I’m planning on moving tomorrow (the person helping me may defer it to Sunday…but that’s another story). One possibility that I’m contemplating is to not get internet at my new place until after prelim exams to force myself to not be constantly distracted. So you may not hear from me for awhile. On the other hand, I’d probably still periodically update from the library or a coffee shop or something.

I warned that I know nothing about representation theory and the point of these posts was to learn more. Well, I must sincerely thank Zygmund (hopefully I’ll receive useful comments like that one throughout the series). Although I’m not surprised about the functorial relation, neither one of the books I’m using talk at all about it. So I dug a little deeper today to figure this out.

It isn’t as straightforward as I thought (we aren’t talking in the category of groups), but it also isn’t hard once you get your head in the right place. We want the representation to be a functor. But the representation is a hom from a group to invertible transformations of a vector space. Thus, our domain category is going to be a category of one object with morphisms given by the group elements. Composition of these morphims is naturally the group multiplication (note that all morphisms are invertible since all group elements have an inverse). Our range space is the category of vector spaces () over k.

Now our representation is a functor . We specify which vector space we are going to by taking our one object there: . The other aspect of the functor (the only data we cared about in the last post) is where morphisms are sent, i.e. group elements (morphisms in *) get sent to invertible linear maps of V. Perfect. Representations are now packaged in the language of categories which gives us something to work with.

Alright, I’ll now say a few words about what I actually promised to post on today. Maybe I’ll come back and rephrase things categorically as much as possible, but it definitely strays from both of the books I’ve been reading.

A quick observation: Any k-representation equips V with the structure of a left kG-module, and any kG-module V determines a k-representation .

Proof of observation: Let be a representation, then for ease let . We have an action by , so given this multiplication V is a left kG-module.

Now let V be a left kG-module. Then is a linear transformation of V with inverse , so call this transformation . Then clearly is a k-representation.

For another two quick examples from the past, we clearly have the trivial kG-module determined by the trivial representation. Another more interesting example is to let where the field extension is Galois. Then E is a kG-module by .

Next time we’ll want to figure out when two kG-modules determined by representations are isomorphic.

Just a brief note to my readers that I’m still alive. It turns out that summer is just as time consuming (if not more so…at the very least it is more stressful since I can’t fudge over details like I could during regular quarter since I’m presenting solutions in class and also there is always the burning in the back of my mind that it is my last chance before qualifying exams to actually make sure I can do it) as regular quarter. Hence, I’ve made essentially no progress on non-class work.

I did check out the Seattle gay pride parade yesterday. It was fun, but kind of disappointing. There were very few absurd floats. Quite tame compared to NYC or San Francisco.

I have settled on both short term and long(er) term plans for the blog. My immediate future will be a week in which I post no math. I will do my mid-point through the year music ranking soon, I am going to finish Ralph Ellison’s Invisible Man in a day or two, so I may talk about that.

For longer plans, I also found Dennett’s Consciousness Explained on clearance at the bookstore for like 4 dollars, and I’ve always wanted to read it. So that is a hefty read that will probably force me to catch up on some philosophy of consciousness, which I plan on doing through here. That is my main non-math project for the summer. Literary-wise, I haven’t decided whether to finish up some of those easier reads that just collect dust on my bookshelf, or to do the one main large work that is sitting there. Either way, that will posted about as well. In fact, the math posting will probably be fairly light during the summer, since my life will be reviewing stuff for prelims rather than learning new stuff. So it will probably just be when I make large revelations about things I probably should have already realized.

Thoughts? For the summer reading, my choices seem to be something along the lines of the large Sot-Weed Factor by Barth, or (after a brief glance at my shelf) it appears The Way the Crow Flies by Ann-Marie MacDonald, Paradise Lost by Milton, Sanctuary by Faulkner, Dune by Herbert, and Jude the Obscure by Hardy.

April 28…hmmm…why does that day sound familiar? Oh yes, that’s right, it is the one year anniversary of my wordpress blogging! So I guess the cliche thing to do is to post my five favorite posts and my five least favorite posts.

Favorites:
5) I guess I should include Abduction as Logical Inference just because it is one of the most active and read posts of my blog. I’m not as fond of it on this re-reading, though.

4) My favorite sequence of posts has to be when I went through the book Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being almost chapter-by-chapter breaking apart and presenting my ideas about this radical way of thinking about math. The first in the series starts here

3) My letter to Sam Harris has to make the list. I think Sam Harris is absolutely fantastic, and almost no one has influenced my thinking as much as him, but I still think I’ve found a slight flaw in his argument. Maybe it can be patched up. Maybe I’m misinterpreting him a little. Either way, I wish he’d let me know. I actually sent a modified version of this letter to him and got no reply.

2) Posting on “Lost in the Funhouse” was probably the best thing I could do for traffic to my blog. Unbeknownst to me, it seems that everyone who reads this book for school is scouring the internet for some help. Anyway. It was a fun post, and now I realize that I promised a second post about it and it never came. Oh well.

1) And the winner is…of course, my series of posts on the meaning behind my blogging name “hilbert theorem 90″. Here is a link to the most general version of the theorem that I posted. The sequence of posts actually starts here. This was one of the other very clearly most active posts.

