New Blog.

April 24, 2013

I love wordpress and how awesome their stuff is, but I needed a redesign.  The new blog [now not hosted on wordpress!] is at


Enjoy!  Leave comments! If I make mistakes [which will be often], let me know!



End of Jan 2011 Update.

January 24, 2011

It’s almost the end of the month, and I haven’t been posting as much as I’d like.  I’ve just started the spring semester, and I’m getting used to juggling my schedule. 

In addition, as it stands, my homology primer has been bothering me; it’s not rigorous enough to be lecture notes and it’s not simplistic enough for it do be of any more use than, say, Hatcher.  This is a significant problem: I set out to compute the cellular homology for certain structures, but without the notion of singular homology (which I don’t even mention!) it’s difficult to justify certain notions and to define new ways of computing and it feels sort of cheap to just say, "it works because it does — trust me." 

I’m not sure when I’ll have a chance to revamp the homology primer, but what I may do is just handwrite a bunch of examples (it’s much faster) and upload them.  The upshot to this is that it’s really quick for me.  The downshot is that any typos will be much harder to fix.  C’est la vie. 


Either way, I will be posting again soon. 

I’m sorry for the long delay in writing — I had a jordan-normal form post, but I made a few errors, and, thus, am saving it for when I have more time to go over Jordan form. 

I am now officially starting up grad school, and I will be attending classes in analysis, complex analysis, topology, and an algebraic geometry class.  I will be TA-ing a calculus class, and a probability and statistics class.  Because this mathblog (mlog?) is, for me, a tool of learning by teaching, I will begin to regurgitate things that I am learning in class, and will address problems that a significant number of my students will have.  This is not to say I will be solving any homework problems — ’cause I won’t!  But, as my old calculus teacher always said,

"If most of the class listens and studies but still doesn’t understand the lesson then it doesn’t make them bad students so much as it makes me a bad teacher."

Does this mean I’m abandoning differential topology?  Potentially; I need to see how much time I have left. 

I’ve been in the process of moving to New Orleans, so there hasn’t been much updating recently.  In fact, I haven’t been able to do much math at all!  I know, very sad.  I will be posting, next, about some random multivariable topics (primarily those that I need for my placement exam.  ha.) and also beginning to follow the (short, but concise) Topology from the Differentiable Viewpoint by Milnor. 

Lemme just note something quickly about TftDV: it’s an older book, but it comes highly recommended and, as far as I know, is still used relatively frequently as an introduction to the subject.  The problem here is that I learned the subject from a different (and slightly stricter) book; so whereas Milnor talks about manifolds sitting in {\mathbb R}^{m}, I will be talking about maps from the manifold M to {\mathbb R}^{m} explicitly and not assuming the manifold is sitting inside of real space.  

Also note that I am in no way an expert in this field.  Far from it, in fact.  I will be learning along with you!  If I make a careless error, please don’t hesitate to correct me.  In addition, I will most likely be going slightly slower and explaining in more detail many of the topics in the book and simply skipping over or mentioning briefly some of the other topics which I feel are slightly less important or slightly less interesting.  I will also (and this is why I haven’t posted it yet…) be experimenting with scanning pictures that I’ve drawn to illustrate proofs and things, and placing them on this blog.  I don’t have a scanner yet, and I feel that explaining open sets and also manifold mappings and things will be much, much clearer with a picture, so I’ve put it off.

Current Update!

August 4, 2010

After the last post on the real spectral theorem, I don’t have too much more to do in linear algebra.  I think this is around all I need to study for my placement exams as far as linear algebra is concerned — and so, for now, this is where I’ll probably end.  I’ve yet to post an example of using the spectral theorems, and I may post something about using complex vector spaces (since I found a bit about how to define them, etc, etc.) but for the most part, I’m not gonna write too much more about linear algebra.

On the other hand, I haven’t really even begun to study multivariable, so I’ve begun by putting up some random tidbits of information.  The reason for this is twofold: because lengthy and well-laid-out texts for multivariable calculus are available (for example, I use Stewart‘s calculus now, but there are many others.) and, in addition, because these concepts as a whole take a ton of time and patience to develop.  I have neither.  Consequently, I will be posting about little things that I find interesting or, perhaps, tricky.  Anything that takes me a little while to "figure out" from Stewart’s text, I will post on here. 

What about my claim (a while ago!) that we’re going to use linear algebra to study groups?  This is still true.  I don’t have my representation theory book anymore, but it’s being sent to me eventually; when I have it, I can begin writing about it.

Last, all I want is to you guys to have fun!  So if you have a topic you want me to explore, tell me!  Please.  I like hearing from you guys!


May 5, 2010


To begin with, what is this blog about?  As I stare deeper and deeper into mathematics, it begins to stare back at me; when it does that, I like to write about it.  Most of the posts I make here will be theorems that I’m currently doing in class, topics that my students are having difficulty with, or just things that I find interesting.

My interest is primarily in Topology, but I’m also partial to Algebra.  I’m not a huge fan of applied maths, but I will be posting a number of things on linear algebra, multivariable calculus, and standard calculus.

Good.  Now who am I?  I’m James.  To answer this more fully, see this page.

Let’s not waste any more time.  Go out there and do some mathematics!  Let me end this post with an old folk saying that I think aptly applies to working in mathematics.


\mbox{May your dreams be more frightening than your nightmares.}