Congruence subroup property. If time will remain, we will also see amenable groups.
Acknowledgement: I thank L.Pyber for his great help in putting the course together
The first problem sheet is available here. I will use the first few classes to go through the basic concepts in group theory, so I have to apologize from these who are already experts in basic group theory.
Problem sheets:
Room:
Monday 1-2 BA2159
Tuesday 12-1 BA2159
Thursday 11-12 BA2139
FIRST CLASS IS ON THIS THURSDAY!
Course requirement
There will be a problem sheet with 5-6 problems every second week. The last one will be considered as a "take home exam"
Literature
Coming soon!