Extra Credit Math talk by Kumar Murty

This was a really fascinating lecture to me. He talked about how the brain is more of a horizontally structured way of thinking instead of as vertical as mathematical thinking is. He brought up certain principles like the Heisenburg Uncertainty Principle where math is not really able to reach it. He brought up the fact that some group was offering money for a good mathematical model of the brain. Then he ended his lecture by saying that in reality, maybe by studying the brain we could come up with a new "field" of mathematics that would function more similarly to how the brain functions. This idea was so cool to me and I really enjoyed this lecture.

Final Blog, due Dec. 8

From looking at the review sheet I know that the concept of prime and maximal ideals is something that I would like to go over. Also, the concept of extension fields. I know that I need to work on better understanding quotient rings and quotient fields and some problems regarding these topics would be helpful. I think it would help to have a couple of examples of these quotient rings/fields and then describe them together (sort of like on our test).
Also, examples of different cyclic, non-integral domain etc. examples of rings/groups/fields would be helpful to me as this is a section on the tests that is always difficult for me. I also had a question: on the review sheet there are lots of theorems we should know well and I was wondering if that is the same list of theorems that we need to be able to prove.

Section 8.5, due on December 6

The most difficult part of this material for me was following the long proofs and connecting all of the ideas. In specific, Theorem 8.34 was slightly confusing to me. I also has a couple of questions regarding Theorem 8.33.

The most interesting part of this section was looking over the proof of theorems we had basically already done or applied in previous exercises and seeing how they proved it. I also round it really neat that so many groups have been classified as they show on page 281.

Section 8.4, due on December 3

The most difficult part of this section for me was pickinga part the different notation so that I got the full meaning of what it was trying to say. Also, In reading th proof of the Second Sylow Theorem, I found parts of it to be confusing. I also found it a little confusing when going over the proof of the First Sylow Theorem.

The most interesting part of this section to me was the fact that conjugacy is an equivalence relation. I was wondering if this means that given any equivalence relation, if that means that the equivalence classes partition whatever mega set you are talking about? I think this would be true, but for some reason it just dawned on me how cool that really is.

Section 8.3, due on December 1

I am confused as to whether Sylow refers to just subgroups or to groups as well. Every group is a subgroup of itself, so is that why it's called Sylow p-subgroup vs. just a Sylow p-group? Also, some of the examples were confusing to me especially following the Applications of the Sylow Theorems subtitle.

The most interesting part of this section was Corollary 8.18 because it seems as though it is a very helpful fact in figuring out what various groups are isomorphic to. Also I was wondering where the term Sylow came from. Is it named after a person?

Section 8.2, due on November 29

The most difficult part of this material for me was following Lemma 8.6. I got a little confused with the difference between a p-group and a cyclic group. Maybe it was just because it is a long theorem but there were some conclusions drawn that just didn't quite make sense to me.

The most interesting part of this material was the Fundamental Theorem of Finite Abelian Groups. When I think about all of the things that come from knowing a group is cyclic and has some characteristic associated with primes, it's very cool to think that every finite abelian group is the direct sum of cyclic groups and each of prime power order.

Section 8.1, due on November 23

The most difficult part of this material for me was understanding how you would write an element of a group that is a direct product of a bunch of different groups that don't have the same operation. Do you do it like a cross product like we had learned before?

The most interesting part of this material for me was the fact that just because G is a group that makes up some cross product, doesn't mean that G is a subgroup of that cross product.