Day 22 (2/26/14)

Time: 8:30 - 3:30

Today, I learned about two very important theorems in Lie algebra theory-- Engel's Theorem and Lie's Theorem. Engel's Theorem gives a powerful criterion for determining when a vector space has a basis that represents a corresponding Lie algebra by a strictly upper triangular matrix, which only has nonzero elements above the main diagonal of the matrix. On the other hand, Lie's Theorem finds a similar criterion for the Lie algebra being an upper triangular matrix, which has nonzero elements on or above the main diagonal. Because these theorems were very complex and difficult to prove, I had to spend a lot of time on them, although eventually I was able to grasp them. Nonetheless, I was not able to study much else, as I spent almost all of the remaining time on exercises provided by the text. I read that these structures and maps have physical applications, which I can't wait to find out!

Day 21 (2/19/14)

Time: 9:30 - 4

Today, as usual, I continued with my studies in Lie algebras. We call two Lie algebras isomorphic if we can find some mapping, or function, that sends each element from one Lie algebra to a unique element in the other, and vice versa. First, I studied the various classifications of Lie algebras, up to isomorphisms. For instance, I discovered that there can only be one Lie algebra with dimension 1, and up to two with dimension 2. However, because the cases for dimension 3 were too difficult, I resorted to the solutions from the textbook. Surprisingly, even though the jump from 2 to 3 is seemingly insignificant, there are infinitely many different Lie algebras with dimension 3!

Afterwards, I began studying the notion of "solvable" Lie algebras. I had previously seen this similar notion when I studied Galois theory, which utilized solvable groups. Both topics were very similar-- they both involve "chains" of sub-algebras that end with the trivial Lie algebra and carry special properties. While I did not get to study this topic extensively yet, I was amazed at how much group theory and Lie algebra theory overlapped. I remembered the discussion I had with Professor Dolbin regarding "category theory" and the notion of studying these different abstract algebraic structures as generally as possible. It's really cool seeing how these various topics-- group theory and Lie algebra theory in today's case-- come together and overlap.

Day 20 (2/12/14)

Time: 8 - 2:30

After a long break, I met up with Professor Dolbin again to continue my internship. We briefly reviewed the basic notions of Lie algebras, and then he let me study independently, with frequent advice and guidance. As I learned previously, Lie algebras form a special family of mathematical structures, with the introduction of a new operator [x,y] for elements x and y. Specifically, these structures are not commutable (i.e. [x,y] is not equal to [y,x]), but rather they have a very similar property: namely, they satisfy the equation

[x,y]=-[y,x].

Also, they are not associative (i.e. [x[y,z]] is not equal to [[x,y],z]), but they satisfy another very similar and cool identity, called the Jacobi identity. It states that for any x, y, and z in the Lie algebra, we have that

[x,[y,z]]+[y,[z,x]]+[z,[x,y]]=0.

These structures carry a lot of significance and cool properties, many of which are derived using linear algebra. And because I never had an extensive experience with linear algebra, I spent most of today reviewing the necessary knowledge that I needed to continue learning about these Lie algebras.

We concluded today's internship with a brief discussion about a very complex branch of mathematics called Category Theory. Category Theory is an extension of all of these abstract algebraic structures that I learned, including groups, rings, and Lie algebras. In a sense, categories are generalizations of all of these structures, and this explains why in a general perspective all of the major theorems and facts about these structures can be proven in similar fashions. Sadly, we had to stop at this point for today, but I can't wait to continue learning more about these abstract notions even more!

Day 19 (2/6/14)

Because FDU and BCA were both closed down today, I was not able to go to internship again. It feels like I haven't been there in ages!

Day 18 (1/29/14)

Today, due to an emergency on my mentor's part, I was not able to attend my internship, and thus I had to go to BCA. I hope I can resume next week!