Good enough math background for a top GRE score?

 Posts: 2
 Joined: Mon Jan 16, 2012 12:49 pm
Good enough math background for a top GRE score?
Hello,
I'm hoping to get in the top 90%. I want to study PDE/Probability. My math background will be:
Calc 13
ODE
PDE
Linear Algebra
Abstract Algebra
Real Analysis
Complex Analysis
Numerical Analysis
Probability
Mathematical Statistics
Capstone: Stochastic Processes
I have no classes in discrete math, number theory, geometry, or topology. How much will this hurt me? Is it easy to pick up what I need to know for these subjects?
Thank you
I'm hoping to get in the top 90%. I want to study PDE/Probability. My math background will be:
Calc 13
ODE
PDE
Linear Algebra
Abstract Algebra
Real Analysis
Complex Analysis
Numerical Analysis
Probability
Mathematical Statistics
Capstone: Stochastic Processes
I have no classes in discrete math, number theory, geometry, or topology. How much will this hurt me? Is it easy to pick up what I need to know for these subjects?
Thank you
Re: Good enough math background for a top GRE score?
I think if you want to study probability and the theory of pde's, you will be taking measure theory/functional analysis. I found that taking a course in topology really helped in understanding the material, especially using Munkre's topology text. It just depends on what part of these subjects you want to study.
As far as the subject exam goes, I do recommend studying topology and abstract algebra (ring and field theory). Having the topology knowledge will also help in the real analysis questions and so forth.
As far as the subject exam goes, I do recommend studying topology and abstract algebra (ring and field theory). Having the topology knowledge will also help in the real analysis questions and so forth.
Re: Good enough math background for a top GRE score?
I wouldn't study topology for the GRE. It's just too much effort for what it's worth.
Judging from past released and my own exam, you are unlikely to encounter more than two topology questions. And it seems that you can miss around a dozen questions before you drop below the 90th percentile. If you invest the time it would take you to master pointset topology into practicing things you already know, you'll probably salvage many more points than the two points you might miss on topology.
I would however very highly recommend that you learn discrete math  if you don't already know it. I assume that you already know combinatorics from your probability classes. The other big part of discrete math is elementary number theory and modular arithmetic. You might already know that too (e.g. from abstract algebra or high school math competitions) and if not, it's very easy to learn.
Judging from past released and my own exam, you are unlikely to encounter more than two topology questions. And it seems that you can miss around a dozen questions before you drop below the 90th percentile. If you invest the time it would take you to master pointset topology into practicing things you already know, you'll probably salvage many more points than the two points you might miss on topology.
I would however very highly recommend that you learn discrete math  if you don't already know it. I assume that you already know combinatorics from your probability classes. The other big part of discrete math is elementary number theory and modular arithmetic. You might already know that too (e.g. from abstract algebra or high school math competitions) and if not, it's very easy to learn.
Re: Good enough math background for a top GRE score?
I do however agree with ackirby that a topology background will help you with your analysis classes in grad school.
Re: Good enough math background for a top GRE score?
While I think the actual formalism of topology is quite minimally useful, I found the concepts of topology really helped in understanding the basic real analysis concepts better. Where I would draw the line is that you should be able to use the specific cases of very basic topological concepts in the setting of the real line. A formal real analysis course is quite (understandably) pedantic about constructing the number system, whereas for a fast exam like the GRE, it is more useful to get an intuitive flavor.
Aside from this, more advanced topology is indeed useful in real analysis classes.
In summary: I think the entire point of most undergraduate real analysis classes is to be very careful with setting up basic theory, but you should go into the GRE ready to tackle questions quickly, and intuition is much more crucial when you don't have to write a full proof of everything in sight.
Aside from this, more advanced topology is indeed useful in real analysis classes.
In summary: I think the entire point of most undergraduate real analysis classes is to be very careful with setting up basic theory, but you should go into the GRE ready to tackle questions quickly, and intuition is much more crucial when you don't have to write a full proof of everything in sight.

 Posts: 2
 Joined: Mon Jan 16, 2012 12:49 pm
Re: Good enough math background for a top GRE score?
Thanks for the replies.
How important would a course in number theory or geometry be for the exam?
How important would a course in number theory or geometry be for the exam?

