A quick note on group actions

June 20, 2010

What’s a group action?

Well, we start with a set A, and a group G. A group action of G on A is a map G \times A \rightarrow A (denoted g \cdot a) with the following properties:

(g_1 g_2)\cdot a = g_1 \cdot (g_2 \cdot a)

and, of course, 1 \cdot a =a for all a \in A.

Now this seems kind of silly and not very useful, but there are some kind of fun things that happen with these. First of all, notice that each group element g can be thought of as a function \phi_{g}: A \rightarrow A, which sends a \mapsto g \cdot a. But it gets cooler! Since \phi_g has a left and right inverse, namely \phi_{g^{-1}}, this function is actually a bijection. Thus, in this way, each group element can be identified with an element of S_{|A|}, which gives us a map G \rightarrow S_{|A|}. It is obvious that this map is a homomorphism of groups. If this map is injection, we call the group action faithful. That is, a group action is faithful if no two group elements induce the same permutation on A.


First Two Paintings

May 24, 2010

So as I said before, I’ve been painting with acrylics!  My first painting took me forever.  I copied it from a picture in black and white, using black and white paints so that I could learn about “value.”  This was HARD.

My second painting I did last night and it only took me about an hour.  This time I used color, and I found it MUCH easier.  I also didn’t paint this one from a picture, just from a bunch of bananas!

Maybe I’ll try to get better pictures up at some point.  These were taken with my phone.

I’ve been linked to!

May 17, 2010

As my mathbff Jmy so kindly and graciously pointed out, I’ve been linked to!  He and Sarah, another math friend also have math blogs.  So everyone who reads this (i.e. probably just jmy and Sarah) should go and read their far superior blogs!  So far I can hardly call mine a math blog, as I haven’t done any math yet.  I have two and a half more weeks left of classes.  Right now I’m struggling to like manifolds enough to continue to do the work.  Too notation-heavy for me!  The funny thing is that I really liked last weeks homework, and I was like “Okay, cool!  Maybe I’m starting to like this stuff!”  But then I realized that this particular homework was almost entirely algebraic topology.  Oh well, I will struggle on.  Speaking of which, manifolds is calling my name as I type.  This week’s homework: wedges!  Exterior algebras!  I’m actually KIND OF excited because maybe this homework will be more algebraic!  I’ll update and let you know as SOON as I finish.  I know you all are at the edge of your seats!

Paint and Apartments

May 12, 2010

I haven’t updated in a while!  But don’t worry, it’s just because I have been very busy, and I will keep becoming less and less busy!  So here are some updates:

I’m going to Ann Arbor to look at apartments tomorrow!  I’m driving there tomorrow after manifolds and then driving back on Friday after I visit MANY places.  I’m kind of anxious about it, just because I’m going to have to drive around to find all these places, but I printed out directions and everything, so hopefully it won’t be too stressful.  After this weekend, I will be done with it, hopefully.

Relatedly, I found a summer sublet in HP!  It’s very convenient–right across the street from HPP and it’s a beautiful studio.  I’m so excited to finally live alone, at least for a few months!

What else?  Oh, I bought a car.  A Honda Fit.  It’s very cute, but getting MN license plates is very stressful, so I’m looking forward to being done with all of that!

And finally, I’ve started painting!  Well, I painted last night.  I used Ian’s acrylics.  It was a lot of fun, and I’m not terrible at it.  I think I might buy my own paints and supplies, and start doing it more frequently!  Especially once school ends.  This will definitely be a fun summer project.

Well, that’s all for now!  I promise I will write many more fun things as soon as I have a little more free time!


May 5, 2010

Just changed the look and feel of the page!  Hope it pleases!


May 4, 2010

So I had this blog a while ago, but didn’t really use it, so I started it up again, completely refreshed!  I think I’m mostly going to use this to talk about my preparations for grad school.  i.e. what I’m learning over the summer.

