## Current reading: neuroscience September 30, 2010

Posted by Sarah in Uncategorized.
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Right now my “interests” are supposed to be limited to passing quals. Fair enough, but I’m also persistently trying to get a sense of what math can tell us about how humans think and how to model it. Except that I don’t actually know any neuroscience. So I’ve been remedying that.

Here’s one overview paper that goes over the state of the field, in terms of brain architecture and hierarchical organization. Neurons literally form circuits, and, in rough outline, we know where those circuits are. We can look at the responses of those circuits in vivo to observe the ways in which the brain clusters and organizes content: even to the point of constructing a proto-grammar based on a tree of responses to different sentences. I hadn’t realized that so much was known already — the brain is mysterious, of course, but it’s less mysterious than I had imagined.

Then here’s an overview paper by Yale’s Steve Zucker about image detection using differential geometry. In his model, detection of edges and textures is based on the tangent bundle. Apparently, unlike some approaches in computational vision, this differential geometry approach has neurological correlates in the structure of the connections in the visual cortex. The visual cortex is arranged in a set of columns; the hypothesis is that these represent $\mathbb{R} \times S^1$, with the column representing position and the slices at different heights of the columns representing orientation.

1. david - October 2, 2010

It would be interesting to know what subjects are part of your PhD quals. Is it the case that most students are looking to just finish off the quals ASAP? Have you decided your thesis topic already?

2. Sarah - October 2, 2010

The quals are on algebraic topology, algebra, and analysis. I’m probably going to be finishing them by the end of this year. Most people want to get them finished relatively fast — partly because there is a deadline, and partly because it’s good to start research once they’re over with.

No, I don’t have a thesis topic.

3. david - October 3, 2010

That is great. Looks like a strong undergraduate program helps in getting past the quals rather early. So, are you giving the quals after only 6 months into grad school?

What are the syllabi? Analysis at the level of Rudin’s R&CA, Algebra at Herstein/Lang?

(Yale’s math department webpage skimps on many details! I couldn’t find any information on the phd qualifiers.)

4. Sarah - October 3, 2010

All the syllabi are paper — I don’t know if this is technological backwardness or some kind of confidentiality. But, yes, analysis is roughly like Rudin, algebra is like Lang, and topology is like Hatcher. Quals are offered every semester and have to be taken in the first or second year; I’m going the conventional route of taking relevant first-year courses and aiming to be done with all the tests by May. Some people go faster (skipping the courses, finishing quals in December of their fist year) and some people go slower (finishing quals in their second year.)

Are you a prospective student?

5. david - October 3, 2010

You guessed right.

I’m enrolled as an EE engineer currently but I am seriously considering an applied math PhD in grad school. I still have a year left before graduating and I’m reading basic algebra and analysis so that I won’t have a terrible math background. But I’m seriously worried if I can convince the grad admissions committee that I’m capable of switching fields. FWIW however, I’m reading a lot of math (some of which I frankly don’t understand at all) and I like it a lot.

I suppose a subject GRE might help in the “convincing”. But the sample test papers in the math GRE are really easy and I don’t think they’ll prove much. If nothing works out then I suppose I have to enroll myself in a Masters programme before attempting grad-committee-appreciable field-switching stunts.

Do you have fellow grad students who have non-math background? I just want to be reassured that it isn’t suicidal to try this 🙂

6. Sarah - October 3, 2010

I don’t, actually. But then, I’m in a pure math program, not an applied math program. The applied math program here has at least one student with a physics background. I don’t actually have a whole lot of evidence to reassure you that it’s going to be fine, but for all I know it’s very possible. I would suspect that applied math programs vary a lot in who they’ll accept and it might even be useful to write them with this question.

Also, I don’t know your precise situation, but don’t write off going to grad school in EE. I know one EE grad student at Yale whose advisor is a mathematician. You can definitely do work in applied math while formally getting an EE degree.

The GRE, as far as I understand, is a gauge to see if you’ve learned enough math in college; a decent score is necessary but not sufficient.

My gut feeling is that you’ll have more success getting into applied math grad schools if you’ve done something that’s essentially math (e.g. research) so that you can present yourself as interdisciplinary rather than as exclusively an engineer.

So, my advice is: talk to anyone at your school who can give you advice and potentially the Directors of Graduate Studies at applied math departments you’re looking at. See if they’ll accept non-math majors. If you aren’t getting reassuring answers, try to find EE departments where there are people working on mathematical topics, or where it’s easy to take a lot of math classes and collaborate with the math department. Yale seems to be really good at leaving those boundaries fluid, and it seems that some other schools are too.

Good luck!

7. david - October 4, 2010

I have developed sufficient interest and background in some varied areas of mathematics (although to not any great depth). I was initially interested in signal processing and the like at the beginning of my EE program but I quickly found out that the most “interesting” (subjective opinion) research seemed to be happening in sparse recovery and inverse problems (compressed sensing for instance) most of which involved a lot of (roughly) linear algebra, matrix analysis, probability and convex optimization. I cursed my stunted math background and read many things on my own to enable me to understand some of the papers in that field. That’s when I decided that I will choose a career in applied math so that I can better understand the arguments and analyses presented in papers, as opposed to feeling hopelessly lost. It seemed feasible at the time (and even now) as some of my friends, here in India, have successfully switched their fields of study. In one case, from EE to theoretical physics (string theory) at a very good school in the US, although no doubt he was in excellent academic standing.

Most of what I wish to learn (the math), is likely offered in some form by the EE department itself and hence I wouldn’t be especially unhappy to get an EE PhD. But my gut feeling is that an applied math degree would serve me better.

I’ve already looked up quite a few applied math departments in the US and Canada, and I’m essentially trying to fit their “ideal” student profile. In any case, I will be applying in Fall 2011 and there is one other student this year who is attempting something very similar (EE to applied math) and I can learn from how it turns out.

If things work out, I will be interning at a “pure math” institute coming summer in discrete optimization. This, like you say, will convey my seriousness about pursuing a PhD program in math.