Math class choice

[SlantedAndDisenchanted]

Hi, I have recently hit a bit of a quandary. So, I'm sure some of you may be able to relate to focusing on a perhaps trivial choice and wondering if it could impact one's chances of getting into grad school. I am looking to apply to Biostats PhD programs in a couple of years.

 

For next year I have the option of taking Advanced Calculus I and II or Real Analysis I or II. Advanced Calculus (essentially undergrad real analysis?) covers similar topics like differentiation, limits, and integration. Analysis covers the whole spiel in R^n so it definitely would be better to take that option. Real Analysis at my school covers baby Rudin where Advanced Calculus uses a textbook at the level of Fitzpatrick's Advanced Calculus. My question is whether it will matter at all what sequence I take? I may not be able to get admission into Real Analysis due to my self-taught (aside from Calc 1-4) background. My supposition is that getting an A in an advanced calculus sequence would be better than getting a B+/A- in real analysis. But, I'm not sure. I am taking these to take up for my poor calculus grades in undergrad (currently a Master's student in Engineering). Any advice would very much be appreciated.

 

Thanks,

Disenchanted

[ParanoidAndroid]

Have you taken a Real Analysis course before? Otherwise, going straight into baby Rudin might be overwhelming. I imagine Real Analysis will be better for PhD applications. If you decide to go that route, make sure to supplement Rudin's book with Apostol's Mathematical Analysis. 

[SlantedAndDisenchanted]

Yeah. I realize it will be a big jump, so I'm self-learning over the summer. I am 75% of the way through "How to Prove It" by Velleman, and then I am going to start "Understanding Analysis" by Abbott (1D Real Analysis book). After that I think I may be ready. Is Apostol on the same level as Rudin? Or is it easier ?

 

Thanks for replying!

[ParanoidAndroid]

 Is Apostol on the same level as Rudin? Or is it easier ?

 

 

They're about the same level. Apostol goes through more topics. 

 

Baby Rudin is very terse for an Analysis text. It becomes especially challenging at the topology section (Chapter 2).  Apostol is more complete, but his exercises aren't as "soul-searching" as Rudin's. My suggestion is to start with Rudin and use Apostol to fill in the gaps. 

[ParanoidAndroid]

And FYI, Apostol also has books called Calculus, Vol. I and II. I think of his Mathematical Analysis book as a sequel to these texts. 

[wine in coffee cups]

I would take the advanced calculus sequence rather than the real analysis sequence.

 

You want to get as good a grade as possible in whichever sequence you take, especially if you're making up for poor grades in lower-level classes. If you've never done proof-based math before and have only self-prepared, you will probably have a tough time keeping up with the rest of the students in a graduate-level analysis class because a lot of mathematical sophistication will already be assumed. They'll be seeing harder versions of things they already know, too, while you'll be learning all the material (as well as just how to write proofs) for the first time.

 

Every biostat PhD program that requires analysis prior to enrolling will be happy with material at the level of advanced calculus. Graduate-level analysis is overkill. You just want to take enough analysis to be able to handle future math requirements in whatever program you end up in (e.g. a measure theory class).

[bsharpe269]

I agree that the grad class is overkill and that getting an A in the undergrad course would be better.

 

Have you taken linear algebra? I think that would also be really important for biostats.

[SlantedAndDisenchanted]

I have just taken Linear Algebra as part of a combined Lin. Alg/ Diff Eq. 

 

I could take a junior/senior level Applied Linear Algebra class instead of the second semester of Advanced Calculus/ Real Analysis. But, I figured analysis would be better. All of my engineering classes/job incorporate a ton of linear algebra, so I already know all about the stuff.

[SlantedAndDisenchanted]

Plus, the Real Analysis class is not really graduate level ( It uses baby Rudin), it is more of a mixed undergrad/grad class. So, maybe for Stats people it would be considered graduate analysis...but math majors at my school take it their senior year if they want to do a PhD, I'm told. Maybe that explains the level of the class with more clarity.

[wine in coffee cups]

You apparently go to UMN. UMN's biostat PhD program is one of the better ones, so let's take a look at exactly which math courses their program requires its students to take since they know the material and level of rigor. Their curriculum page says that incoming students without a master's degree in statistics or biostatistics need to take Math 5615H or Math 4603 their first semester.

• Math 5615H: Honors: Introduction to Analysis I 4.0 cr; Prereq-[[2243 or 2373], [2263 or 2374], [2283 or 3283]] or 2574; fall, every year. Axiomatic treatment of real/complex number systems. Introduction to metric spaces: convergence, connectedness, compactness. Convergence of sequences/series of real/complex numbers, Cauchy criterion, root/ratio tests. Continuity in metric spaces. Rigorous treatment of differentiation of single-variable functions, Taylor's Theorem.
• Math 4603: Advanced Calculus I 4.0 cr; Prereq-[2243 or 2373], [2263 or 2374] or 2574 or # ; fall, spring, summer every year. Axioms for the real numbers. Techniques of proof for limits, continuity, uniform convergence. Rigorous treatment of differential/integral calculus for single-variable functions

From this it is clear that the math UMN biostat needs its students to acquire during the PhD program is advanced calculus, not real analysis. In terms of applying, coming in with advanced calculus already completed ought to be good enough.

[SlantedAndDisenchanted]

Thanks for your help and responses. I guess my *somewhat flawed* line of thinking was that if I aced a harder math sequence it may make up from my B- in multivariable back when I used to smoked weed instead of going to class. But, I see now that the easier sequence makes the most sense!

[bsharpe269]

Thanks for your help and responses. I guess my *somewhat flawed* line of thinking was that if I aced a harder math sequence it may make up from my B- in multivariable back when I used to smoked weed instead of going to class. But, I see now that the easier sequence makes the most sense!

 

I think that an A in the easier sequence will make up the B- anyway!

[reinhard]

Did your last calculus class mentioned something like "sup/inf", some basic set theory (like proving A= ? If so Baby Rudin should be okay, otherwise no...

[cyberwulf]

Thanks for your help and responses. I guess my *somewhat flawed* line of thinking was that if I aced a harder math sequence it may make up from my B- in multivariable back when I used to smoked weed instead of going to class. But, I see now that the easier sequence makes the most sense!

 

Well, there's no doubt that getting an "A" in a harder course will do more for you; but I think the concern is that's much less likely in a class you are under-prepared for.

[reinhard]

Look through the first few chapters of Spivak, try to do some of the problems. If you can't, don't even touch Rudin.