I wanted to point out two ICM2018 plenary lectures which are closely related to
the contents of our seminar.
1) Have a look at G. Williamson't talk here:
http://www.youtube.com/watch?v=-3q6C558yog. You will recognize many of the ;
themes that we discussed in the seminar (e.g. SU(2), finite reflection groups,
Coxeter groups, Kazhdan-Lusztig polynomials) and learn more about the
connections between them.
2) If you want to know more about knots you might be interested in P.
Kronheimer's and T. Mrowka's talk: http://www.youtube.com/watch?v=UW8KmtY6LI8. ;
The first couple of minutes are spent on discussing the fundamental group of
the knot complement and how homomorphisms from the fundamental group to SO(3)
can be used to detect knottedness (SU(2) and dihedral groups also appear in
this talk). Note that we will discuss the fundamental group next week.
Don't worry if you do not understand everything that is presented to you in
these lectures. The plenary talks are given by speakers who have developed some
of the most cutting-edge mathematics in the four years preceding the ICM. In
particular, don't expect the contents to be easy. Maybe you get stuck at about
10 minutes into a lecture because you simply don't know what a "flat
connection" is. At this point you should consult the literature or talk to
other people if you are curious. By using the above lectures in this way you
can experience what it is like to be a researcher. After all, not understanding
something but looking for answers anyway is exactly how research works.
P.S. At the end of the talk last week I briefly discussed the characterization
of Coxeter groups in terms of the exchange property. If you want to know more
about this and connect the recent talks with some of the earlier ones have a
look at Section 1.5 of this book
https://sites.math.washington.edu/~billey/classes/coxeter/EntireBook.pdf or ;
Humphreys book (the reference is on our website).