<p>Can you guys give me some info about the math department? I know some of the basic information about majoring in math, like course requirements, etc., but I would like to know more.</p>
<p>What are the hardest required math courses (by required I mean required for math majors)?
What kinds of mathematical research is going on that is accesible to undergraduate students?
Specifically, what kind of research have you done?
Are there many students doing research that applies theoretical math to physics?
I've heard that for each of the major branches of math (analysis, geometry, algebra, and topoloy) there are three year-long courses at Caltech. Is this all at the undergraduate level? Do people tend to specialize somewhat in one of these areas, in multiple areas, etc?
How much collaboration do you see between the math and physics departments?</p>
<p>I know it's a lot of questions, just answer as many as you want/can. Thanks =)</p>
<ol>
<li>People disagree about what the hardest required math major course is, I personally found Math 109c (Intro to Riemannian Manifolds) to be harder than the other requirements.</li>
<li>Almost every prof in the math and ACM departments has supervised at least one SURF student. The greatest number of SURF's tend to be in combinatorics and algebraic number theory since these require the least background.
Personally, in the summer after my freshman year I did a (mostly theoretical) SURF in the CS department on incremental methods for detecting change in data streams. In the summer of 2006 and much of the past year I did research in applications of logical model theory to topological dynamics of certain automorphism groups under Prof. Kechris. Now I am working under Barry Simon (who is cross listed in the math and physics department) on the problem of finding Hausdorff dimensions of limit sets of certain Fuchsian groups (its a problem that can be stated completely in terms of hyperbolic geometry but seems to require largely ergodic-theoretic methods), which will the subject of my SURF this summer and senior thesis next year.</li>
<li>There are some. I know a student who last summer did a SURF last year in topological field theory in the physics department. The problem I am working on this summer is directly related to the spectra of finite gap Jacobi-matrices, which according to my advisor has applications in quantum field theory, but I don't know enough about QFT to understand them. I also know a current senior who is majoring in math, but has done research only in experimental physics at Caltech (he decided that although he wanted to eventually be a theoretical physicist, understanding experiments would be a useful skill)- and is going to grad school at Harvard in string theory next fall.</li>
<li>There are required 3-term undergrad sequences in Algebra, Analysis, Top./Geometry, and corresponding 3-term graduate sequences (which a lot of undergrads take). There are also 3 term "graduate" sequences in combinatorics, algebraic geometry, logic/computability (alternate years), functional analysis/ODE's (alternate years), dynamical systems (every other year) and a bunch of topics classes in various subjects offered either annually of semiregularly. I put "graduate" in quotes since a lot of these classes that I've taken have had between 0 and 1 grad student and the rest undergrads. Apart from the requirements for the major, people classes that interest them- since a lot of people arent sure what they want to do yet, they take graduate classes in a variety of subjects.</li>
<li>There are two people Barry Simon (functional analysis/applications to QFT) and Sergey Gukov (string theory) with dual apppintments in the math and physics departments. Some other topologists in the math department also collaborate with physics and CDS people.</li>
</ol>
<p>thanks for the info</p>