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Need Course Selection Advice

  1. May 31, 2012 #1
    Hi!
    I usually choose my courses in advance then prepare for those courses before term starts.

    Background: I have taken: Multivariable Calculus (little spivak), Real Analysis (Pugh), Abstract Algebra (Dummit and Foote chapters 1-5,7-9,13,part of 14(including FTGT)), Ordinary Differential Equations (Hirsch & Smale & Devaney, half being linear system, half nonlinear (existence & uniqueness, some other stuff, linearization, bifurcation, Lyapunov)

    Note: I did not take point-set Topology by Munkres, but I have self-taught myself enough point-set Topology.

    Before september, I will have undergraduate probability background from a course (from Whittle's Probability via Expectation)

    Goal: Since I am becoming 3rd year in upcoming September, I would like to get as much background grad level math as possible, so that it will help me decide my specialization.

    Achevements: I will take Complex Analysis (Stein & Shakarachi) 1 year, Graduate Differential Topology (John Lee's Smooth Manifolds), Graduate Algebraic Topology (Hatcher)

    Constraints: I do not know whether it is beneficial for me to still take point-set topology course (in terms of applying to graduate school)

    Possibilities:
    1. I have prerequisite to take 2 half-year courses in Graduate Algebra (Dummit and Lang), where 1st term does Groups/Rings/Modules, 2nd Term does D&F chapters 13-19
    2. I have prerequisite to take 2 half-year courses in Real Analysis (textbook of choice is usually Stein/Shakarachi, Folland and/or Lieb/Loss), where 1st term covers Measure Theory, Hilbert Space, L^p space?, and 2nd term covers Banach Space, Baire Category Theorem and its cousins (Open mapping, uniform boundedness), Fourier Analysis, L^p space?
    3. I have prerequisite to take 1 year course in Undergraduate Partial Differential Equations (Strauss). Alternatively, I can take 2 half-year courses in Graduate PDE (Evans), where 1st term is pretty much condensation of 1 year undergraduate course (but this still is intended for people without any background), and 2nd term is Evans chapter 5 and beyond. I can try graduate option safely if I do option 2 at the same time (or I have to make sure I work on Folland/Rudin regularly without falling behind as 2nd term requires a great deal of Real Analysis)
    4. I will have prerequisite to take 2 half-year courses in Graduate Probability (Rosenthal, https://www.amazon.com/First-Look-Rigorous-Probability-Theory/dp/9812703713/ref=dp_ob_image_bk), where two terms combined pretty much goes through whole book
    5. Miscalleneous courses such as K-theory and C* algebras (where course description says I have all prerequistie, but I do not believe this), Set Theory and Model Theory, Linear Operator Course (Peterson), which I am unlikely to try out.

    Sorry to write such a lengthy post that is not even done yet!
    Up to now I showed positive sides pretty much, but:

    Problems (Constraints)
    1. Undergraduate PDE overlaps with Algebraic Topology for 1 out of 3 hours (in 2nd term), which I am willing to sacrifice (so this is not too much big deal)
    2. 1st Term Graduate Real Analysis overlaps with 1st term of Complex analysis (undergraduate level) for 2 out of 3 hours! This is a serious problem. (maybe I should wait till 4th year to take graduate real analysis). But then I cannot take 2 real analysis if I do not take 1st term one, which means that it is likely that I have to give up on Graduate PDE
    3. 2nd Term of Graduate Algebra overlaps with Algebraic Topology for 1 out of 3 hours, which seems to be somewhat big deal (also because 2nd term of Graduate Algebra gets to cover all interesting algebraic geometry, representation theory of groups)
    4. 2nd Term of Graduate Probability overlaps with Algebraic Topology for 1 out of 3 hours! To make it worse, this overlaps with 2nd term of Graduate Algebra ENTIRELY.

    Why are these problems? Because when things are grouped together, 2nd term of whatever is usually pickes up where 1st term left off (sometimes profs try different organization in 1st term, then 2nd term gets greately influenced). So it is somewhat risky to take only 1 term counterpart instead of doing all year thing (also because I have to wait till 4th year winter from 3rd year fall, to finish the 2-term combination). Of course, probability is somewhat exception, because I am doing probability as one of sidekicks, so maybe just taking 1 term might be okay.

    I am sighing myself so much, because I thought core graduate courses (Real Analysis, PDE, Complex Analysis, Algebra, Differential Topology, Algebraic Topology) would not overlap, but I was wrong.

    What are your advices? Thank you for reading.

    P.S. If my course description is vague, then here is link:
    http://www.math.toronto.edu/cms/tentative-2012-2013-graduate-courses-descriptions/
    where MAT1000/1001 are Graduate Real Analysis, MAT1002 is Complex Analysis, MAT1100/1101 are Graduate Algebra, 1060/1061 Graduate PDE, 1300/1301 Topology, STA2111/2211 Grad Probability
    and http://www.artsandscience.utoronto.ca/ofr/calendar/crs_mat.htm#APM351Y1 for undergraduate PDE with course website: http://www.math.toronto.edu/almut/APM351.html
     
    Last edited by a moderator: May 6, 2017
  2. jcsd
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