YES. A lot of what I have been learning on my own has applications in theoretical physics and quantum physics. I was lucky to have a professor in my undergraduate that actually did some of the research into the Calabi-Yau manifold. It's nice to be able to email her when I have questions about actual group tessellations onto a manifold.
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!
@@TheMathSorcerer bro how to solve this. I like mathematics, but when I want to study other things like economics, I am afraid that I will lose my skills/interest in mathematics
@@kalebjosuasaragih981 Math is the core of all science. No matter how much you might like a different subject, you will be using / returning to math one way or another.
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.
SOME do... it's hard to find them though... Depending on what you want to learn about. I have an entire text on Fuchsian groups by Svetlana Katok. There are answers to select exercises...it isn't COMPLETE answers to proofs...but it spells out the basic proof and how to solve it. I want to buy it...
It's extremely hard to study math or science on your own while working a full-time technical job. Your brain is focused on something hard all day, and having to come home to it can really burn you out in a hurry. I've found myself having to drop graduate classes for that reason. In that vein, it's very easy to purchase books, thinking "I want to learn that!" In practice, they tend to pile up. Always buy an e-copy of technical books!
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'.
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
sometimes its about being at the right place at the right time! because sometimes they have a class but then the one professor who can teach it leaves....
@@tablecloth1752 Functional Analysis, Measure Theory, and Abstract Algebra. They were all requirements for my major, but my major is really weird. It's basically 40% engineering, 30% comp sci, and 30% pure math
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...
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)
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.
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!! 😄😄
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.
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
Abstract algebra looks a lot like what I was seeing in my engineering grad class Linear Systems Theory. The last 3-4 weeks of the semester was good when we finally got to the point of applying it after our prof (who was a math major then went into engineering when he discovered that there's no money in being a math major) was done beating the theory to death.
Bruh, Kreyszig is such a great author. That functional analysis text and his text 'Advanced Engineering Mathematics' are the best in their subject, imo
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
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
You don’t do metric spaces and topology in undergrad at your college? That’s really surprising to me, we do both metric spaces and topology in the first semester of 2nd year undergrad at my university, with Algebraic Topology 1 and 2 available in 3rd year
@@stephencurran2284 I study maths and stats at a uni in glasgow so my classes are quite stats heavy when it comes to 3rd and 4th year… I believe it is covered lightly in an optional class in 4th year but for the mandatory classes it is not covered
@@awlay3092 Im at Edinburgh and none of them are required but theyre decently popular, didnt think it'd be that different, are you at uni of or strathclyde? Even for maths and stats here you only have 2 Stats classes in 3rd year, 4th year is a bit heavier but you still have loads of room for other stuff.
And abstract algebra was a required class for my computer science degree... but they got rid of the requirement for any language lower level than JAVA. Go figure!
Hey, we do see Abstract Algebra in engineering, specially for Error Correcting Codes. I agree it may not get deep into the subject, but we do study abstract algebra and Galois Fields
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.
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.
I just started teaching grade school math again. I introduce some abstract algebra (non-rigorously, obviously), graph theory, combinatorics and symbolic logic. So much more math can be introduced as wonderful, simply stated problems than is introduced in the standard math curriculum. Math is much more a toy than a game, and the insistence of memorizing algorithms for complicated arithmetic kills the sense of math as a toy.
I started my BS as a Computer Science student, wandered into Applied Mathematics for a year, then switched back. I've definitely found the knowledge from that Abstract Algebra book you showed useful, but the most fun elective I took was on the knot theory sub-field of topology. The class was taught using the The Knot Book by Colin Adams. The thing that grabbed me was the multiple levels you could think through the problems at: topology, knot polynomials, and a physical piece of string.