 Posts: 44
 Joined: Tue Aug 09, 2011 6:18 pm
Re: Good enough math background for a top GRE score?
I do not believe that knowing the modern formulations of algebraic/differential geometry will directly help you on the exam. Being able to solve Putnamlevel questions on Euclidean geometry will definitely help. (I only have experience with differential geometry, not algebraic, but still.)
On the October 2011 exam there were a couple questions on Euclidean geometry. Of course they weren't PutnamHard, but they weren't trivial either. (I got 1 out of two.) To prepare for this, knowing the geometry part in "Cracking ..." will help.
I would say the best you'd get out of knowing differential geometry would be in improving your ability to do the calculus problems with a geometric flavor (e.g. find the length of this wonky parameterized curve). Although having a solid ability to solve calculus problems will get you the same thing.
I never took a course on Number Theory. I think that if you have a solid background on Abstract Algebra, then the work you do with modular arithmetic should suffice. Make sure you do practice problems so you know what is coming before Test Day.
Knowing an epsilon's worth of pointset topology will be useful. I think on one of the practice exams they combined a pointset topology question with a combinatoricslike question. (Something like "How many different topologies can you put on the set of 3 points such that blah blah blah blah blah.")
They might also try to trick you by asking topological questions and then including words like "Bounded" or "Complete" in the possible answers. Don't be fooled!
There once was an aspiring grad student who was taking his qualifying exam. Unfortunately he was not doing so well. The examiners decided to throw him an easy question. The analyst on the panel asked the poor soul "If you will, please name a compact set". The student pondered the question and after quiet contemplation replied "The real numbers. Definitely the real numbers." The analyst reached for the giant "FAIL" stamp to stain the student's evaluation with red ink. However, the topologist in the group stayed her colleague's hand and asked the student "under which topology?"
On the October 2011 exam there were a couple questions on Euclidean geometry. Of course they weren't PutnamHard, but they weren't trivial either. (I got 1 out of two.) To prepare for this, knowing the geometry part in "Cracking ..." will help.
I would say the best you'd get out of knowing differential geometry would be in improving your ability to do the calculus problems with a geometric flavor (e.g. find the length of this wonky parameterized curve). Although having a solid ability to solve calculus problems will get you the same thing.
I never took a course on Number Theory. I think that if you have a solid background on Abstract Algebra, then the work you do with modular arithmetic should suffice. Make sure you do practice problems so you know what is coming before Test Day.
Knowing an epsilon's worth of pointset topology will be useful. I think on one of the practice exams they combined a pointset topology question with a combinatoricslike question. (Something like "How many different topologies can you put on the set of 3 points such that blah blah blah blah blah.")
They might also try to trick you by asking topological questions and then including words like "Bounded" or "Complete" in the possible answers. Don't be fooled!
There once was an aspiring grad student who was taking his qualifying exam. Unfortunately he was not doing so well. The examiners decided to throw him an easy question. The analyst on the panel asked the poor soul "If you will, please name a compact set". The student pondered the question and after quiet contemplation replied "The real numbers. Definitely the real numbers." The analyst reached for the giant "FAIL" stamp to stain the student's evaluation with red ink. However, the topologist in the group stayed her colleague's hand and asked the student "under which topology?"
Re: Good enough math background for a top GRE score?
A course in number theory or geometry will not be useful, except in making you use ideas you have encountered in abstract algebra or multivariable calculus. I'd take owlpride's advice regarding discrete math and elementary number theory ideas involving solutions to very, very basic Diophantine equations. What you find in a basic prep book such as what the Princeton Review book offers is more than enough.
I agree with the above advice that a solid grounding in calculus will get you through any geometry you encounter on the exam. Well, except the occasional obscure question that may or may not show up (the examiners reserve the right to include a few questions on almost any topic; however, you aren't meant to be able to study for this part, more than just having a good background in math in general). Your ability to score exceptionally high is not dependent on any specialized material at all.
I agree with the above advice that a solid grounding in calculus will get you through any geometry you encounter on the exam. Well, except the occasional obscure question that may or may not show up (the examiners reserve the right to include a few questions on almost any topic; however, you aren't meant to be able to study for this part, more than just having a good background in math in general). Your ability to score exceptionally high is not dependent on any specialized material at all.
Re: Good enough math background for a top GRE score?
Hey, I don't mean to hijack your thread, but what about a course in numerical methods, how useful would it be for the subject test?

 Posts: 35
 Joined: Sat Oct 15, 2011 2:08 pm
Re: Good enough math background for a top GRE score?
for the subject test, not useful at all. I could see maybe an approximation question (like Euler's method) popping up, but you've probably seen that in any number of math courses: calc, odes... I doubt you'll see anything beyond that (approximation methods, not those courses).Ell wrote:Hey, I don't mean to hijack your thread, but what about a course in numerical methods, how useful would it be for the subject test?
Most numerical methods are iterative algorithms involving messy arithmetic, not the easiest thing to fit into an already fiercly timed test with no use of calculators.
However, this is just my opinion, and other's experiences may differ.
Re: Good enough math background for a top GRE score?
Besides Euler, this is the ONLY numerical method you might need:
http://en.wikipedia.org/wiki/Newton_Raphson
http://en.wikipedia.org/wiki/Newton_Raphson
Re: Good enough math background for a top GRE score?
Thanks guys, I appreciate your help.
Re: Good enough math background for a top GRE score?
definitely saw an incomplete divided difference table on one of the released math GREs....