Time: 10:00 - 4:30
Today, I further studied the implications of root systems. First, I learned about the geometric structures that root systems provide. For example, using the inner product, one could determine the possible angles between two elements in a root system, thereby arriving at a series of vectors stemming from the origin outwards symmetrically. I showed that there were surprisingly not much variety to the angles-- for example there are less than ten possible angles total between two base vectors in the regular xy-plane!
Afterwards, the book shifted to a more graph-theoretical approach, as it began to relate root systems to "Dynkin diagrams," which is a graph with nodes and edges representing the value <a,b><b,a>. Through such a notion, we showed that to show that root systems were equivalent, it is enough to simply compare different Dynkin diagrams, of which there are only 4 kinds and 5 exceptions. Through this, we eventually were able to demonstrate that all but two of the "classical" Lie algebras are simple. While I learned a lot, I think another review day next week would help me recover some more materials that I might have missed today...We went over a lot!
In addition, today Professor Dolbin took us out to lunch, and we talked about his past life, careers, and academic choices. He told me and gave me a lot of advice about both undergraduate and graduate schools, and how he dealt with studying theoretical mathematics. While the situations that he described seem incredibly difficult and stressful, I'm nonetheless thankful for the advice, as I now know more about how to achieve my career goals. Overall today was an amazing day, as I got to bond more with Professor Dolbin!
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