And the #1 math subject most people never see is... 🥁🥁🥁 Machine learning! 🤣🤣 But seriously, here's some of my favorites: 1. equivariant degree theory (see the green book by Dr. Balanov, Dr. Krawcewicz, Dr. Steinlein) 2. variational analysis (a more general form of convex analysis, applied to set-valued mapping) 3. complex analysis of several variables (you have that one in your list too) 4. functional differential equations (in particular delay differential equations) There's plenty more, but these are top 4. For 1, I was taught by both Dr. Balanov and Krawcewicz themselves. So yeah, if you don't go to a certain school/program, you may not get to see some of these.
Would also add Larry Wasserman’s ‘All of Nonparametric Statistics’ for a more contemporary nonparametrics text. Covers some essential material past the 90s on resampling methods like the jackknife and bootstrap, nonparametric regression, some minimax theory, and the normal means problem to highlight some of the high points of the text. Lots of great references for each topic, and was my jumping off point into studying functional analysis!
His course on Statistical Learning Theory is excellent - there’s a full set of recordings of the lectures for the class, and the notes can be readily found online
I feel like the field of Clifford Algebras is a branch of math that isn't as commonly known as it probably should be. They generalize real numbers, complex numbers, quaternions, and other hypercomplex number systems. Geometric algebra (and by extension geometric calculus) is a real Clifford algebra that has applications in computer vision and theoretical physics. Also, perturbation theory is an important field of applied mathematics and it deals with quantifying deviations in mathematical problem-solving.
1:03 Abstract Algebra: "This is something that anyone who goes to college to study... physics they're not going to see" I went to Purdue in and around 2010. My physics undergrad had an abstract algebra class - not part of the core curriculum but as part of what we students called "mandatory electives" Taking hours to write a proof was *NOT* what I wanted to be doing with my time when sweet, sweet thermo was calling from around the corner. 4:16 This book was core, though. I have it on the shelf behind me, in fact, right next to Griffiths Intro to Quantum and Bartle's Intro to Real Analysis.
I bought "Tensor Analysis on Manifolds" a while back to try to help me understand general relativity better, but found it more even more impenetrable than the GR textbook I was using ("Gravitation" by Misner, Wheeler, and Thorne).
It's always weird to hear this is not undergraduate material. I learned functional analysis and measure theory as an undergrad at the faculty of engineering in the third and fourth exam of mathematical analysis (so in my third and fourth year). Functuonsl analysis comes before distribution theory, and spectral theory. Measure theory before variational methods. (Europe here)
I also saw almost all of these during my mathematics undergraduate studies, and I also studied in a European country. US undergraduate majors in math are a lot less advanced, but it all evens out at the graduate level eventually.
A statement of scope of The Companion straight from the author: "The word “companion” is significant. Although this book is certainly intended as a useful work of reference, you should not expect too much of it... In this respect, the book is like a human companion, complete with gaps in its knowledge and views on some topics that may not be universally shared.... Indeed, the structure of the book is such that it would not be ridiculous to read it from cover to cover, though it would certainly be time-consuming." I'd say good to read through, but I wouldn't take notes on the subjects unless you would like to pursue that topic more and find more literature on it.
Actually in my discrete math class at cs major we see a lot of abstract algebra, after number theory we learn about groups, rings , and on recent classes, fields. That's so crazy, but is fun.
Halmos! Now that brings back fond memories. I wished the internet existed when I first got into math. I despised math in high school, but fell in love with it in my 20s, and gobbled up as much as I could find (esp. in analysis). But it was always tricky to find the right recommendations, and I ended up reading stuff of very varying quality. I got that Nonparametric Statistics book too, but didn't find it worth the investment (but that says more about my tastes - I just didn't find the topic very interesting). Never heard of Garrity, but it looks like a book that needed to be written. I'll check it out. It sounds like a great gift for young people getting into the topic.
