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Will A Good Undergrad Program Help?

  1. Jun 14, 2010 #1
    Will A "Good" Undergrad Program Help?

    This Fall I'll be starting my freshman year at the University of Chicago as a prospective physics major. From what I can tell, research opportunities for undergrads are pretty good and the faculty are good as well.

    But what I want to know is, will having gone to a "good" school like UChicago really help come grad school? Will it give me even a small boost? Or is the boost that a "good" school gives solely found in the undergrad research opportunities that it allows one to pursue?

    Sorry if this post comes off as entitled, but I'm curious! This forum has surprised me many times before, and made me a bit less starry-eyed about the future, but as long as it's realism and not pessimism...

    Thanks in advance!
     
  2. jcsd
  3. Jun 14, 2010 #2
    Re: Will A "Good" Undergrad Program Help?

    Wow, are you serious? UChicago is a prestigious school, are you having doubts? If you are already in, why are you worried?

    I think going to a "good" undergrad program means the quality of education is better and probably because the school is rich

    Good luck!
     
  4. Jun 14, 2010 #3
    Re: Will A "Good" Undergrad Program Help?

    In short, yes, it will help--perhaps very significantly--but maybe not in the most obvious ways. The name UChicago alone will provide a little boost, but not much.

    What will help will be the atmosphere--the other students there, the more rigorous courses, the greater opportunities. I've been an undergraduate at both an average public school and a top 20 private university, and I really, really think the atmosphere at the latter is significantly more conducive to really mastering a subject in depth, not to mention the courses go at a faster pace and cover material in greater depth.

    That said, reaping these benefits will require a huge amount of work from you. UChicago isn't magical. You'll get out of it what you put in. The difference is that UChicago can handle your putting in a lot more than most schools can handle, and it's capable of giving back accordingly. UChicago will have lots of mechanisms in place for students wanting to pursue a subject to the maximum; most schools will just make those students drift through the years without challenging them at the appropriate level.
     
  5. Jun 14, 2010 #4
    Re: Will A "Good" Undergrad Program Help?

    Well, I've been reading a lot on this forum, and many people seem to say that the school is very secondary to the person attending it. Given that my parents are paying significantly more for this school...I'd like to make sure I'm getting (their) money's worth.

    That's good to hear, since it's one reason I picked the school. I figure that being around a lot of people who are both smart and intense will in turn bring out the best in me.
     
  6. Jun 14, 2010 #5
    Re: Will A "Good" Undergrad Program Help?

    UChicago is considered by some to be "sub-Ivy."
     
  7. Jun 14, 2010 #6
    Re: Will A "Good" Undergrad Program Help?

    True, but if you compare the majority, the prestige does matter to a little.
     
  8. Jun 15, 2010 #7
    Re: Will A "Good" Undergrad Program Help?

    Well, I think that's still a very true statement, despite what I wrote earlier.

    I suppose the bottom line is that UChicago wouldn't be worth it for most people, because most people don't care about really immersing themselves in academic studies and pursuits, and even many of those who do care about such things maybe aren't actually willing to put forth the effort to make those things into a reality.

    But if you do care about those things and you are willing to put forth the effort, UChicago and places like it will have a lot to offer you that a typical school may not--and it will mostly be in terms of a much different atmosphere. One thing that just pops right into my head is UChicago's (somewhat infamous) Honors Analysis course and the culture that springs up around it.

    None of this is to say that you couldn't go on to be a master of your subject if you don't go to UChicago: that of course is an absurd thing to think. :D Like I said, UChicago isn't magical; it is just a school, and everything will ultimately depend on you. You could easily come out of UChicago more incompetent than the average physics graduate of... well, any other school on the planet. It's not going to do any of the work for you.
     
  9. Jun 15, 2010 #8
    Re: Will A "Good" Undergrad Program Help?

    If you want to get the most from UChicago as a physics major, you'll want to plan on taking the hardest math courses and do well in them. Learning calculus over again with rigor is the first basic step. In a nutshell, this means placing into and staying in honors analysis your first year.

