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VIXtrick


Total Posts: 11
Joined: Mar 2020
 
Posted: 2022-02-05 21:51
Gentlemen, apologies if this is in the wrong forum: I'm re-learning maths for personal and employment reasons.

The below is the 1 to 2 years curriculum I've made for myself. Would you comment on its sanity, and recommend any changes?

My intention is to achieve a well-rounded education in pure maths, so that I can later pick up the applied-side of things (e.g. signal processing, risk modeling, pricing, etc.) with relative ease -- and to build a "mile wide, meter deep" foundation for whatever luck decides will be built atop it.

In order:

1. Proofs

2. Matrix Algebra

3. Calculus

4. Ordinary Differential Equations

5. Vector Spaces

6. Partial Differential Equations

7. Abstract Algebra

8. Topology

9. Analysis

The books I've chosen for these have already been vetted by a snob with a sense of taste.

There are a handful of more areas that I'll be exploring, but those are a secondary goal.

princeofbelair


Total Posts: 7
Joined: Feb 2020
 
Posted: 2022-02-06 05:34
jesus, you won't see any statistics for 4 years

VIXtrick


Total Posts: 11
Joined: Mar 2020
 
Posted: 2022-02-06 17:49
Hah! Good point.

Might as well learn something useful in parallel, when the fundamental stuff gets too much.

Probability and Statistics (DeGroot) is now the first one on that list.

pj


Total Posts: 3645
Joined: Jun 2004
 
Posted: 2022-02-06 19:35
@VIXtrick

Would you care to share the titles of the chosen books?

The older I grow, the more I distrust the familiar doctrine that age brings wisdom Henry L. Mencken

VIXtrick


Total Posts: 11
Joined: Mar 2020
 
Posted: 2022-02-06 21:33
@pj

Of course:

1. Proofs
- A Transition to Advanced Mathematics (Smith, Eggen, and Andre)

2. Matrix Algebra
- Linear Algebra Done Wrong (Treil)

3. Calculus
- Calculus, vols. I & II (Apostol)

4. Ordinary Differential Equations
- Ordinary Differential Equations (Pollard & Tenenbaum)

5. Vector Spaces
- Linear Algebra (Hoffman & Kunze)

6. Partial Differential Equations
- Partial Differential Equations: An Introduction (Strauss)

7. Abstract Algebra
- Topics in Algebra (Herstein)

8. Topology
- Topology (Monkres)

9. Analysis
- Functions of One Complex Variable (Conway)

Not all-encompassing in any sense, but these are enough for my purposes.

pj


Total Posts: 3645
Joined: Jun 2004
 
Posted: 2022-02-07 10:23
Thank you!
I didn't know some of them, but
those I know, I endorse them heartily.
Cool

The older I grow, the more I distrust the familiar doctrine that age brings wisdom Henry L. Mencken

VIXtrick


Total Posts: 11
Joined: Mar 2020
 
Posted: 2022-04-07 12:57
Yeah, I was a fool.

Most of this math has zero real-world applicability, and all the finance stuff is straight up schizophrenic, made-up gibberish.

I'm reading a gambling theory and applications book (Theory of Gambling and Statistical Logic, Epstein), for a crash course in applied start&prob, without having all the math dumbed down (or pumped up). It's leagues more interesting than dry, context-stripped "pure" "maths" texts.

After this one is finished, I'll look into real analysis, more advanced probability, measure theory, game theory, and some more CS-focused maths (combinatorics, number theory, etc.) -- maybe even actuarial science.

@princeofbelair was right. The shit I picked out was absolutely pointless.

frolloos


Total Posts: 137
Joined: Dec 2007
 
Posted: 2022-04-07 21:01
On the pure maths side, and specifically regarding analysis, I'd start with Ross "Elementary Analysis", then Axler "Measure, Integration & Real Analysis", followed by Oksendal "Stochastic Differential Equations" and if you're still alive by then, end with the bible Karatzas and Shreve "Brownian motion and stochastic calculus".

I have only read one of the books listed above, and most of the time I still have no clue what I am doing when integrating or differentiating.

I have some recommendations on the differential geometry side as well, but not sure you'd be interested or even need diff.geom.


No vanna, no cry

VIXtrick


Total Posts: 11
Joined: Mar 2020
 
Posted: 2022-04-08 01:13
@frolloos

I'll probably end up only skimming through all of those books; anything that has stripped away the context for the mathematics makes me restless and agitated (though I'm sure I can get the gist of these fields through them).

Maybe I'm just a naive, but seeing Brownian motion and calculus used in a financial context (dealing with people, not particles) strikes me as stupid. Maybe I'll be learned and humbled by the latter two tombs.

Probably won't need any differential geometry.

pj


Total Posts: 3645
Joined: Jun 2004
 
Posted: 2022-04-08 07:47
@froollos
> I have some recommendations on the differential geometry side
Could you post anyways?

On Oksendal, I wasn't not new to the field when I read it,
but I found it quite clear and beginner friendly for
Ito calculus and stuff.




The older I grow, the more I distrust the familiar doctrine that age brings wisdom Henry L. Mencken

nikol


Total Posts: 1451
Joined: Jun 2005
 
Posted: 2022-04-08 14:12
Do Carmo:
https://www.amazon.com/Differential-Geometry-Curves-Surfaces-Mathematics/dp/0486806995

This man (M.Penn) makes things very easy (as intro is incomparable, I watched him at breakfast):
https://www.youtube.com/watch?v=PaWj0WxUxGg&list=PL22w63XsKjqzQZtDZO_9s2HEMRJnaOTX7

Books from V.Arnold are great. Sometimes he makes non-obvious shortcuts, but that's the sin of many Russian scientists (e.g. Landau).

Currently I am struggling with this one from Souriau. French diff.geometrists adore him for giving geometric interpretation of entropy. I am not sure yet.
https://www.amazon.com/Structure-Dynamical-Systems-Symplectic-Mathematics/dp/0817636951

And this one is an encyclopedia. Hyperlinks allow quick check of concepts. Very high level. Sometimes, I do not get their explanations, and have to consult other sources. Not very homogeneous level of description because it is written by many people.
https://ncatlab.org/nlab/show/HomePage

I confess, sometimes I quickly go through Wiki to check ideas. Quite neat. But one has to cross check things if it makes sense, of course.

... What is a man
If his chief good and market of his time
Be but to sleep and feed? (c)
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