What an awesome collection. I love how you call them " old", I used several of these in college where a lot of our math was accomplished with slide rules and statistics were done using tables - only calculators we had were mechanical Frieden calculators which if you set it for 0 divided by 1 they were crank for infinity. Lab professors loved us. Overall some of these books seem like giant white papers. Even today those old text books can still sell for over a hundred dollars. I go to book store and estate/garage sales to collect them. Just think we went to the moon using similar math and built the SR 71 without computers. Now our cell phones have more capacity than Apollo space vehicles. Thanks for the " memories" :-)
Here's a bit of fun: calculating intervals between Primes. Case 3: Pa+1 is Composite, Pa+2 is Prime factor. Add first term of Pa+1 and first term of Pa+2: 2+5= Count 7 Count =7 Pa = 23 (23) 1 Pa+1 = 24 (2!3*3) 2 Pa+2 = 25 (5!) 3 26 (2*13) 4 27 (3!3) 5 28 (2!*7) 6 29 Pb (29) 7 Case 5: Pa+1 is Composite with three terms and a Factor, Pa+2 has two terms. Add first terms of both Composites for Count. 2+7=9 Count=9 Pa = 89 (89) 1 Pa+1 = 90 (2*3!*5) 2 Pa+2 = 91 (7*13) 3 92 4 93 5 94 6 95 7 96 8 97 Pb (97) 9 Still working out the other Cases..."Prime Pairs" like 11 and 13, are proving tricky, but I have a general Case that works. More later...
Thanks for the recommendation of "All the math you missed". I have a physics degree and a bunch of math classes, so I've at least seen some of it, but definitely haven't really worked problems in all of them. Math methods in physics textbooks (Boas, Kreyszig) are also good in that respect.
These days, you pretty much always see some group theory as a physics undergraduate. Things like representation theory in quantum mechanics... are very much abstract algebra.
Lol. That book of math you are supposed to know before graduate school is a hoot. The Riemann Mapping Theorem?? As an undergrad? Don’t be ridiculous. That said, it’s a really cool bit of math.
In probability theory and mathematical finance I see mostly analysis. Integrals and partial/stochastic DEs from here to eternity. Obviously some linear algebra, and graph theory as well since since a lot of it is done numerically on a computer.
The real frustrating thing about most math textbooks is they just glance over the important bits. Developments and proofs skip over any number of steps and claim "this is easily seen" - no, it's not! Also, I really wish there was a complete anthology of math. By one author, or one group of authors, following a common approach and common presentation style, instead of having to piece stuff together piecemeal from any n books on individual subjects. With the broad strokes big picture included, I might say.
There are a lot of powerful ideas out there if you just know the name the last person used when he talked to the first person that wrote the idea. That math book with all the math you missed in it would have a word in there you could search with the computer and answer your question.
@@TheMathSorcerer Even doctoral degree students struggles and can't solve the problems. It's the cradle of Abstract Algebra. I got the solutions of almost all the chapters. This is a book that must be read with many other books. It is NOT easy. It is THE Algebra book.
I think the super foundational topics are the ones the fewest people learn. ZFC set theory doesn’t have much real world applications but is super interesting philosophically. Same with mathematical logic. These topics force you to think in a very precise way, taking absolutely nothing for granted, but weird and seemingly paradoxical results abound. I honestly didn’t get through more than the first 3 chapters on these because it gets kind of tedious, but the foundational stuff like building the real numbers from scratch is interesting to me.
I ran into nonparametric statistical methods more than a few times as an AI researcher in the 90s. It's a basis for anomaly detection, decision-making under uncertainty, model selection, natural language processing, and image and signal processing. All of these applications operate on data spaces that cannot be assumed to be normally distributed. The Mann-Whitney U test (determines if two samples came from the same population) is not particularly complex. Implementing it anyway. Understanding what it tells you is a whole 'nother story. (:
It always stuns me what some people can do in math without never knowing things like what a group is. I guess Mathematics is a most theoretical discipline while being the most pratically useful, too.
