10:00 - 4:30
Today during internship, I encountered a subtopic that I found was extremely difficult to understand. Thus, with not having thought about Lie algebras over the last two weeks, I decided to spend today reviewing all of the materials that I have learned prior to this specific section. I revisited old exercises that I was not able to solve first, and surprisingly I was able to solve them more readily the second time. I also got a much better grasp of Lie algebras, their homomorphisms, weight spaces, and other concepts than before. Therefore, while I am partly disappointed that I was not able to make much new progress today, I am still very happy also that I was able to solidify my understanding in my studies.
Wednesday, March 26, 2014
Friday, March 21, 2014
Day 25 (3/19/14)
Because of a sudden illness, I was not able to attend the internship today. However, I was still able to work on the exercises from the text, which was fun by itself!
Wednesday, March 12, 2014
Day 24 (3/12/14)
10:30 - 5:00
Today I explored the specific structure of Sl2(C), the set of 2x2 matrices of complex numbers whose diagonal elements add up to 0. I learned about a very cool and intricate theorem, which used a basis of that Lie algebra to classify any irreducible module. While reading about this, I thought about how these irreducible modules act as Lego pieces, which look boring by themselves but can be fit together to yield very cool and complicated structures.
Because this theorem was very hard to conceptualize, I had to spend a lot of time on this chapter, and unfortunately I was not able to finish the exercises available. However, I was able to fully grasp this specific theorem, which the author contends has tremendous amounts of applications and possible generalizations. I can't wait to find out more!
Today I explored the specific structure of Sl2(C), the set of 2x2 matrices of complex numbers whose diagonal elements add up to 0. I learned about a very cool and intricate theorem, which used a basis of that Lie algebra to classify any irreducible module. While reading about this, I thought about how these irreducible modules act as Lego pieces, which look boring by themselves but can be fit together to yield very cool and complicated structures.
Because this theorem was very hard to conceptualize, I had to spend a lot of time on this chapter, and unfortunately I was not able to finish the exercises available. However, I was able to fully grasp this specific theorem, which the author contends has tremendous amounts of applications and possible generalizations. I can't wait to find out more!
Sunday, March 9, 2014
Day 23 (3/5/14)
10:00 - 4:30
Today, I studied the core of basic Lie algebra representation theory. Representation theory, as stated before, revolve around "homomorphisms," a kind of functions, from a Lie algebra to the set of self-transformations of a vector space. One of the most important theorems that my mentor described to me was Schur's Lemma, which requires too much background information to be described in a single blog post. However, even though it is considered to be very "elementary," it has a lot of applications in proving stronger and much more complicated theorems. Lastly, towards the end of the internship, my mentor referred again to the very general "category theory," and how even Lie algebra representations could be generalized and extended to other algebras such as groups and rings. While I still don't have a satisfying grasp on the topic, I realized that this topic is one of the most interesting ones that I have studied so far!
Today, I studied the core of basic Lie algebra representation theory. Representation theory, as stated before, revolve around "homomorphisms," a kind of functions, from a Lie algebra to the set of self-transformations of a vector space. One of the most important theorems that my mentor described to me was Schur's Lemma, which requires too much background information to be described in a single blog post. However, even though it is considered to be very "elementary," it has a lot of applications in proving stronger and much more complicated theorems. Lastly, towards the end of the internship, my mentor referred again to the very general "category theory," and how even Lie algebra representations could be generalized and extended to other algebras such as groups and rings. While I still don't have a satisfying grasp on the topic, I realized that this topic is one of the most interesting ones that I have studied so far!
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