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Quant View on Pure Math

Daniel Duffy

C++ author, trainer
Um, I'm not blaming them for doing PDEs and not homeomorphisms. Someone has to do the gritty and unpleasant work of PDEs so that the more poetic and artistic types can prove theorems in etale cohomology. It's that old Morlock/Eloi divide.
It's the way ye tell it.
Have you studied the PDE works of Lagrange, Laplace, Euler, Hamilton, Navier-Stokes, Poincare?
PDE doesn't mean what you think it means.
 
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Daniel Duffy

C++ author, trainer
No way. Come on now, a homeomorphism is nothing but a bijective continuous map with continuous inverse. If someone says they do PDEs but don’t know what a homeomorphism is, then damn sure they have never heard of a Sobolev space, i.e. they don’t actually do PDEs.
In the case of bbw, I think he is way behind the curve. A little knowledge is a dangerous thing.

Around 1900 mathematics fractured into pure maths and physics and both were damaged. I have the books by Wallace and Fulton on Algebraic Topology .. lots of pictures and flow diagrams but little real life. Henri Poincare was the last great mathematical physicist.
One of my research degrees was "PDE and Finite Elements in Sobolev Space".
 
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Daniel Duffy

C++ author, trainer
Another remark: after 4 years of undergrad you should be ready to embark on baby research instead of yet 1 1/2 years of cramming other's work. It's like a straightjacket.
 
Another remark: after 4 years of undergrad you should be ready to embark on baby research instead of yet 1 1/2 years of cramming other's work. It's like a straightjacket.


The research frontier has moved on. After undergrad courses in linear algebra, differential equations, differential geometry, complex analysis, etc., no way are you prepared to start even baby research. Not in pure math at least.
 
I have the books by Wallace and Fulton on Algebraic Topology .. lots of pictures and flow diagrams but little real life.

Wallace is an old text and not very inspiring. Fulton is more interesting in that it has a foray into algebraic curves as well, including Riemann-Roch.
 
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The research frontier has moved on. After undergrad courses in linear algebra, differential equations, differential geometry, complex analysis, etc., no way are you prepared to start even baby research. Not in pure math at least.
I don't think this is correct. Math majors that mean business get exposure to a lot more math than I think you are giving them credit for. If you aren't reading at least 2-3 math books every winter and summer and doing end-of-chapter exercises, you aren't really a math major in my book. 4 years is plenty sufficient to get up to speed. Classes don't prepare you for research, they just help you build out your toolbox and become familiar with new tools in toy cases. What prepares you for research is intense self-study of well-known texts, reading of papers, doing challenging exercises, mingling with other students that are engaging in the same activities, and discussing with professors at your uni that specialize in the area(s) you are learning about.
 
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Daniel Duffy

C++ author, trainer
The research frontier has moved on. After undergrad courses in linear algebra, differential equations, differential geometry, complex analysis, etc., no way are you prepared to start even baby research. Not in pure math at least.
Your words, not mine.
Those topics I did by the end of my 2nd year and most was 1st year stuff for us. And then I did 2 more years of more advanced topics.

For the record in years 3 and 4 I did functional analysis, several complex variables, numerical analysis X 2, statistics, Topology of surfaces, Probability in Hilbert space, QM, General Relativity, Electromagnetics, fluid mechanics,. Finite elements. Galois, group representation, Lie groups, spectral theory.

.. and group theory at school when I was 16 (my maths teacher was a freshly-minted PhD graduate in number theory).
 
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I have them on my bookshelf. The topics are pretty dull. Where is this stuff used?

Nowhere. Pure math. Some theoretical physicists take a stab at using some of this material (cohomology, sheaf theory, fibre bundles). Over the last 40 years or so a major research direction has been using physics to understand mathematics (rather than the other way around). People like Atiyah, Singer, Drinfeld, Manin, Donaldson, and Witten.
 
Nowhere. Pure math. Some theoretical physicists take a stab at using some of this material (cohomology, sheaf theory, fibre bundles). Over the last 40 years or so a major research direction has been using physics to understand mathematics (rather than the other way around). People like Atiyah, Singer, Drinfeld, Manin, Donaldson, and Witten.

I'm guessing you would have written differential geometry off too before Einstein. Come on, if you don't know, why make it out to seem that you do. Algebraic topology has been gaining traction in neuroscience over the last several years.