I really disliked Abstract Algebra, probably my worst upper division/graduate course, but did enjoy my time in special functions [ Gamma, Bessel, etc... ], and Fourier Analysis. Had some Differential Geometry working in the area of physics / Relativity, and did write a pretty basic paper on Topology in undergraduate school. But never did cover much of the other topics here. It is amazing how much more Math there is out there, considering all the time I spent on it ! thanks
Hello I have a question, if you can answer I would appreciate. What is the branch of mathematics that deal with divergence, grad, curl, laplacian of curvilinear coordinates (spherical, cylindrical etc.)? Is it differential geometry? Algebraic geometry?? Topology? I am electrical engineer, we used spherical coordinates in electromagnetic and robotics classes but I always tried to memorize the formulas which is hard (they usually give them in exams for cylindrical coordinates though). I wanna know which branch /math class deals with this. Is it shown in undergraduate to all math students? Thanks in advance
I worked out some math. Without ever being good at it. I made a homemade Mandalorian board game. Worked out how to apply twenty cards to work out on three out of the for games that correspond to twenty double sided tokens half are one type the other half are a different type. To work for tokens on two different games. Between two to four players. With one game being a cross between memory and poker.
What? No Algebraic Curves or Knot Theory?! Is there any graduate math program which doesn't require abstract algebra? Few may get to Galois Theory, but groups, rings, fields and polynomials are as fundamental as general topology. FWIW I came across the Nonparametrc Statistical Methods text in an undergrad course, but then again I also came across Box and Jenkins in an undergrad course. And to get picky, you haven't really done differential geometry and manifolds if you lack Spivak's multiple volumes. That is unless you enjoy the 19th century approach immersed in differential forms.
Temperature was one and had negative measurements but so was acceleration when it stood singular from both acceleration and the steady state velocities of special relativity Each was also another one reigning like separate yet equal infinities ( Google wrote very very small like an Incredible Spring Band ballad but the geometry had sunk two cubic square points into one as if they were pubic as if hiding the points of the square. Perspective would flatten that line of information requiring some movement or deviation from absolute zero which was still a quantum problem but they were getting closer but it would evoke florene
How do I problem solve? I try to solve as much problems as I can, but when I face a completely new idea I have never seen before , it takes me more time to solve it. Practicing alot makes me solve problems similar to the ones I have practiced much faster , but when I see a problem I have never seen before ,then either it takes me a really long time , or I end up not being able to solve it. How do I train my self to solve new problems?
Maybe its cause I specialise in theoretical computer science but I have had plenty of abstract algebra in my CS education so I'm not really sure if it holds that computer scientists never touch on abstract algebra.
I'm currently in the final year of my network engineering degree and planning to go to grad school for telecommunications engineering. Can you suggest a good book for learning information theory and math related to signal processing? I see these two topics come up a lot in most telecom engineering syllabuses.
For Signal processing you probably want to read real analysis and Fourier analysis. I haven't studied Fourier anaysis so I can't recommend anything, but some good intro texts for real analysis would be Understanding Analysis by Abbott, or Real analysis by Gaughan. I think Terence Tao has a two book series on analysis which coveres some fourier anaysis, but from the bit I've seen it is not as introductory as the other books i mentioned.
What is even crazier. The entirety of theoretical physics relies on these abstract concepts, they're not just some made up bogus or freetime activity for bored mathematicians. This crazy abstraction level is indispensable as it clearly seems to be fundamentally embedded in our world
@@maalikserebryakov Maybe not, but we all had to study it anyway, and I actually found it very useful. You do need to be able to solve differential equations as an engineer (e.g., material mechanics, fluid dynamics, etc.) and it is very useful to know that the solutions to certain linear differential equations form a vector space and that you only need to find a base (i.e., a certain set of functions) of such vector space and then apply the initial conditions in order to find the particular solution (a certain vector) to your problem.
Students interested in mechanics ( the normal force et al ) classes should comence in the fitness gymnasium arena with the springs and pullys obeying Sir Issac Newton. Mathematics reminds me of hand sewing.