    If you were planning on taking honors calculus and then honors analysis, forget it, it's a waste of time. The honors calculus sequence at UChicago is typically not that difficult, and it certainly won't be as difficult as say, working out the most difficult problems in Spivak. The very essentials of basic analysis that Spivak covers in 30 chapters or so is all contained in the first 8 chapters of Rudin's Principles of Mathematical Analysis. However, if you actually want to take anything away from Rudin, you need to devote a lot of time to understanding his extremely elegant and concise proofs and exposition in general. Thankfully, for the first 5 chapters or so, there is a set of online lectures here: http://www.youtube.com/user/HarveyMuddCollegeEDU" [Broken]

    The lecturer is exceptional, and his lecturing style is very similar to one of the professors who taught honors analysis this year. Essentially, you have the resources to understand the more important points in an honors calculus sequence that you'll need for more advanced analysis.

    I'll reiterate that if the thought of taking honors analysis has ever crossed your mind (and if you're a physics major at UChicago, it's kind of ubiquitous), prepare to take it in the fall. You'll have to do a lot of self-studying in honors analysis anyways. For instance, throughout the third quarter, you'll be exposed to Banach spaces, Haar measure, Hilbert spaces, and Fourier analysis on locally compact abelian groups. Paul Sally won't prove much and his book will probably still be unfinished, but you'll know by then to check out Big Rudin and other texts to pursue the lines of inquiry that Sally provides in class.

    Note that taking honors analysis will in all likelihood mean that you are precisely taking the class where the most intense physics majors are anyways. It's not just about taking a difficult math course for the sake of a good looking transcript. The only reason mathematical analysis is so sophisticated now is due to physical considerations in the first place, and it shouldn't take too long to figure out that it is extremely fundamental to physics and to mathematics in general.
     
    Last edited by a moderator: May 4, 2017
  10. Jun 15, 2010 #9
    Re: Will A "Good" Undergrad Program Help?

    Can you put some more context into this? Saying that "some" people think something doesn't really mean much on its own.

    Well, at this point I've had basically no experience with proofs, which doesn't really bode well here. I started the summer thinking that I'd work through Spivak, found the textbook pretty hard on its own (especially since I'm used to the lame, Geometry-style fractured, numbered proofs, and am having difficulty knowing whether the proofs I write are correct when his take a different route) and this week decided to just do a Multivariable MIT OCW course, which is going better. Frankly, given that the "highest" math I've had is the math-lite AP Calc BC, Honors Analysis (even if I study hard over the next ~3 months) doesn't seem terribly likely or wise.

    At any rate, some 15 freshmen place into the course yearly; are you really saying that the physics students that don't are distinctly second-tier?

    I mean, I'm willing to work hard if I need to, but it just seems like rigor and proofs are things that are very easy to learn the wrong way, if that makes any sense. I've also read from a variety of sources that one should not try and game the placement exam to get into Honors Analysis, since this is a recipe for really struggling in the class/dropping it.

    EDIT: I'm assuming you are/were a Chicago student. I was thinking of taking the IBL section of Honors Calculus. Is that any better?
     
    Last edited by a moderator: May 4, 2017
  11. Jun 15, 2010 #10
    Re: Will A "Good" Undergrad Program Help?

    http://en.wikipedia.org/wiki/Ivy_League#Other_Ivies
     
  12. Jun 15, 2010 #11
    Re: Will A "Good" Undergrad Program Help?

    Well, the Wikipedia excerpt credits the label to "marketing groups, journalists, and some educators" so I don't know how much weight I should place on that.

    Anyway, I thought that you were saying that UChicago's physics program was inferior to that of any Ivy League school, but seeing MIT and Stanford lumped in their as well kind of makes this moot I guess.
     
  13. Jun 15, 2010 #12
    Re: Will A "Good" Undergrad Program Help?

    Don't bother with the multivariable calculus course too much. You'll learn it regardless of which analysis course you take. Spending an entire day to understand a few pages of Spivak is going to much more useful than watching multivariable calculus lectures. The more time you spend actually reading and understanding proofs (I'm assuming you have an idea of the basic techniques of proof), the quicker you will improve. However, you really do have to spend time understanding every detail. Spivak is a very conversational expositor, so it's sometimes easy to gloss over his explanations without grasping the intricacies of the argument at hand.

