Day 10 (11/20/13)

Fairleigh Dickinson University (Part 1 - Professor Dolbin)

(8:00 - 12:00)

           Today, Professor Dolbin taught me some extra information in Galois Theory that was not covered in the textbook. I also began my studies in Lie algebras, in which several unusual properties hold. For instance, x+(y+z) is not always equal to (x+y)+z. I spent a majority of my time solving exercises from the text, which helped me get a better grasp in the topic. I am really glad to be in this internship, where I have the freedom to choose whatever I want to learn, guided by a professional in this field.

(Part 2 - Professor Farag)

(1:00 - 3:00)

            Afterwards, in Professor Farag's class, we discussed more ring theory and explored special kinds of polynomials. We separated into groups and discussed problem set questions, and another test was scheduled. Luckily for me, the topics I studied with Professor Dolbin again resurfaced in Professor Farag's class today, allowing me to compose notes and solution sets for the week more easily. In general I learned more knowledge in various types of algebras and I am very thankful for this internship!

Day 9 (11/13/13)

Fairleigh Dickinson University (Part 1 - Professor Dolbin)

(8:00 - 12:00)

           Today I further studied the intricate structure of Galois groups. I learned the Fundamental Theorem of Galois Theory, which says that there is a direct correspondence between extensions of fields and subgroups of Galois groups. I also became used to drawing Galois correspondence diagrams, which shows this relationship pictorially. It’s amazing how the two types of mathematical structures, which seem entirely different at first, can be related in such a simple way. Moreover, I learned that all of this can be further related to the one type of groups called the symmetric groups, which are just the set of ways to permute a set of objects. I see this in two ways—mathematicians have either shown that these seemingly complicated structures in mathematics boil down to the simple group of rearrangements, or that the seemingly simple symmetric group is, in fact, monstrous and obscenely complex. I guess it depends on whether one is a pessimist or an optimist. Nonetheless, after studying these structures, I moved on to the proof for why there is no generic solution to the cubic equation, which utilizes the material that I have learned so far. In addition, under the guidance of Professor Dolbin, I have learned several interesting side details along the way. Nonetheless, by the end of the day I was able to finish yet another chapter of the text, thus taking a gigantic step towards my algebraic research project.

(Part 2 - Professor Farag)

(1:00 - 3:00)

            Next, in Professor Farag’s class, we further studied rings. We covered a special case of rings called ideals, which have a unique "absorption" property. Afterwards, we discussed some more practice problems and went over the last problem set, for which I had already written up solutions. Today I gained further knowledge about abstract algebra, and this class has been helpful especially for the first part of my internship, as one supports and adds more information to the other.

Day 8 (11/06/13)

Fairleigh Dickinson University (Part 1 - Professor Dolbin)

(8:04 - 12:03)

            Today I continued to study from the textbook that Professor Dolbin had given me. I finished a brief chapter on an algebraic notion called “fields,” which are groups of elements under two operations. An example is the set of real numbers under addition and multiplication. After studying this branch of algebra, I moved on to studying Galois Theory, which studies the relationships between fields, field extensions, and groups. This was used to eventually show that there are NO ways to solve general equations of degree 5 or higher; for instance, the famous Quadratic Formula can solve any quadratic equation, but Galois showed that there are no such formulas for polynomials of degree 5 or higher. I find it fascinating how this is one of the greatest examples of how mathematics had proven the fact that something does not exist, which really sounds impossible at first thought. This will likely be included in my research later on. Professor Dolbin frequently checked up on me and guided me when needed, and I have already learned more than I possibly could have by myself in such a short amount of time. The past couple of weeks had been invaluable to me, as they showed me that I really can survive while doing just math all day, which is what most mathematicians in the field do. I am grateful that Professor Dolbin is willing to spend so much time to mentor me in this manner.

(Part 2 - Professor Farag)

(1:00 - 3:00)

            Next, in Professor Farag’s class, he taught us more about ring theory, which is another structure in algebra like “groups” or “fields,” as well as how rings relate to those other structures. Throughout his lecture today he emphasized the need to stop ourselves from invoking our previous knowledge on “numbers,” and the need to stick with the axioms given only. This mindset is crucial for the study of abstract algebra, as I learned through the various classes that our intuition may not always hold. For instance, x plus y might not always equal y plus x in a lot of algebraic structures. Boy, does this class give me a headache!