Group theory is pretty central to theoretical physics.
Differential geometry is pretty central to general relativity.
Yeah most of these math topics are covered at some length in physics undergrad at some schools
YES. A lot of what I have been learning on my own has applications in theoretical physics and quantum physics. I was lucky to have a professor in my undergraduate that actually did some of the research into the Calabi-Yau manifold. It's nice to be able to email her when I have questions about actual group tessellations onto a manifold.
Yeah calculus in manifolds/tensor calculus/differential geometry and group theory is definitely needed for theoretical physics
Lmao
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!
That is awesome!
@@TheMathSorcerer bro how to solve this. I like mathematics, but when I want to study other things like economics, I am afraid that I will lose my skills/interest in mathematics
@@kalebjosuasaragih981 Math is the core of all science. No matter how much you might like a different subject, you will be using / returning to math one way or another.
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.
SOME do... it's hard to find them though... Depending on what you want to learn about. I have an entire text on Fuchsian groups by Svetlana Katok. There are answers to select exercises...it isn't COMPLETE answers to proofs...but it spells out the basic proof and how to solve it. I want to buy it...
i love the schaum's series for this exact reason
lots of exercises + demonstration explained in full details
@@madtrade Awesome! I'll have to take a look at that.
It's extremely hard to study math or science on your own while working a full-time technical job. Your brain is focused on something hard all day, and having to come home to it can really burn you out in a hurry. I've found myself having to drop graduate classes for that reason.
In that vein, it's very easy to purchase books, thinking "I want to learn that!" In practice, they tend to pile up. Always buy an e-copy of technical books!
Well said...
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'.
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
sometimes its about being at the right place at the right time! because sometimes they have a class but then the one professor who can teach it leaves....
May I ask which math you covered? Did you take them as separate courses from your major requirements?
@@tablecloth1752 Functional Analysis, Measure Theory, and Abstract Algebra. They were all requirements for my major, but my major is really weird. It's basically 40% engineering, 30% comp sci, and 30% pure math
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...
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)
Yeah physics covers alot of complex math at this point
It's also the basis of Fuzzy Logic, which I think is due to make a comeback in AI.
Differential geometry, tensor analysis, fourier analysis, topology, real and complex analysis are also all used in fluid dynamics.
When I was at UGA, the department head was the author listed in topology of manifolds. JC Cantrell .
oh wow!
Oh how I wish I was at a school 1/10 that strong in math lol
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.
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!! 😄😄
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.
4:19 2nd year physics, Quantum Mechanics courses, every chapter.
Yup
I agree, Kreyszig wrote a beautiful intro to functional analysis.
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
Abstract algebra looks a lot like what I was seeing in my engineering grad class Linear Systems Theory. The last 3-4 weeks of the semester was good when we finally got to the point of applying it after our prof (who was a math major then went into engineering when he discovered that there's no money in being a math major) was done beating the theory to death.
Bruh, Kreyszig is such a great author. That functional analysis text and his text 'Advanced Engineering Mathematics' are the best in their subject, imo
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
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
That’s interesting, at my uni metric spaces is a 3rd year class and you can take both general and algebraic topology in 4th year
You don’t do metric spaces and topology in undergrad at your college? That’s really surprising to me, we do both metric spaces and topology in the first semester of 2nd year undergrad at my university, with Algebraic Topology 1 and 2 available in 3rd year
@@stephencurran2284 I study maths and stats at a uni in glasgow so my classes are quite stats heavy when it comes to 3rd and 4th year… I believe it is covered lightly in an optional class in 4th year but for the mandatory classes it is not covered
@@nope110 hopefully my reply above also applies to yourself
@@awlay3092 Im at Edinburgh and none of them are required but theyre decently popular, didnt think it'd be that different, are you at uni of or strathclyde?
Even for maths and stats here you only have 2 Stats classes in 3rd year, 4th year is a bit heavier but you still have loads of room for other stuff.
