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!

Day 7 (10/30/13)

Fairleigh Dickinson University (Part 1 - Professor Dolbin)

(8:10 - 12:05)

            Today was the first day of my new internship with Professor Dolbin. I began studying from a textbook for undergraduates recommended by the professor, called “Abstract Algebra: an introduction” by Thomas W. Hungerford. I began studying the concept of field extensions, which are essentially fields that act as vector spaces over other fields. (It sounded just as complicated to me when I first saw it.) Throughout the day, the professor talked with me about the concepts, provided additional exercise problems for me to solve or prove, and clarified certain notions. At first, I had a difficult time grasping the higher-level concepts, but with some guidance I managed to understand an entire chapter of the textbook. I have already learned so much more about abstract algebra under his guidance, and I hope to continue this learning process so that I will be well-equipped to handle even more difficult problems in abstract algebra later on. I’m really thankful for Professor Dolbin, as he devotes a significant amount of time to help me learn. 

(Part 2 - Professor Farag)

(1:00 - 3:00)

            In class, we tackled another interesting and difficult problem in the field of group theory. We further discussed the "First Homomorphism Theorem," which relates groups that are deceptively different. Also, in the middle of class the students and the professor suddenly entered a gigantic, almost-philosophical debate about whether things are "well-defined"; the problem arose when a student suggested a function that might not always map a value of "x" to the same value of "y." However, after this heated discussion, we settled down and had a good, long laugh about it, and the rest of the day passed by normally. I finished the notes for the past week in time, although I have yet to write the solution set for the newest homework assignment. Surprisingly, I learned that these problem sets might not be as difficult as I had originally assumed!

Day 6 (10/23/13)

          Unfortunately, I was again too sick today to attend my internships. However, I gathered enough strength to go to the informal interview for my second, new internship at FDU, at which I would undertake a small mathematical research project and a directed study project with Professor Dolbin at the university; in short, I was accepted! More details to follow next week, but I am very thankful for this opportunity. Thank you, Professor Dolbin, and also thank you, Professor Farag, for helping me attain this internship! I do feel sad that I will not be able to work with Mr. Sincak at River Dell Regional HS, and I hope he understands.

Day 5 (10/16/13)

River Dell Regional High School

(7:30-12:00)

            Today I began creating my own lesson plan for class. Mr. Sincak provided me with reading materials and sample lesson plans from various teaching colleges, and I learned how teachers can prepare for each class. I specifically focused on creating a lesson on solving non-routine problems, such as those involving mathematical modeling or problems that might appear on math competitions. I designed the overall purpose of the lecture, outlined prerequisite knowledge and expanded on how the lesson would actually be carried out. I found example problems and practice questions for the class from the reading materials provided, and I specifically chose to incorporate problems that involved higher-level thinking in geometry and probability theory. Specifically, for example, I began the first day’s lesson plan with a coin-flipping game for students, in order to experimentally calculate the probability that more heads than tails show up when five coins are flipped. Then, I continued by asking students to think of ways to prove that their resulting probability was correct, with several different suggestions on how to approach the problem. Finally, I ended the lesson plan by demonstrating how one can theoretically calculate the required probability, emphasizing the methods utilized in attaining that answer.

Throughout the process, I learned and realized how I can also organize my thoughts and lessons for when I tutor students throughout the year. I have really enjoyed all of the processes and techniques associated with teaching that I experienced so far, and I hope to use these skills that I learned when I grow up.

Fairleigh Dickinson University

(1:00-3:00)

            Today’s class was devoted to reviewing for the test. We discussed problems from an old Midterm from the course, and several students presented solutions on the board. Professor Farag also outlined possible topics that would be on the test, and encouraged students to remember the main concepts and definitions.

            After the class, Professor Farag introduced me to Professor Dolbin, another mathematics professor at the university. While I could not speak with him for long, I learned that his interests in research were in Lie algebras, representation theory, and associated combinatorics, which I also find very fascinating. Professor Farag suggested that I ask him to mentor me in a research or independent reading project regarding that field of mathematics, and I am very thankful for that suggestion. I will soon email Professor Dolbin for an opportunity to work with him.

Day 4 (10/09/13)

Due to a sudden illness, I was unable to attend my internships in this week. Coincidentally, my mentor for River Dell Regional High School was sick as well.

Day 3 (10/02/13)

River Dell Regional High School

(7:30-12:00)

            Today, Mr. Sincak and I prepared a domino circuit activity to demonstrate logical operators and their effects on statements. Specifically, we designed a circuit for the logical statement “If P or Q is true, and if R or S is true, then T is true.” Although I felt that the business-casual outfit made me feel a bit out of place, the class and the domino exercise were enjoyable as a whole. After the class, I spent the day constructing methods of teaching indirect proofs and their applications to geometry. Mr. Sincak and I researched optimal strategies from various teaching textbooks, and he lent me a copy of one of the textbooks for me to study further if I chose to do so. I am very appreciative of how much Mr. Sincak cares about my career goal of becoming a teacher or professor, by constantly giving me advice and explanations on the best methods of teaching and working with other students. I feel like I am already attending a teaching school!

Fairleigh Dickinson University

(1:00-3:00)

            Today in class, Professor Farag taught about a special kind of maps from groups to groups called “homomorphisms.” We also learned about several branches of groups that revolve around these functions, as well as different kinds of homomorphisms. We began to discuss the importance of “isomorphisms,” or maps that relate each item in the first group to a unique element in the second group. He showed us that this is an especially important concept, as groups that are “isomorphic” to each other are essentially the same group. 

            After the class, my temporary office was moved to a new building, but the rest of the day was routine. I finished typing up the class notes and solutions for the past week accordingly, with ample time left for revisions.