Relevant Mathematics Courses?

[George Lambeth]

Hi, I'll be a sophomore next year and I'm starting to plan my courses and i'm interested what courses are relevant to financial engineering? I plan on perusing either a MFE or PhD in economics after graduation. My freshman year I took Linear Algebra and Analysis 1, and a multivariable calculus based course in probability theory.

As of now I plan on taking Analysis 2, A year long graduate course on statistics/probability, a graduate course in stochastic calculus , and discrete mathematics in the fall of 2013.

But I was wondering what the main courses I should be taking are? There's a plethora of mathematics courses at my uni so it's hard to decide: Abstract Algebra, ODE/PDE, Applied Lin. Algebra, Complex/Numerical Analysis, Metric/topological spaces, and then countless graduate courses.

So what would admissions officers be looking for? I'll definitely have the standard Calc sequence/lin alg/analysis sequence but past that what is important for top MFE/PhDs?

Oh and another question I have is how does my degree program sound? I plan on doing a BS in Economics/Finance and then a MS in statistics in my 4 years (Came in with many AP/Community College credits).

Sincerely,

George.

 

[bigbadwolf]

But I was wondering what the main courses I should be taking are? There's a plethora of mathematics courses at my uni so it's hard to decide: Abstract Algebra, ODE/PDE, Applied Lin. Algebra, Complex/Numerical Analysis, Metric/topological spaces, and then countless graduate courses.

Abstract algebra and complex analysis are probably redundant but still useful for developing that elusive "mathematical maturity." The problem with the applied courses is that though they may be more "relevant," you're always at risk of having no real theoretical understanding and merely having mastered a cookbook of techniques with no conceptual focus. So ideally you want your numerical analysis and applied linear algebra to have a theoretical backbone, something that gives intellectual coherence to the subject matter.

 

[Barny]

Completely agree with bbw. Learn something that will force you to think in a different way about maths. Abstract algebra and analysis will do that. Complex analysis did that for me, too, so I'd recommend that too. If you want to be a quant you can certainly do too much pure maths, though. Once you've learnt how to think like a pure mathematician there is little utility in taking lots of pure maths options. Advanced courses in pure maths tend to be rote learning and regurgitation of proofs rather than concepts, so avoid that.

Admissions officers will just want to see you've taken the right applied courses though, so make sure you do ode/pde etc.

 

[bigbadwolf]

Complex analysis did that for me, too, so I'd recommend that too. If you want to be a quant you can certainly do too much pure maths, though.

Complex analysis is one of the most beautiful subjects around and even now, jaded and blase as I am in the autumn of my years, the theorems elicit gasps of wonder on my part. But when people ask in mercenary fashion what will please philistine admissions officers, what can I say?

 

[George Lambeth]

Awesome I really appreciate the advice. I just have 1 more question relevant to course scheduling. How important is accounting? I feel like I wouldn't be using it frequently, but I haven't had an internship yet. Should I try to take a year or two of it? I'm mostly interested in Sales & Trading, with a particular interest in creating new securities.

 

[Daniel Duffy]

Abstract algebra and complex analysis are probably redundant but still useful for developing that elusive "mathematical maturity." The problem with the applied courses is that though they may be more "relevant," you're always at risk of having no real theoretical understanding and merely having mastered a cookbook of techniques with no conceptual focus. So ideally you want your numerical analysis and applied linear algebra to have a theoretical backbone, something that gives intellectual coherence to the subject matter.

A nice route is to do linear operators on Hilbert spaces ---> finite dimensional vector spaces --> matrices and vectors --> Numerical Linear Algebra --> etc.

 

[Barny]

Awesome I really appreciate the advice. I just have 1 more question relevant to course scheduling. How important is accounting? I feel like I wouldn't be using it frequently, but I haven't had an internship yet. Should I try to take a year or two of it? I'm mostly interested in Sales & Trading, with a particular interest in creating new securities.

There is a problem here. The set of ideal knowledge is so vast that no one person can cover it all in sufficient detail. What do you want to do? What is your skill set? In life you work as part of a team to achieve goals. What part of the team will you play?

Consider that people do entire degrees in accounting. Accountants spend 3 years of 10+ hour days and 14 examinations before they can call themselves an accountant. Do you think you will get to their level of knowledge by taking one course? You need to focus at this stage and not spread yourself too thinly. When I did my degree, I was only allowed to take maths and physics courses and I still wished that I had done a European style 5 year degree so that I could have learnt other areas of physics and maths that I didn't get the chance to.