    Another thing to keep in mind is to not worry too much about the formalism of proof writing for now. Take the very first chapter of Spivak for example. Most of the problems ask you to verify inequalities or equalities in basic analysis. It's okay to write down enough to convince yourself that you can do the problem. You should be telling yourself which axioms you need to use, but there is no particularly good reason to write down a lot of english. In fact, you'll probably notice that Spivak does the same for a lot of the proofs in the first chapter.

    Also, it's not necessary to do all of the problems in the textbook. Work on the exercises that you find interesting or challenging, but don't worry if you can't solve them immediately. If you think you can immediately see a solution to a particular problem, convince yourself this is the case and move on. You get better by working on exercises that are challenging but not too challenging. You can always come back later to the more difficult problems.

    The same idea applies to the order in which you read the chapters. Skipping chapter one is in most cases a pretty bad idea, but after that, there is no particularly good reason to read the text in a linear fashion. For Spivak, induction won't be a central tool throughout the text, so you can certainly skim chapter 2 and come back to it if needed. You also probably know what a function is and how to graph a function, so that's chapters 3 and 4. This leads you up to the chapter on limits of functions, which is probably something you have some intuition for, but do not know how to work with rigorously. This will hopefully be something you find interesting, and how well you understand this concept will affect everything else you do in analysis. From there you can skip ahead to chapter 22 which ties the limit of an arbitrary function to limits of sequences. If you get to the monotone convergence theorem in that chapter, you'll want to read chapter 8 to understand least upper bounds before learning the theorem. Then you can go back to chapter 6 for the definition of continuity, or skip forward to series and sequences of functions.

    Really, the only way you could learn things "the wrong way" is if you get caught up with the formal details of mathematics at this early stage. Also I would never advise you to game the placement test. In any case, it really doesn't tell you if you're prepared for honors analysis or not. Actually learning how to do basic analysis is obviously more important than simply memorizing the definition of limit, derivative, integral, field axioms, etc. which is more or less enough to get you on the borderline of placing into analysis.

    I would also never call any students "second-tier". There are probably physics majors who didn't take honors analysis and did just fine. I do know however that a few of the physics majors in my graduating class (one of whom I know personally) came here because of the math department and any one of them would tell you that honors analysis was their most challenging course freshman year.
     
  14. Jun 16, 2010 #13
    Re: Will A "Good" Undergrad Program Help?

    @snipez90: Since I'm assuming you took HA Freshman Year, what was your math background beforehand? I'm a little iffy about really trying for HA because I've yet to see anyone who placed into it (and stayed) without already having experience with multivariable, linear algebra, number theory, etc., of which I have basically none.
     
  15. Jun 16, 2010 #14
    Re: Will A "Good" Undergrad Program Help?

    Huh no, I didn't because no one who took honors analysis told me it was a waste of time to take honors calculus if the eventual plan was to take honors analysis. My background was AP Calculus BC and reading chapters 1 and 5 of Spivak the Friday before the placement exam. Diane Hermann told me I was a few points away from placing into honors analysis. My professor for the first and third quarter of honors calculus was fairly lousy. My second quarter professor was a good lecturer, but the exams were too easy (regardless of the fact that the difficulty of the problems is subjective, he only asked us to do 4 out of 5 problems for full credit, and many people solved at least 4). The majority of the time, there was little incentive to go to class since reading Spivak was enough to understand everything that was going on.

    I don't know where you get your sources from, but I bet most people who have taken honors analysis would disagree with your sentiment. First of all, we basically never use any number theory unless we're talking about the p-adics, and that is a small part of the course (it's only part of the course because that's Paul Sally's specialty). The elementary number theory you need for that is covered in a project in Paul Sally's textbook, and you will probably have to do it for homework.

    The linear algebra you need for the course is also covered in 1 or 2 lectures. It is also chapter 2 of Paul Sally's textbook (tools of the trade), and you can read it in a day. You really only need to understand the basics, which is essentially what chapter 2 covers.

    Multivariable calculus is certainly not needed. It will be covered extensively in honors analysis. The nature of multivariable calculus is computational, and very few theorems will be anything beyond straightforward extensions of similar single-variable real analysis theorems. The problems on the exams also reflected this, as we were asked maybe two theoretical questions, one of which involved knowing the definition of the derivative in R^n and the other involved making estimates via the multivariable Taylor's theorem. The computational problems were often pretty straightforward, but it will all be covered in the course.

