Math Most People Never See

This video will show you math subjects that most people never see. Many of these subjects are graduate level but some are also undergraduate level. What other areas of math do you think most people never see? Leave a comment below:)

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[stephenaustin]

Group theory is pretty central to theoretical physics.

[jules]

I love to collect ebook/pdf versions of the books you share! I passed my differential equations for engineers last semester. Your videos motivated me and helped a lot. Thank You!

[newwaveinfantry]

Here are some advanced and obscure topics in mathematics that I've had books on recommended, know researchers of, have researched personally, or have seen videos and papers on, that I nevertheless found interesting:
1) Model theory and proof theory. Some of my favourites topics in all of math and logic.
2) Spectral theory, which is the name of actually three theories. One on spectral sequences in alg. topology, one on spectra of operators in functional analysis and one on prime spectra of rings and schemes in alg. geometry
3) Stochastic polynomials in alg. geometry for computer graphics. Basically, random perturbations in varieties used to code in computer graphics and animations
4) Morse theory and its applications to numerical methods and data analysis
5) Cohomological methods in homotopy theory (book by Mosher & Tangora)
6) Infinity categories
7) Topos theory
8) Categorification of fourier operators
9) Orbifolds, vertex operator algebras & representation theory
10) Cohomology for complex & projective alg. varieties
11) Homotopy type theory
12) Differential forms in (sheaf and de Rham) cohomology theory
13) Representations of quivers
14) Discrete differential geometry
15) Applications of tensor algebra in statistics
16) Topological methods for dynamical systems and mathematical modelling of the brain
17) Representation theory for semisimple Lie algebras
18) Adelic groups in algebraic number theory
19) Topological problems with election systems
20) Tropical geometry
21) Infinite trees & hypergraphs
22) Determining if an antiderivative is elementary based on the orientability of its representative Riemann surface
23) Random matrices in computer graphics

[joshhallam]

I love the Dover Books on Math. After I graduated, I thought I would just keep studying math, so I bought myself a handful of them. I never did have the time or motivation once I started working full time and had a super long commute. I really wish upper division math books had answers and solutions to all the problems though. That would have helped. I had 1 semester of real analysis and tried to self study the 2nd half, but when you write a proof and aren't sure if the logic follows and can't see the solution, it is really frustrating and demotivating.

[terrencedaniels]

At school I struggled with math, despite being keen. But with depression, social isolation and an in ability to focus, I did very poorly.
About 5 years later I picked up my old algebra text book and worked thru it cover to cover, checking with my physics major friend occasionally. He thought I was doing very well, just overdoing it. Then came the burnout and the brain fog and the 'blues'. And I became stupid again. But you've inspired me! Will look into 'the math i missed'.

[SuperNovaJinckUFO]

This video makes me feel extremely privileged to have been able to have been able to cover some of these subjects during my undergraduate studies. As an engineering major, no less

[ntesla]

Needham has written a book "Visual Differential Geometry and Forms: A Mathematical Drama in Five Acts". I picked it up because of how much I enjoyed his "Visual Complex Analysis". And of course the 5 vol self published Spivak's "Comprehensive Introduction to Differential Geometry" Then for a good Dover book, I certainly enjoy having Shahshahani's "An Introductory Course on Differentiable Manifolds". Thanks for all you do and taking the time to share with us! I was not aware of Garrity's "Electricity And Magnetism For Mathematicians : A Guided Path From Maxwell's Equations To Yang-mills" All those things we find searching for the Hopf fibration floating in a Dirac Sea...

[stereographicpencilsharpener]

Abstract algebra is used in particle physics, tensor analysis on manifolds and differential geometry are used in general relativity, and some things like topology on manifolds are used in string theory, and fourier analysis of several complex variables is also used throughout theoretical physics. Hope I don't seem rude. I love math. I am actually kind of unsure if I want to do physics or math in grad school (If I can get in hopefully)

[borincod]

Nice content!
Would be also really interesting if you start video series about how some obscure math has found its applications somewhere else, like in engineering.

[MadScientyst]

Brilliant selection...once u get a basic BSc /BA in Math & according to IF the curriculum was mainly Pure with minimal Applied, then the syllabus would've covered a few of these Topics at an introductory level.

However, once u enter a Master's (Pure Math) Program with Thesis specialization, chances are some of the more advanced disciplines mentioned WILL be a part of the syllabus.

Some random fun facts:
*Combinatorics /Discreet Math & Graph Theory go hand in hand for Comp Sci foundation studies

*Real, Complex, Fourier & Functional Analyses is a natural 'sequence' of progressional topics

*Einstein relied heavily on Differential Geometry /Tensor Calculus for developing Relativity Theory

& That's just scratching the surface folks!
Now, isn't Mathematics simply beautiful!! 😄😄

[joeldick]

I'm not a mathematics major or graduate student. I did an engineering bachelor's and master's (dynamic control), and I saw some of these topics. For example linear algebra in my linear control courses, topology in non-linear systems, combinatorics and abstract algebra in my cryptography and networks courses. And obviously a lots of differential equations and statistics.

[randomvariable]

This is all completely awesome!!! I got a dual major in physics and math, and took many courses I did not have to for a degree. I studied most of these topics and I still have my quantum physics book

[davidmarshall]

I agree, Kreyszig wrote a beautiful intro to functional analysis.

[byronwilliams]

When I was at UGA, the department head was the author listed in topology of manifolds. JC Cantrell .

[owenwillard]

very interesting to hear someone talk about something they are passionate about, even if i not interested in the subject. i’m a senior in highschool and will likely never take a math class beyond basic algebra. not even trig. my major has little to do with math, but i still have all the respect in the world for the people willing to put the effort in to such a difficult area of study.

[baronvonbeandip]

Bruh, Kreyszig is such a great author. That functional analysis text and his text 'Advanced Engineering Mathematics' are the best in their subject, imo

[renesperb]

It is interesting to see how many topics you mention with the remark '' most people never see ''. Almost all of these subjects where compulsory
in my time at the ETH in Zürich in the Sixties.

[varsityathlete]

4:19 2nd year physics, Quantum Mechanics courses, every chapter.

[awlay]

I just finished my dissertation in my final year of a bachelors in maths stats. I did my dissertation on topology and metric spaces which is not covered at all during my course. I ended up enjoying learning about it more than some of my classes however the actual math exercises behind were far more rigorous than any area i had covered so i can understand why it isnt taught at undergrad level. Researching it though was extremely fun, the concepts of metrics and how they are applied topologically was very interesting and i do encourage people to look into it if they are interested in this sort of area

[stereographicpencilsharpener]

I will say, as a physics and math double major, abstract algebra counted as a required math credit for both my physics and math degrees.