That said, a basic knowledge of accounting is extraordinarily valuable in life, not just in finance. But I learnt my accounting on the job, rather than through a course, so I can't tell you whether courses are useful.

 

[Ken Abbott]

I have found basic accounting to be extremely useful. I encourage all financial math students to take it.

 

[Jae Sang Shim]

Abstract algebra and complex analysis are probably redundant but still useful for developing that elusive "mathematical maturity." The problem with the applied courses is that though they may be more "relevant," you're always at risk of having no real theoretical understanding and merely having mastered a cookbook of techniques with no conceptual focus. So ideally you want your numerical analysis and applied linear algebra to have a theoretical backbone, something that gives intellectual coherence to the subject matter.

Abstract algebra and complex analysis are probably redundant but still useful for developing that elusive "mathematical maturity."

I got quite surprised by reading this line because this comment was exactly the same as my prof. in abstract algebra said...

 

[Daniel Duffy]

'Abstract Algebra' is essentially linear operators in finite-dimensional space. You are locked into matrices! A better and official name is "Finite-dimensional Vector Spaces".

A more useful study is Functional Analysis and is the basis for much of modern pure, applied and numerical maths. IMO it is the single most important course in undergrad maths. For sure.

http://en.wikipedia.org/wiki/Functional_analysis

http://en.wikipedia.org/wiki/List_of_functional_analysis_topics

 

[Jae Sang Shim]

'Abstract Algebra' is essentially linear operators in finite-dimensional space. You are locked into matrices! A better and official name is "Finite-dimensional Vector Spaces".

A more useful study is Functional Analysis and is the basis for much of modern pure, applied and numerical maths. IMO it is the single most important course in undergrad maths. For sure.

http://en.wikipedia.org/wiki/Functional_analysis

http://en.wikipedia.org/wiki/List_of_functional_analysis_topics

A lot of times I saw the name Hilbert or Banach space in financial mathematics class...

but I haven't studied them enough I think ... if I do get into one of MFE programs

I think I should study them beforehand

 

[bigbadwolf]

'Abstract Algebra' is essentially linear operators in finite-dimensional space.

You mean linear algebra, not abstract algebra. Though the methods and ways of thinking of linear algebra percolate throughout abstract algebra (e.g., homological algebra, group characters and representation theory generally, module theory, etc.)

 

[CasanovaJ]

A lot of times I saw the name Hilbert or Banach space in financial mathematics class...

but I haven't studied them enough I think ... if I do get into one of MFE programs

I think I should study them beforehand

How real analysis'y courses get can often just depend simply on the professor... If you were to take time series statistics at Columbia with Richard Davis you'd use Hilbert spaces because he thinks it "makes solving prediction equations easier," but in the other section of the exact same course at the exact same university, you wouldn't.

In general I do wish I'd taken some kind of real analysis course before doing MFE, though-- probably not a bad idea if you have the chance.

 

[bigbadwolf]

In general I do wish I'd taken some kind of real analysis course before doing MFE, though-- probably not a bad idea if you have the chance.

To be done properly, it's covered in measured steps. The first step is a basic course in real analysis -- and there are enough good elementary texts around that one can teach oneself (e.g., Ross's Elementary Analysis or Howie's Real Analysis). These books just work in R (real numbers), using the key result that every Cauchy sequence converges in R. The next step is the multivariable case, and things become more complex. Notions of open and closed sets from metric spaces -- the first step into general topology -- make their presence felt. Good texts for this might be Marsden's Elementary Classical Analysis, or Apostol's Mathematical Analysis. The third step is the infinite-dimensional setting of function spaces -- but it's important this be introduced via examples rather than axiomatically. For example, the method of successive approximations for solving differential equations involves the convergence of a sequence of functions to the solution in a function space (see Coddington's book on differential equations for more on this). These function spaces are a blend of vector spaces and topological spaces. A math graduate from a good uni will have covered these steps sequentially. The rest will have gaps in their background and try to remedy this by doing too much too fast, and without the appropriate clarifying examples (there's a plethora of functional analysis books without motivating examples).

 

[Daniel Duffy]

Linear algebra and differential equations are good subjects which are very related to mathematics.

I agree. They are like methods.

Analysis is more fundamental.