8:00-12:00, 5:00-7:00
Because of a major competition during the middle of the day, I had to split my times of internship. Hence, today was another review day. I specifically reread the sections on simple, semisimple, and nilpotent algebras, and I also redid many of the exercises from those chapters. I also referenced another text that dealt with Lie algebras and representation theory, which provided another perspective on the implications of nilpotency and solvency. Now that I was able to review the basics and fundamentals of those concepts, I feel more equipped to handle the more complicated arguments involving root systems!
Thursday, May 1, 2014
Wednesday, April 23, 2014
Day 30 (4/23/14)
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!
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!
Thursday, April 17, 2014
Day 29 (4/16/14)
9:00 - 3:30
Today in internship, I learned about root spaces and how they could be used to determine even more information about the structure of semisimple complex Lie algebras. Root spaces are sets of "roots" or functions of weights, which as described earlier are generalizations of eigenvalues in linear algebra. Essentially, a "root" is a linear mapping "f" from the Lie subalgebra L of gl(V) to its scalar field such that there exists at least one element v in V such that for any element x in L, x(v) = f(x)v. This set of roots, the root space, is especially important because any Lie algebra over the complex numbers can be decomposed as a direct sum of weight spaces that stem from the roots.
This new notion leads to interesting results, as these root spaces eventually lead to another kind of inner products, which are operations of two vectors that output a scalar. Such a valuable tool can be used to geometrically describe specific Lie algebras in terms of symmetric vectors. After my mind was blown by the tremendously symmetric representations of these Lie algebras like sl(3,C), Professor Dolbin talked to me about how these notions could be further generalized. I saw a glimpse of such ideas when I learned about buildings and chamber systems with Dr. Abramson in school, and I was struck by the resemblances. I can't believe how cool math is and how it ties in such different branches together!
Today in internship, I learned about root spaces and how they could be used to determine even more information about the structure of semisimple complex Lie algebras. Root spaces are sets of "roots" or functions of weights, which as described earlier are generalizations of eigenvalues in linear algebra. Essentially, a "root" is a linear mapping "f" from the Lie subalgebra L of gl(V) to its scalar field such that there exists at least one element v in V such that for any element x in L, x(v) = f(x)v. This set of roots, the root space, is especially important because any Lie algebra over the complex numbers can be decomposed as a direct sum of weight spaces that stem from the roots.
This new notion leads to interesting results, as these root spaces eventually lead to another kind of inner products, which are operations of two vectors that output a scalar. Such a valuable tool can be used to geometrically describe specific Lie algebras in terms of symmetric vectors. After my mind was blown by the tremendously symmetric representations of these Lie algebras like sl(3,C), Professor Dolbin talked to me about how these notions could be further generalized. I saw a glimpse of such ideas when I learned about buildings and chamber systems with Dr. Abramson in school, and I was struck by the resemblances. I can't believe how cool math is and how it ties in such different branches together!
Tuesday, April 15, 2014
Day 28 (4/9/14)
10:00 -4:30
Today, I had to review more materials for a harder topic. I read more about linear algebra, and then worked towards some vector analysis. Because today was solely meant for review, I was not able to learn anything new in terms of Lie Algebra Theory. However, I learned to appreciate the subject of linear algebra, as I found out about the tremendous ties between the two fields of study. I can't wait to get back to Lie Algebra theory next week!
Edit: I apologize for publishing this blog entry late - I had written it on the day of the internship, but I forgot to press the submit button!
Today, I had to review more materials for a harder topic. I read more about linear algebra, and then worked towards some vector analysis. Because today was solely meant for review, I was not able to learn anything new in terms of Lie Algebra Theory. However, I learned to appreciate the subject of linear algebra, as I found out about the tremendous ties between the two fields of study. I can't wait to get back to Lie Algebra theory next week!
Edit: I apologize for publishing this blog entry late - I had written it on the day of the internship, but I forgot to press the submit button!
Wednesday, April 2, 2014
Day 27 (4/2/14)
Time: 10:30 - 5:00
After a much-needed review on previous topics like representations and modules, I continued with my journey on Lie algebra theory. Specifically, I was pleased to make much progress, as I studied both representations of Sl(2,C) and criteria for determining whether a Lie algebra is semisimple. First, through a series of very rigorous and complex exercises and lemmas, I was able to classify the irreducible modules of Sl(2,C), which again is the set of 2-by-2 matrices with complex numbers and whose diagonal elements sum to zero. This area was especially important, as irreducible modules, in one way, become a sort of building blocks for other modules.
Afterwards, after a nice lunch break, I moved on to studying the various useful conditions that are both necessary and sufficient for a Lie algebra to be semisimple. To recap, a Lie algebra is semisimple if it has no non-zero solvable ideals. While ultimately I had to consult the book multiple times, I eventually discovered two very crucial criteria, which are officially called Cartan's Criteria. The first of these criteria asserts that the complex Lie algebra L is solvable if and only if the sum of the diagonal elements of the matrix of [x,[y,-]] is equal to 0. The second and more applicable theorem states that L is semisimple if and only if the "Killing form," or the function K defined as K(x,y)=sum of diagonal elements of the matrix [x,[y,-]], is non-degenerate. While I learned a significant amount of information regarding Lie algebra theory, today's portion of my internship especially showed me how complicated and difficult mathematics can get. However, I also realized that these difficulties don't discourage me, but rather make me even more hooked into the subject!
After a much-needed review on previous topics like representations and modules, I continued with my journey on Lie algebra theory. Specifically, I was pleased to make much progress, as I studied both representations of Sl(2,C) and criteria for determining whether a Lie algebra is semisimple. First, through a series of very rigorous and complex exercises and lemmas, I was able to classify the irreducible modules of Sl(2,C), which again is the set of 2-by-2 matrices with complex numbers and whose diagonal elements sum to zero. This area was especially important, as irreducible modules, in one way, become a sort of building blocks for other modules.
Afterwards, after a nice lunch break, I moved on to studying the various useful conditions that are both necessary and sufficient for a Lie algebra to be semisimple. To recap, a Lie algebra is semisimple if it has no non-zero solvable ideals. While ultimately I had to consult the book multiple times, I eventually discovered two very crucial criteria, which are officially called Cartan's Criteria. The first of these criteria asserts that the complex Lie algebra L is solvable if and only if the sum of the diagonal elements of the matrix of [x,[y,-]] is equal to 0. The second and more applicable theorem states that L is semisimple if and only if the "Killing form," or the function K defined as K(x,y)=sum of diagonal elements of the matrix [x,[y,-]], is non-degenerate. While I learned a significant amount of information regarding Lie algebra theory, today's portion of my internship especially showed me how complicated and difficult mathematics can get. However, I also realized that these difficulties don't discourage me, but rather make me even more hooked into the subject!
Wednesday, March 26, 2014
Day 26 (3/26/14)
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.
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.
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!
Wednesday, February 26, 2014
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!
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!
Wednesday, February 19, 2014
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.
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.
Wednesday, February 12, 2014
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
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
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!
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.
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!
Thursday, February 6, 2014
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!
Friday, January 24, 2014
Friday, January 17, 2014
Day 16 (1/15/14)
Professor Dolbin had told me that he will not be able to return to FDU until Thursday of next week. Hence, I interned again at BCA for today with Mr. Samarakone.
Wednesday, January 8, 2014
Day 15 (1/8/14)
Because my professor was not present, I interned at BCA for today in Mr. Samarakone's room. I hope that I can return to my mathematical internship soon!
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