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Academic Guidance Academic Guidance - Academic guidance for those pursuing a college degree... what college? Grad school? PhD help?


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April 29th, 2019, 10:48 AM   #1
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lack of direction

Hello,

I have completed a BSC, majoring in math. I also studied classical mechanics/ electromagnetism/ and a third year introduction to quantum mechanics. I enjoyed classical mechanics/ electromagnetism, as it tended to appeal to my visual style of learning, and I enjoy using math to model physical systems.

In math I have taken two courses in analysis, one course in general topology, one course in algebra, and a number of applied courses in numerical methods/differential equations/mathematical statistics and probability/calculus of variations. I enjoyed analysis quit a bit. Topology was..challenging. I am not sure that I am motivated to read James Munkres book past the first four chapters.

I have been admitted to a master program starting next year, My supervisor specializes in differential equations.

At this point I am simply unsure what to focus on. I would like to learn about some applied topics, such as game theory, particularly evolutionary game theory. I would also like to study topics in optimization.
At the same time, as I enjoyed analysis, I thought I'd pick up a book on Functional analysis.

Although I have constructed a number of computer programs designed to solve PDE's/variational/ optimization problems, I am using the "as long is it works method," I have no formal understanding of any programming language, furthermore I cannot seem so decide which programming language would be best to learn first.

To summarize, I don't know where to focus my time right now, and would appreciate some advice if anyone has some to give, thanks.
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April 30th, 2019, 01:18 AM   #2
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Since you have passed well enough to be admitted onto a Masters (congratulations) you should be able to make some sense of these thoughts.

Functional analysis does indeed point towards applied techniques and a good book bridging that gap is

Applied Functional Analysis by Griffel.

Griffel has a readable introduction to distribution theory to help. This enables understanding of generalised operational mathematics and (integral) transform theory. Also look at the terms 'operational methods/calculus'.

However applied maths very very often replaces the difficult function/expression with a simple one, repeated many times. You will have seen this in your numerical methods.
This is nowhere more so than in finite element/boundary element methods. You should certainly study these.
There is an important link directly from the fundamental theorem of calculus to modern finite element and boundary methods via Gauss' and Green's theorems.

A modern area you haven't mentioned is called concrete mathematics or discrete mathematics - a motley collection of the mathematics computers and computing. Very useful and simple to dip into. You will find many useful algorithms here.
This is better than learning any specific computing language. These go in and out of fashion and you will also be constrained by the programs your future employer/ institution possesses.

All the best with your future career.
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