    Having gotten through honors calculus, my background was everything in Spivak and some basic linear algebra knowledge, nothing else. But I've already stated that you don't need all of Spivak, just the more important topics. In fact, I think chapters 1, 5, 6, 8, 22, 24 are crucial if you want to stick to Spivak. Rudin is better preparation because unlike Spivak, Rudin introduces metric topology right away, and this is the framework for more advanced analysis. Metric topology also happens to make up a large portion of the first quarter of honors analysis. Nevertheless, the chapters I mentioned in Spivak will give you a good feel for the topology of the real line and just as importantly, it will help you become fluent in epsilon-delta arguments.
     
  16. Jun 16, 2010 #15
    Re: Will A "Good" Undergrad Program Help?

    Sorry if I sound like a noob here guys but I am very interested in a few details about what you guys are discussing.

    One thing I got from your posts snipez seems to be that if you know most of single variable calculus, you're ready for analysis. What exactly is analysis? To me it sounds like going more in depth into the theory, what with having to write proofs and all. I'm interested in taking a lot of math classes in college and possibly getting a math minor, but I want to make sure I don't get myself into any GPA troubles. When would you recommend someone to take this analysis class? Could I even learn it on my own while I study calculus? And lastly, which of the books (I think you mentioned Rudin and Spivak) would you recommend in this sort of situation? I'm currently taking a summer class on Calc. 1 (though I taught myself calculus during high school last year because I took calculus based physics) and I'd like to know what my options are as far as math courses go.

    Sorry if I'm derailing the topic or something OP, I just thought it would be more appropriate to post here.
     
  17. Jun 16, 2010 #16
    Re: Will A "Good" Undergrad Program Help?

    Not nearly as much as the costs. I dont know how much that school costs, but I sure wouldn't pay more than a public university for 'prestige'. If you have a scholarship or rich parents then go for it, otherwise dont put yourself into debt over prestige and pedigree. The caliber of the student is far, far more important than the prestige of the school. All of the smartest students in my graduating class started at community college. The ended up in top ten grad schools and one got the marshal scholarship. In my opinion, expensive universities are a rip off every time (unless you dont have to pay).
     
  18. Jun 17, 2010 #17
    Re: Will A "Good" Undergrad Program Help?

    I would refer you to wikipedia, but essentially introductory analysis is a rigorous study of all the limiting processes you learned in calculus. I think it's fairly typical for analysis to be the first course that a math major/minor takes. Other possibilities for the first course may be a course on basic set theory and proof writing or a course in abstract algebra or maybe a proof-based linear algebra course. For obvious reasons, it's probably best to follow the sequence of courses the college suggests.

    I think it's a very good idea to start studying analysis on your own as you're taking calc 1 now. I think taking AP Calculus gave me an intuitive feel for the more rigorous treatment of the same topics studied in analysis, but the typical calculus course can probably introduce misconceptions as well. Thus, studying analysis on the side will help you gain a deeper understanding of the topics you absorb in class and possibly even rectify the form of a particular concept you have encountered already.

    I would recommend Spivak in your situation. Incidentally, the textbook is titled Calculus, but for all intents and purposes it's a basic analysis text (others call it "rigorous calculus", but it's same thing). There is also no particularly good reason to get the latest 4th edition (which is probably considerably more expensive than a previous edition) since I think only two chapters were modified, and neither modification was significant. Even if you don't plan on developing the proof-writing skills right away, you can gain a lot from reading Spivak. For instance, you have certainly encountered the notion of a limit before, but you might not have thought about how to actually give a definition for such an intuitive concept that will allow you to prove properties regarding limits. This is the kind of scenario for which Spivak will be useful. Of course you could just simply read Spivak, but then the course would be a waste.

    It's probably best to borrow Spivak if you can find a library that has it. Also there are a lot of introductory analysis textbooks that are somewhere between the difficulty of reading of Spivak and Rudin. If you post or PM the contents of your calc course, I can probably make a better suggestion. Lastly, the wikipedia pages for undergraduate mathematics are usually exceptional. It's a good idea to heed to usual precautions regarding misinformation, but otherwise most of the pages have a great deal of mathematical sophistication. You can learn just by looking through the relevant wikipedia articles.
     
    Last edited: Jun 17, 2010
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