Before I can do any serious studying, though, I do have about 5 weeks of school left.  I’m currently taking Topics in Geometry and Differential Manifolds.  Topics in Geometry is sort of silly.  We’re sort of following Artin’s Geometric Algebra.  It’s kind of fun in the same way combinatorics is kind of fun, but I don’t think that it’s really a hot field right now.  It’s very classical stuff–affine and projective geometry and geometric interpretations of matrix groups.  Anyway, this class is full of much younger people who I don’t really know.  This makes sense I guess, since I pretty much only know the math majors my year (and the ones who graduated last year, of course).  So I took the midterm for this class last Friday after not really studying at all, and I hadn’t really been paying attention in class at all either.  The questions were all *very* easy.  Two of them I could’ve done before taking the class, and the other two were really easy applications of the stuff we’d learned.  I didn’t really have enough time though, and I wrote basically nothing (I would say nothing, but somehow I got points for what I wrote…) for one of the problems, and was very hand-wavy with all the others.  I was pretty sure I’d done really badly, at best a B-, but then in class yesterday, he said the average was 45/100, and it turned out that I got 75!  I’m not really sure how this is possible, since I felt like I did the bare minimum of trivial things.  I couldn’t even on the spot think of a normal subgroup of S_6.  I just scribbled one down in the last second, and it wasn’t normal.  How did the majority of people do so much worse than I did?  There are even a few math majors my age in the class, who I’m assuming did just as well as I did, which means that the other people must have done even worse to get the average so low!

I also just took my midterm for my other class, manifolds.  Manifolds is a much more important subject to know, apparently.  I will need to know all this stuff if I want to do just about any kind of math (except maybe finite group theory or something).  The thing is, this class is HARD.  Nothing is intuitive for me.  I get the “idea” of everything, but when it comes to actually writing proofs, I can’t do it.  It’s so notation-heavy, and I feel like the notation of everything takes away all the elegance, whereas in algebra, once you figure out the idea of why something works, it’s really easy to write down.  Anyway, the test was VERY easy.  There were three problems and we had an hour and 20 minutes to do it.  I finished second (there were 30 minutes left).  However, in two of the problems I was maybe not rigorous enough.  The first problem was something like: Is there an immersion from every closed n- without boundary into R^n?  I gave the counterexample of S^1 into R: every nonvanishing vector field on S^1 must have a vector map to 0 under the differential, by continuity.  The other problems were trivial, and kind of hard, respectively.  The trivial one isn’t worth mentioning, but the other was: Is RP^2 x RP^3 orientable?  I said “no” and tried to show that every atlas has a transition function that does not preserve orientation, but I’m not sure how correct all of the assumptions I made were.  I probably should have done something using volume forms, but I’m a little afraid of them.  The people in my class seem to know their stuff, though, so the curve might not be so much in my favor this time, but I guess I’ll see.

When I get to Michigan, my goal is to pass both the algebra and the topology quals.  The algebra should be sort of easy to pass once I review over the summer, and the topology shouldn’t be too bad either.  I will just have to learn more homology over the summer, since I think the manifolds class will have prepared me pretty well for that part of the class.  So my plans for review start with Dummit and Foote (pretty much all of the stuff up through Galois Theory) and Hatcher.  In Hatcher, I’ll try to get as far as I can, but definitely through homology.  If I am able to pass both of those exams, then the classes I’ll take will be: Complex Analysis (then I won’t have to take the analysis qual), Algebraic Geometry, and ADVANCED algebraic topology!!  I think the latter two classes will help me decide sort of what I want to do.  I think, though, that algebraic geometry will be very very hard.  In the course description, he says: “A solid background in commutative algebra is very useful; working out the exerises in the Atiyah-Macdonald book Commutative Algebra is a good summer project to prepare for the course.”

So I guess I’ll be going through Atiyah-Macdonald also!  The algebraic topology class shouldn’t be too bad.  According to the description, there are no exams, and about three to four problems assigned each week.  I think that this class will be useful even if I wind up doing commutative algebra or something.  I’ll learn lots of homological algebra!