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.
And abstract algebra was a required class for my computer science degree... but they got rid of the requirement for any language lower level than JAVA. Go figure!
im in highschool struggling with calculus AB and am astonished that the math I struggle with is easy compared to all this.
Hey, we do see Abstract Algebra in engineering, specially for Error Correcting Codes. I agree it may not get deep into the subject, but we do study abstract algebra and Galois Fields
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.
That Abstact Algebra book cover and fonts just screams late Eighties and early Ninetees.
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.
Wow that's interesting, and here you are watching this video. I think that's really cool:)
I just started teaching grade school math again. I introduce some abstract algebra (non-rigorously, obviously), graph theory, combinatorics and symbolic logic. So much more math can be introduced as wonderful, simply stated problems than is introduced in the standard math curriculum. Math is much more a toy than a game, and the insistence of memorizing algorithms for complicated arithmetic kills the sense of math as a toy.
differential geometry seems so interesting
I started my BS as a Computer Science student, wandered into Applied Mathematics for a year, then switched back. I've definitely found the knowledge from that Abstract Algebra book you showed useful, but the most fun elective I took was on the knot theory sub-field of topology. The class was taught using the The Knot Book by Colin Adams. The thing that grabbed me was the multiple levels you could think through the problems at: topology, knot polynomials, and a physical piece of string.
And the #1 math subject most people never see is... 🥁🥁🥁
Machine learning!
🤣🤣
But seriously, here's some of my favorites:
1. equivariant degree theory (see the green book by Dr. Balanov, Dr. Krawcewicz, Dr. Steinlein)
2. variational analysis (a more general form of convex analysis, applied to set-valued mapping)
3. complex analysis of several variables (you have that one in your list too)
4. functional differential equations (in particular delay differential equations)
There's plenty more, but these are top 4. For 1, I was taught by both Dr. Balanov and Krawcewicz themselves.
So yeah, if you don't go to a certain school/program, you may not get to see some of these.
Would also add Larry Wasserman’s ‘All of Nonparametric Statistics’ for a more contemporary nonparametrics text. Covers some essential material past the 90s on resampling methods like the jackknife and bootstrap, nonparametric regression, some minimax theory, and the normal means problem to highlight some of the high points of the text. Lots of great references for each topic, and was my jumping off point into studying functional analysis!
His course on Statistical Learning Theory is excellent - there’s a full set of recordings of the lectures for the class, and the notes can be readily found online
That topology brings me back to my 4th year. I had that stats book in post grad in the 90s.
I bet you can cast Math Skill Holy from Final Fantasy Tactics in real life...
I feel like the field of Clifford Algebras is a branch of math that isn't as commonly known as it probably should be. They generalize real numbers, complex numbers, quaternions, and other hypercomplex number systems. Geometric algebra (and by extension geometric calculus) is a real Clifford algebra that has applications in computer vision and theoretical physics. Also, perturbation theory is an important field of applied mathematics and it deals with quantifying deviations in mathematical problem-solving.
Another beautiful entry into an advanced subject is Berberian's Introduction to Hilbert Space.
That sounds interesting, I will check it out, thank you very much!
This is the math of Bigby, Leomund, Mordenkainen, Otiluke, and Vecna. What an adventure! 😊
Kreyszig long time no see
"The more you know, the more you realize you don't know." - Aristotle
I had no idea such topics existed! Thanks!
1:03 Abstract Algebra: "This is something that anyone who goes to college to study... physics they're not going to see"
I went to Purdue in and around 2010. My physics undergrad had an abstract algebra class - not part of the core curriculum but as part of what we students called "mandatory electives"
Taking hours to write a proof was *NOT* what I wanted to be doing with my time when sweet, sweet thermo was calling from around the corner. 4:16 This book was core, though.
I have it on the shelf behind me, in fact, right next to Griffiths Intro to Quantum and Bartle's Intro to Real Analysis.
