Theoretical math vs. applied math

[onetime]

Hello, I was just looking for thoughts on how deeply into theoretical math one should go in order to prepare for the MFE. For instance, are real analysis, measure theory and the like required knowledge?

Is it alright to stick to courses that strictly teach you the methodology of solving ODEs and PDEs or is it necessary to know existence and uniqueness and other theory?

 

[Daniel Duffy]

"theoretical maths" does not compute; it's called "pure maths"

Numerical analysis and methods are vital.

 

[snipez]

Eh, doesn't really matter what you call it.

Also, having a decent grasp on real analysis can help a lot. Obviously it's crucial if you want a decent grasp on stochastic calculus. Since you mentioned measure theory, I'll say that measure theory is useful, but not as crucial as basic analysis concepts (which you'll need to understand measure theory anyways). You can do a lot of stochastic calculus with a good grasp on the basic intuitive notion that a sigma algebra (the basic foundation of measure theory) represents information, but you might find yourself struggling a lot if you've never seen Riemann integration theory before. Indeed, the dozens of important theorems regarding the Ito integral (any of which may be touched upon according to quant interview books, e.g. Joshi) are rather simple if you've done basic real analysis before. Well at least that's my impression.

Regarding ODEs and PDEs, knowing how to solve them is probably a priority. Existence and uniqueness are exercises in basic analysis that are interesting, but not necessarily useful in an MFE program. I would understand the statements, but don't worry too much about the proofs. Numerical analysis is certainly vital, but in this realm, existence and uniqueness of differential equations is often taken for granted.

 

[Daniel Duffy]

Eh, doesn't really matter what you call it.
I agree. The only people who might raise an eyebrow are mathematicians :D

Regarding ODEs and PDEs, knowing how to solve them is probably a priority.

Replace 'probably' by 'absolutely'.

 

[onetime]

I suppose what for the first part of my question it boils down to this. I am currently taking a course with the following description,

Higher dimensional calculus, chain rule, gradient, line and multiple integrals with applications. Use of implicit and inverse function theorems. Real number axioms, limits, continuous functions, differentiability, infinite series, uniform convergence, the Riemann integral.

It is an 8 month course titled Calculus and Introductory Analysis II that uses Advanced Calculus by Taylor.

After completion of this course I will have the option of taking a course with the following description,

Metric spaces and their topologies, continuous maps, completeness, compactness, connectedness, introduction to Banach spaces

It is a 4 month course titled Real Analysis that uses I believe Carothers book.

Would it be recommended that I take the second course or would time be better served taking other more applied classes?

 

[bigbadwolf]

It is an 8 month course titled Calculus and Introductory Analysis II that uses Advanced Calculus by Taylor.

After completion of this course I will have the option of taking a course with the following description,

Metric spaces and their topologies, continuous maps, completeness, compactness, connectedness, introduction to Banach spaces

It is a 4 month course titled Real Analysis that uses I believe Carothers book.

I have both books (somewhere in my basement ...). The first one is okay and you need it. The second one (published by Cambridge, I think) is something you do not need and you can safely avoid it.

 

[onetime]

I have both books (somewhere in my basement ...). The first one is okay and you need it. The second one (published by Cambridge, I think) is something you do not need and you can safely avoid it.

are you speaking just interms of the books or the classes as well?

 

[bigbadwolf]

are you speaking just interms of the books or the classes as well?

Only in terms of the books (obviously). You don't need general topology (open sets, closed sets and definitions of continuity and compactness based on such sets; a fortiori you don't need Banach spaces either). The book should be called "Real Analysis II." It won't hurt you but you'd probably be better off with something more related to quant finance.

 

[onetime]

Only in terms of the books (obviously). You don't need general topology (open sets, closed sets and definitions of continuity and compactness based on such sets; a fortiori you don't need Banach spaces either). The book should be called "Real Analysis II." It won't hurt you but you'd probably be better off with something more related to quant finance.

This is what I was getting at. From what I have gathered the course I am taking now would often be classified as a first course in Real Analysis at other schools where as the class that is called Real Analysis at my school would be called intro to Topology at most other schools.

 

[Daniel Duffy]

Do people still study the books of Walter Rudin? They were compulsory for 1st year maths majors.

 

[bigbadwolf]

Do people still study the books of Walter Rudin? They were compulsory for 1st year maths majors.

Some places still use Principles of Mathematical Analysis. There are so many other introductory analysis books around today.

 

[Barny]

Just an aside, I've studied real analysis up to open/closed sets, compactness etc. How much more analysis would I need to do to understand the measure theory behind modern probability theory?

 

[bigbadwolf]

Just an aside, I've studied real analysis up to open/closed sets, compactness etc. How much more analysis would I need to do to understand the measure theory behind modern probability theory?

Pick up a copy of Capinski and Kopp's Measure, Integral and Probability. You should be fine. If you want to move at a faster clip, try the first few chapters of Rudin's Real and Complex Analysis.