Resources for learning mathematics
This thread is for links to high quality resources that are useful for learning mathematics (at any level).
To start, here are the recordings of an introductory course on Topology & Geometry (subtitle: Pictorial Thinking) given by Dr. Tadashi Tokieda of the University of Cambridge (UK) when he visited the African Institute for Mathematical Sciences (South Africa) in 2014. [URL]https://www.youtube.com/playlist?list=PLTBqohhFNBE_09L0ilf3fYXF5woAbrzJ[/URL] 
If I may, Nick, what exactly is your mathematical background? Over the last year or two you've continually heightened my impression of your skills, but I know hardly anything about what exactly your areas of expertise are.

My main interests at present are algebraic, but good ideas in mathematics often come from new interactions between specialisations, so it is important not to let research become too narrow!
For anyone interested in discovering the correct position from which to view a painting, or how modular arithmetic can help us prove that two knots are genuinely distinct, Christopher Zeeman's introduction to 3dimensional geometry is illuminating: [URL]https://www.lms.ac.uk/sites/lms.ac.uk/files/2000%20Recommended%20theorems%20in%203dimensional%20geometry%20%28preprint%29.pdf[/URL] 
Here is a list of public lectures in mathematics that have been recorded and made available online by the University of Oxford (UK).
They are intended for a nonspecialist audience. [URL]http://www.maths.ox.ac.uk/events/publiclecturesevents#PublicLecturesOnline[/URL] 
Introduction to surfaces (no expert knowledge required):
[URL]http://www.maths.ed.ac.uk/~v1ranick/surgery/zeeman.pdf[/URL] 
First year bachelor students in mathematics at Oxford University get a couple of introductory lectures on the complex numbers.
The Oxford Maths Institute has recorded the second of these and made it public: [URL]http://www.maths.ox.ac.uk/node/30464[/URL] 
Not sure if any high school students are around but I remember on my first year at uni when I found out about Piskounov, Apostol or Demidovitch, I was so upset not having known about these resources to have even better grades at math during my high school.
I understand that now with all the internet of things everything is shared faster than before so nowadays people have the advantages of that but the disadvantage of not thinking by themselves. Good stuff Nick. 
Several chapters of the book "A Friendly Introduction to Number Theory" by the wellknown mathematician Joseph Silverman are available free online at his page about it here:
[URL]https://www.math.brown.edu/~jhs/frint.html[/URL] 
Thank you so much for these links...fascinating material!
Lumpy 
The Mathematics Institute at the University of Oxford (UK) have now followed up on their filmed lecture about Complex Numbers with several other single lectures
(usually the first lecture) from 1st and 2nd year undergraduate courses for mathematicians. [URL="https://www.maths.ox.ac.uk/node/30464"]Complex Numbers[/URL] [URL="https://www.maths.ox.ac.uk/node/34054"]Introductory Calculus[/URL] [URL="https://www.maths.ox.ac.uk/node/31744"]Dynamics[/URL] [URL="https://www.maths.ox.ac.uk/node/32464"]Integration[/URL] [URL="https://www.maths.ox.ac.uk/node/34201"]Differential Equations[/URL] [URL="https://www.maths.ox.ac.uk/node/34349"]Quantum Theory[/URL] 
I follow Oxford on YouTube and sometimes at night, instead on reading, I’m watching their lecture, my wife thinks I am crazy. Haven’t seen the last one.

I collect such sources here: [url]http://philomath.boards.net/board/93/mathematics[/url]

Here is a link to Huang's paper on the Sensitivity Conjecture as noted within the recent issue of Discover magazine: [url]https://arxiv.org/pdf/1907.00847.pdf[/url]
A related (possibly dated around 194955) paper from the University of Illinois Urbana which I photocopied years ago used 3D cubes of unit resistances to develop some interesting results..Fibonacci, etc.. Some original papers by G. Kron are also worth a look. These kinds of papers showed me at that time how circuit theory and certain kinds of mathematics are essentially interchangeable. Computer Science extends this methodology. 