I bought "Tensor Analysis on Manifolds" a while back to try to help me understand general relativity better, but found it more even more impenetrable than the GR textbook I was using ("Gravitation" by Misner, Wheeler, and Thorne).
It's always weird to hear this is not undergraduate material. I learned functional analysis and measure theory as an undergrad at the faculty of engineering in the third and fourth exam of mathematical analysis (so in my third and fourth year). Functuonsl analysis comes before distribution theory, and spectral theory. Measure theory before variational methods. (Europe here)
The knowledge requirements varies a lot from major and country. grad school in Europe tend to be a lot more rigorous.
I also saw almost all of these during my mathematics undergraduate studies, and I also studied in a European country. US undergraduate majors in math are a lot less advanced, but it all evens out at the graduate level eventually.
I perfectly agree with your statement " a lot of mathematics knowledge is needed in order to start understanding... 😁😁😁😅
I am on chapter 3 of "Introductory Functional Analysis with Applications" for Reinforcement Learning
I hope your channel exist when I was studying this... 15 yrs.ago...
Do you think the Princeton Mathematics Companion is a worthwhile study?
A statement of scope of The Companion straight from the author: "The word “companion” is significant. Although this book is certainly intended as a useful work of reference,
you should not expect too much of it... In this respect, the book is like a human companion, complete with gaps in its knowledge and views on some topics that may not be universally shared.... Indeed, the structure of the book is such that it would not be ridiculous to read it from cover to cover, though it would certainly be time-consuming." I'd say good to read through, but I wouldn't take notes on the subjects unless you would like to pursue that topic more and find more literature on it.
no
Actually in my discrete math class at cs major we see a lot of abstract algebra, after number theory we learn about groups, rings , and on recent classes, fields. That's so crazy, but is fun.
Halmos! Now that brings back fond memories. I wished the internet existed when I first got into math. I despised math in high school, but fell in love with it in my 20s, and gobbled up as much as I could find (esp. in analysis). But it was always tricky to find the right recommendations, and I ended up reading stuff of very varying quality. I got that Nonparametric Statistics book too, but didn't find it worth the investment (but that says more about my tastes - I just didn't find the topic very interesting). Never heard of Garrity, but it looks like a book that needed to be written. I'll check it out. It sounds like a great gift for young people getting into the topic.
1 video on IMO
International mathematical Olympiad
What an awesome collection. I love how you call them " old", I used several of these in college where a lot of our math was accomplished with slide rules and statistics were done using tables - only calculators we had were mechanical Frieden calculators which if you set it for 0 divided by 1 they were crank for infinity. Lab professors loved us. Overall some of these books seem like giant white papers. Even today those old text books can still sell for over a hundred dollars. I go to book store and estate/garage sales to collect them. Just think we went to the moon using similar math and built the SR 71 without computers. Now our cell phones have more capacity than Apollo space vehicles. Thanks for the " memories" :-)
Here's a bit of fun: calculating intervals between Primes.
Case 3: Pa+1 is Composite, Pa+2 is Prime factor.
Add first term of Pa+1 and first term of Pa+2: 2+5= Count 7
Count =7
Pa = 23 (23) 1
Pa+1 = 24 (2!3*3) 2
Pa+2 = 25 (5!) 3
26 (2*13) 4
27 (3!3) 5
28 (2!*7) 6
29 Pb (29) 7
Case 5: Pa+1 is Composite with three terms and a Factor, Pa+2 has two terms.
Add first terms of both Composites for Count. 2+7=9
Count=9
Pa = 89 (89) 1
Pa+1 = 90 (2*3!*5) 2
Pa+2 = 91 (7*13) 3
92 4
93 5
94 6
95 7
96 8
97 Pb (97) 9
Still working out the other Cases..."Prime Pairs" like 11 and 13, are proving tricky, but I have a general Case that works.
