VIXtrick


Total Posts: 11 
Joined: Mar 2020 


Gentlemen, apologies if this is in the wrong forum: I'm relearning 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 wellrounded education in pure maths, so that I can later pick up the appliedside 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. 





jesus, you won't see any statistics for 4 years 



VIXtrick


Total Posts: 11 
Joined: Mar 2020 


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: 3664 
Joined: Jun 2004 


@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 


@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 allencompassing in any sense, but these are enough for my purposes. 




pj


Total Posts: 3664 
Joined: Jun 2004 


Thank you! I didn't know some of them, but those I know, I endorse them heartily.

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 


Yeah, I was a fool.
Most of this math has zero realworld applicability, and all the finance stuff is straight up schizophrenic, madeup 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, contextstripped "pure" "maths" texts.
After this one is finished, I'll look into real analysis, more advanced probability, measure theory, game theory, and some more CSfocused 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 


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 


@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: 3664 
Joined: Jun 2004 


@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: 1474 
Joined: Jun 2005 


Do Carmo: https://www.amazon.com/DifferentialGeometryCurvesSurfacesMathematics/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 nonobvious 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/StructureDynamicalSystemsSymplecticMathematics/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) 