Worst things I did and must remember to avoid for the next year:

5) Avoid titles that could come up in a search for the opposite thing. My post “When Art is Meaningless” has brought in so many hits for “art is meaningless” that I am actually the top google hit for that search phrase.

4) I should try to avoid posting on things I know nothing about. Er uhm…my 4 post long attempt at noncommutative geometry

3) Random speculation should be kept to a minimum. I’m scared to think about how many people took this post to have some merit (just based on search hits, it has got to be more than just a couple).

2) I have sort of a toss up when it comes to what my worst decision was. So in second place I’m going to put “heat of the moment” posts. The three that immediately come to mind were all posted after really heated debates. On parenting is sort of along the lines of The Ethics of Manners in which I conclude that having good manners is unethical. And the third one being Ethical Voting Habits which I at least think had quite a bit of merit, but probably should have been posted under a cooler head.

1) The winner for worst decision of my first blogging year was what I called SSS, for Strange Sunday Specials. The first one is here. For those that weren’t following at the time, I tried to post a random fact I found on wikipedia using the random page generator that somehow related to a topic in my blog (sub)title. The attempt was to keep me posting at least once a week during a pretty busy point in my year. It sort of worked, but only to the dismay of my readers who had to read dumb randomness.

Well. I hope you enjoyed the year as much as me. It has been great fun so far and I don’t plan on stopping anytime soon.

I feel I owe a little explanation, since my original plan was to pick something cool like Morse theory and do a series of posts on it for a spring break project. Instead, my spring break became a rather intense experience.

My parents came to visit me, which was in the original plans. We did a few basic touristy things through Tues., then disaster struck. My dad we’ll just say started losing a lot of blood and ended up in the hospital. You should note that this is literally on the other side of the continent from where my parents actually live. So it is not a good place for this type of thing to happen.

He ended up receiving 4 units of blood over a course of 2 transfusions. They were due to leave extremely early on Friday morning. But in order to figure out what caused the bleeding, the doctors needed to do a colonoscopy (scheduled, you guessed it, Thursday afternoon). Thus, they had to extend all their reservations (rental car, hotel, plane tickets, etc) for an unspecified amount of time. They guessed it would be over by this morning, and it was luckily.

So without going into much detail, that is the general story of my spring break (and hence for any of you that read this that wanted me to do things with you over break and I promised I would, that is why I didn’t. I wasn’t just ignoring you). On a more positive note, I’ve made a lot of progress over this break on one of my life-long goals: learning to play the banjo. Really, do instruments get much cooler than the banjo?

Alright, I had this vow that I would post at least once a week no matter how crazy things got, but I guess I didn’t realize just how much could stand in the way. So instead of a “math, physic, philosophy, or art” post, I’ll just explain why I haven’t been posting.

This weekend I will be moving. As with any time you move, it is highly stressful and time consuming. Speaking of time consuming, I am a grad student and have midterms approaching and for the most part classes that go faster than any human should be able to keep up with. Not to mention trying to set aside the time to grade the midterms I’ll be giving my students as a TA.

On top of that, my computer decided to break. Or Ubuntu decided to break my computer…or maybe this should somehow be worded that doesn’t involve a machine without consciousness making a decision to do something. In any case, I’ve spent numerous hours fixing it. It is finally back to normal, except for getting all the software back in place which will take at least a few weeks (yes, I formatted my hard drive).

On the plus side, I did get the new Andrew Bird album and may post on that soon. I’ve also read about half of Barth’s Lost in the Funhouse which would make a great compare/contrast post with Wallace’s Girl With Curious Hair.

I may also start a series on Morse theory, since I’ve become fascinated by it. Tallyhoe! (spelling on that?)

I didn’t fall off a cliff or anything, but the semester did start, and I tend to lose motivation for posts will all the coursework. I am going to try to post once a week, but usually I think of things that seem not worth mentioning or are so overly complicated that I just don’t have the energy to hash it all out. I never strike on a good idea to post that is of appropriate length and importance. I will keep my eye out, though.

This picture thing turned out to be a pain. Maybe there was some easy fix, like, I don’t know, clicking “Use your old picture,” but that didn’t seem to work. So then I just tried re-uploading it, but the size was too big. Then no matter how I shrunk the size and dimensions, when I would re-upload it would just blow it back up to the original size. Thus, I had to change the picture to one in which I was further away from the camera in order to have a head small enough so that after cropping you could see more than just a single eye.

I have this nice post all typed up on every finite group arising as the Galois group of number fields, but there is some weird LaTeX bug that is cutting out some sentences. I also have a square bracket problem, which I think I’ve narrowed down to: you cannot start your LaTeX code with “[". The other weird thing might be that you cannot end with "!"?

Both of these are bad considering I'm dealing with the group , so it has order n!, and I'm dealing with field extensions which means I need [K : F], but I’ve subscripted, so I need it inside of LaTeX code…

I know I haven’t posted for awhile. I have some good ideas on what I want to do next, but it is finals week, so…I should be back up and posting regularly again starting say Thursday.