Here's some more for you https://www2.math.upenn.edu/~ghrist/preprints/ATSN.pdf
 
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Daniel Duffy

C++ author, trainer
Nowhere. Pure math. Some theoretical physicists take a stab at using some of this material (cohomology, sheaf theory, fibre bundles). Over the last 40 years or so a major research direction has been using physics to understand mathematics (rather than the other way around). People like Atiyah, Singer, Drinfeld, Manin, Donaldson, and Witten.
My Topology prof was a PhD student of Hodge.
Now, homotopy is useful in solving nonlinear systems of equation F(X) = 0 by embedding it in a larger H(x,t) = 0 and managed to use it later in industry.
 

Daniel Duffy

C++ author, trainer
Nowhere. Pure math. Some theoretical physicists take a stab at using some of this material (cohomology, sheaf theory, fibre bundles). Over the last 40 years or so a major research direction has been using physics to understand mathematics (rather than the other way around). People like Atiyah, Singer, Drinfeld, Manin, Donaldson, and Witten.
Too late. It ended with Poincare.

It's like trying to teach judo to a boxer.
 

I'm guessing you would have written differential geometry off too before Einstein. Come on, if you don't know, why make it out to seem that you do. Algebraic topology has been gaining traction in neuroscience over the last several years.

Here's some more for you https://www2.math.upenn.edu/~ghrist/preprints/ATSN.pdf

The use of differential geometry was in the air after Riemann, Levi-Civita, and Ricci.

These other links you're posting can be dismissed. You're arguing for the sake of arguing and scoring some points.
 
The use of differential geometry was in the air after Riemann, Levi-Civita, and Ricci.

These other links you're posting can be dismissed. You're arguing for the sake of arguing and scoring some points.
You're not addressing the point I'm making. I'm pointing to the fact that you're naively writing off an area of mathematics as one with no applications outside of mathematics.

Arguing because I genuinely disagree with the content of some of your posts in this thread, could not care less about scoring points lol.
 
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You're not addressing the point I'm making. I'm pointing to the fact that you're naively writing off an area of mathematics as one with no applications outside of mathematics.

Arguing because I genuinely disagree with the content of some of your posts in this thread, could not care less about scoring points lol.

You don't know what you're talking about. I've seen Ghrist's work before. I don't know whether you got this in a hurried Google search to prove me wrong. This is not a serious application of cutting-edge algebraic topology. I'm wondering what kind of background you have in the subject.
 
You don't know what you're talking about. I've seen Ghrist's work before. I don't know whether you got this in a hurried Google search to prove me wrong. This is not a serious application of cutting-edge algebraic topology. I'm wondering what kind of background you have in the subject.
I found his Elementary Applied Topology book a few years back, certainly isn't something I found in a rush just to prove a point. Nowhere did I say it was an application of the cutting edge, nor do I think it reasonable to expect such. I was merely trying to say that there do exist applications of algebraic topology. I've been aware of the apps to neuroscience because I was interested in pursuing computational psychiatry before I began focusing on quantitative finance. While the applications may not call on the most advanced tools, I think it is still interesting nonetheless that there are people applying it in the first place. I have no formal exposure to algebraic topology, only undergrad abstract algebra and some grad + multilinear algebra at the level of Greub. I was trained in algebra by an algebraic K-theorist. I'm certainly a novice, but I'm not googling things on the spot in an attempt to prove you wrong.

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Daniel Duffy

C++ author, trainer
Why not familiarize yourself with, for example, index theory so you have a clearer idea of what you're talking about? Or the more recent work of Witten?
I was referring to persuading pure mathematicians to show more inferest in mathematical physics! Academics tend to live in silos in the sense that they focus on very narrow subject areas.

As I mentioned, this is not my area. And I don't see many real applications. And I have so many other things to do,

However, I have seen that Python has a library SHGO to do global optimisation using simplicial holmology



And that's an interesting development

And at one stage there was discussion on TDA for Machine Learning

 
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Daniel Duffy

C++ author, trainer
Algebraic geometry seems to have acquired the reputation of being esoteric, exclusive, and very abstract, with adherents who are secretly plotting to take over all the rest of mathematics. In one respect this last point is accurate.

David Mumford
 
What is the prejudice against pure math grads for quant jobs? surely common sense would say that they could learn what they need to on the job?
 
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