[QUOTE=jwaltos;534453]Here is a link to Huang's paper on the Sensitivity Conjecture[/QUOTE]
Nice! 
Paul Pollack's Number Theory books.
The first one can be downloaded free! [URL]http://pollack.uga.edu/index.html#books[/URL] 
In the [url=https://www.mersenneforum.org/forumdisplay.php?f=132]Number Theory Discussion Group[/url] subforum of the Math forum, there is a series of threads whose titles begin "Basic Number Theory" followed by numbers  1&2, 3, 4, 5, ..., 22.
Unlike Andre Weil's book infamously titled [u]Basic Number Theory[/u], these consist of introductory material. 
[QUOTE=Dr Sardonicus;563050]In the [url=https://www.mersenneforum.org/forumdisplay.php?f=132]Number Theory Discussion Group[/url] subforum of the Math forum, there is a series of threads whose titles begin "Basic Number Theory" followed by numbers  1&2, 3, 4, 5, ..., 22.
Unlike Andre Weil's book infamously titled [u]Basic Number Theory[/u], these consist of introductory material.[/QUOTE]As in texts which cover [u]Elementary Particle Physics[/u]. 
[QUOTE=xilman;563086]As in texts which cover [U]Elementary Particle Physics[/U].[/QUOTE]
Algebraic Number Theory can legitimately be thought of as both the algebraic branch of number theory and the theory of algebraic numbers. 
[QUOTE=Nick;563088]Algebraic Number Theory can legitimately be thought of as both the algebraic branch of number theory and the theory of algebraic numbers.[/QUOTE]One time a grad school classmate referred to Analytic Number Theory as "the theory of analytic numbers." Oops.
Of course, the title [u]Basic Number Theory[/u] was somewhat in jest  it [i]is[/i] "basic" in the sense that it covers the basics of Class Field Theory. But it is definitely [i]not[/i] an introductory text. I don't know whether Weil ever became exasperated with the jokes about the book title, but it is a fact that 12 years later, he came out with another book called [u]Number Theory for Beginners[/u]. Reminds me, my third grade math class was really [i]hard![/i] The textbook was [u]A Course in Arithmetic[/u] by JeanPierre Serre... 
[QUOTE=xilman;563086]As in texts which cover [u]Elementary Particle Physics[/u].[/QUOTE]
The ideas of particle physics: An Introduction for Scientists. Coughlan and Dodd. Cambridge University Press The above book is a decent introduction for "beginners" and is a great starting point for further inquiry. This is one of those books where even if you skim through it superficially you will retain something..which is always a good thing. 
A professor who was (and maybe still is) a forum member has written
a leisurely introduction to some foundational concepts of universitylevel mathematics, and made it freely available [URL="https://math.byu.edu/~pace/Transitionereader.pdf"]here.[/URL] 
[QUOTE=Nick;578658]A professor who was (and maybe still is) a forum member has written a leisurely introduction to some foundational concepts of universitylevel mathematics, and made it freely available [URL="https://math.byu.edu/~pace/Transitionereader.pdf"]here.[/URL][/QUOTE]
This is absolutely amazingly awesome!!! Personally, I will never get to the level of many of you when it comes to deep maths. But even just understanding the nomenclature and notation a little bit can go a long way. I will be sharing this around. A nonzero percentage will find it valuable! :tu: 
I don't see anyone mentioned
[url]https://ocw.mit.edu/index.htm[/url] which covers all topics. There's a number theory course on YouTube [url]https://www.youtube.com/watch?v=19SW3P_PRHQ[/url] based around a Dover book. I have not watched much of the video. As the author points out, Dover books are quite reasonably priced. I have bought the book, and intend watching the video, using that in conjunction with the book. The Open University put a bit of a course on number theory on their website. [url]https://www.open.edu/openlearn/sciencemathstechnology/numbertheory/contentsection0?intro=1[/url] However, I think the amount is pretty small compared to other resources mentioned where people have put whole books, or several chapters of books freely available. As such, the OU course might not be worth bothering with. Dave (engineering graduate). 
Thanks to Mike for the following tip:
[URL]https://artofproblemsolving.com/articles/files/SatoNT.pdf[/URL] Slight caveat on this one: the puzzles are good but the theory is not always completely precise or optimal. 
[QUOTE=Nick;579038]
Slight caveat on this one: the puzzles are good but the theory is not always completely precise or optimal.[/QUOTE] You got me thinking, that perhaps a forum that is devoted to comments/errata about number theory text books would be useful. One thread per text book. Dave 
For anyone looking for a fun bit of summer reading:
[URL="https://www.ams.org/journals/bull/20195603/S02730979201901653X/S02730979201901653X.pdf"] The discrete yet ubiquitous theorems of Carathéodory, Helly, Sperner, Tucker, and Tverberg[/URL] I think the above link is open access, but if you have any trouble there is also an arXiv version [URL="https://arxiv.org/abs/1706.05975"]here.[/URL] 
[QUOTE=Nick;583999]For anyone looking for a fun bit of summer reading..[/QUOTE]
Hi Nick, I haven't been doing any math for the last several months with the exception of occasional sporadic jottings of approaches to solving a multivariate equation of degree 4. ie X^4*(a*b*c*d*e+...)+X^3*(b^2*d*e...)+..+X^0*(c*d..). The variables (there are up to five with their associated combinations) within the brackets where each variable does not exceed degree 2 but they may all be of degree 2. Do you know of any math books, notes or approaches that apply to solving such expressions? I have a few that apply but I'm wondering about some other clever ideas that may exist. I've considered operational calculus, umbral calculus, trig. substitutions, iterative programming... Since this is a fourth degree form, I was also considering moments of inertia, chemical equations.. any thoughts you may have would be appreciated. Just to provide some context, all of the variables exist as part of a Cartesian coordinate frame (with a slight modification). The expression exists under a radical so it represents solution(s) which may be +ve/ve or complex...algebraic in any case. The variables composing the expression were combined from several merged general equations. There are some interesting facets regarding the interpretation of this degree four equation and I would like to obtain as broad a perspective as possible by encapsulating it via mathematical theory and utility. The coordinate frame mentioned previously represents the scaffolding for actual measurable figures. Further, that coordinate frame exists in conjunction with another required coordinate frame which has its own special features, some of which extend directly into lattice theory. Perhaps this context helps or perhaps not but at the very least I hope it provides you with an idea that a solid theoretical foundation exists within some elementary mathematics. Without being disingenuous, I'd like the final construct to display some ingenuity when the crank is turned and meaningful numbers are either turned out in one direction or meaningful concepts are turned out in the reverse direction...yes, there is also some thermodynamic reasoning involved with a 50/50 probability of some quantum logic involved as well. 
The short answer is that if you are working with a single polynomial equation of degree
at most 4 over the complex numbers, then you can choose any values you like for all variables except one, and then find the possible values of the remaining variable by the historical methods passed down to us by Cardano and Ferrari. To gain more insight into such problems means delving into Algebraic Geometry, which is a vast and fascinating area of modern mathematics. It takes time to master, however: where Differential Geometry relies on Calculus (which everyone knows already), Algebraic Geometry relies on Commutative Algebra (which is new to most people). A good introduction is Miles Reid's [URL="https://homepages.warwick.ac.uk/staff/Miles.Reid/MA4A5/UAG.pdf"]Undergraduate Algebraic Geometry.[/URL] For the number theory point of view, the book [URL="https://www.springer.com/gp/book/9783319185873"]Rational Points on Elliptic Curves[/URL] is a good place to start (but won't cover quartic equations, of course). 
[QUOTE=Nick;584596]The short answer is that if you are working with a single polynomial equation of degree
at most 4 over the complex numbers, then you can choose any values you like ..[/QUOTE] Thanks for responding Nick. Unfortunately it didn't really help me out. I'm a little more advanced than the books you recommended (or at least think I am) and where you state "you can choose any values you like" that is the crux of the matter. There must be a guiding principle which must limit my options (free will concept should not be in play) to specific numeric values of the parameters related to the value(s) I'm seeking. I'm looking initially at "real world" applications, here is mathstackexchange link: [url]https://math.stackexchange.com/questions/3469546/whatcanbearealworldapplicationforsolvingquarticequations[/url] Theoretically any univariate polynomial can be considered as a base number system. Consider the single parameter as 10 and simplify the coefficients to represent a base 10 number..and so on. Multivariate systems..same deal. Extending this concept within base 10..consider digit sums and the picene hydrocarbon. Then consider the inverse tessellation of the same. Then consider the geometric form as an infinite sheet or surface...and so on down this particular rabbit hole. Considering the difference of squares, sum of squares, primes that are and aren't the sum of two squares geometrically, such relations provide some simple figures. Embellishing these figures properly leads to reciprocity and a few other gems. Given that the general quartic can be solved via radicals is a particularly interesting limit point. In any case, dimensional analysis and measurable objects/influences is what I'm looking at presently. Prior to this I was examining the general cubic parabola from which I have since elucidated some interesting facts. Just to keep in line (pun intended) with the purpose of this thread here are a few "light read" books containing some of the above ideas: Geometry and the Imagination, Hilbert and the other guy. Explorations in Mathematical Physics, An Elegant Language...,Koks Curves for the Mathematically Curious, Havil Proofs from the Book, 6th Edition ..and not so light.. [url]https://scholarworks.umb.edu/cgi/viewcontent.cgi?article=1340&context=masters_theses[/url] [url]https://uma.enstaparis.fr/files/publis/2018/2018theseuma1930Alicia.SimonPetit.pdf[/url] 
[QUOTE=jwaltos;584755]Thanks for responding Nick. Unfortunately it didn't really help me out...[/QUOTE]
Well you have a very individual perspective so any suggestions are bound to be a bit hitandmiss! Mathematically, quartic polynomials tend to arise from 4dimensional spaces, for example as the determinant of a 4x4 matrix. And a key difference between those and 3dimensional spaces is that 4 is even  yes, it sounds trivial, but it has deeper consequences. Calculate the volume of the unit ball in n dimensions, for example, and you will see one pattern for the odd dimensions and another pattern for the even ones. 
Ok Nick. I’m home on A/L and I’ve already packed my Piskunov books. I will re study them but I want then to think about next step. I always wanted to take a math degree but to get a job it would be better to have a engineer degree. Need to go back to my dream. Please I want your guidance.

[QUOTE=pinhodecarlos;586792]Ok Nick. I’m home on A/L and I’ve already packed my Piskunov books. I will re study them but I want then to think about next step. I always wanted to take a math degree but to get a job it would be better to have a engineer degree. Need to go back to my dream. Please I want your guidance.[/QUOTE]
Apologies for the late reaction: we have been away on holiday and I have only now seen your post. Universitylevel mathematics for mathematicians differs from mathematics for engineers or natural scientists in its emphasis on abstraction and rigour. So alongside Piskunov it would be a good idea to read "A Companion to Analysis" by T. W. Körner. Chapter 1 is available free [URL="https://bookstore.ams.org/gsm62/"]here.[/URL] Similarly, you probablly have a good knowledge of linear algebra from an applied point of view, but may gain something from a more abstract perspective. Prof. Ronald van Luijk has written his lecture notes in English and made them freely available [URL="http://websites.math.leidenuniv.nl/algebra/linalg1.pdf"]here,[/URL] with followup notes by him and Michael Stoll available [URL="http://websites.math.leidenuniv.nl/algebra/linalg2.pdf"]here[/URL]. There are various good books on the theory of groups, for example the book "[URL="https://link.springer.com/book/10.1007/9781475740349"]Groups and Symmetry[/URL]" by M. A. Armstrong. This is where you start to get into the mathematics that engineers do not usually see! If you are looking for a broad, accessible overview of modern pure mathematics, then the "[URL="https://press.princeton.edu/books/hardcover/9780691118802/theprincetoncompaniontomathematics"]Princeton Companion to Mathematics[/URL]" edited by Prof. Tim Gowers is worth dipping into. 
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