More later...
Thanks for the recommendation of "All the math you missed". I have a physics degree and a bunch of math classes, so I've at least seen some of it, but definitely haven't really worked problems in all of them.
Math methods in physics textbooks (Boas, Kreyszig) are also good in that respect.
Quasigroup and loop theory!!!!
Real analysis is pretty important for economics. Look at MIT's mathematical economics major requirements for example.
These days, you pretty much always see some group theory as a physics undergraduate. Things like representation theory in quantum mechanics... are very much abstract algebra.
Can someone recommend me book for geometry&algebra formulas ?
Lol. That book of math you are supposed to know before graduate school is a hoot. The Riemann Mapping Theorem?? As an undergrad? Don’t be ridiculous. That said, it’s a really cool bit of math.
:)
In probability theory and mathematical finance I see mostly analysis. Integrals and partial/stochastic DEs from here to eternity. Obviously some linear algebra, and graph theory as well since since a lot of it is done numerically on a computer.
There is no way I can live without this channel, The math sorcerer, I am ♾ gratefull!
The real frustrating thing about most math textbooks is they just glance over the important bits. Developments and proofs skip over any number of steps and claim "this is easily seen" - no, it's not! Also, I really wish there was a complete anthology of math. By one author, or one group of authors, following a common approach and common presentation style, instead of having to piece stuff together piecemeal from any n books on individual subjects. With the broad strokes big picture included, I might say.
I have the DOVER book and another by KREYSZIG, as well as a library of similar books.
Dummit & Foote's Abstract Algebra textbook is classic 👍
I got to say that it's funny to hear you say "If you have some math background..." because almost no one has, but often they think they do.
There are a lot of powerful ideas out there if you just know the name the last person used when he talked to the first person that wrote the idea. That math book with all the math you missed in it would have a word in there you could search with the computer and answer your question.
Dummit and Foote IS the best Abstract Algebra book if you find the solution of the problems. I think they are available somewhere.
Yeah it really is epic isn't it? It has so much and it's so well organized the exercises rock. I agree, if you can find the solutions it is amazing.
@@TheMathSorcerer Even doctoral degree students struggles and can't solve the problems. It's the cradle of Abstract Algebra. I got the solutions of almost all the chapters. This is a book that must be read with many other books. It is NOT easy. It is THE Algebra book.
I think the super foundational topics are the ones the fewest people learn. ZFC set theory doesn’t have much real world applications but is super interesting philosophically. Same with mathematical logic. These topics force you to think in a very precise way, taking absolutely nothing for granted, but weird and seemingly paradoxical results abound. I honestly didn’t get through more than the first 3 chapters on these because it gets kind of tedious, but the foundational stuff like building the real numbers from scratch is interesting to me.
I ran into nonparametric statistical methods more than a few times as an AI researcher in the 90s. It's a basis for anomaly detection, decision-making under uncertainty, model selection, natural language processing, and image and signal processing. All of these applications operate on data spaces that cannot be assumed to be normally distributed.
The Mann-Whitney U test (determines if two samples came from the same population) is not particularly complex. Implementing it anyway. Understanding what it tells you is a whole 'nother story.
(:
That Abstract Algebra book is still in my nightmares... that and my Number Theory class made it clear I had hit my limit! LOL
I like Linear Algebra. It's fun and interesting. I'm working on my masters in statistics now. One of my classes is Linear Algebra 🥰
It always stuns me what some people can do in math without never knowing things like what a group is. I guess Mathematics is a most theoretical discipline while being the most pratically useful, too.
Where's Abraham Robinson's book.......'Non-Standard Analysis'
Topology is a staggeringly beautiful subject.
I really disliked Abstract Algebra, probably my worst upper division/graduate course, but did enjoy my time in special functions [ Gamma, Bessel, etc... ], and Fourier Analysis. Had some Differential Geometry working in the area of physics / Relativity, and did write a pretty basic paper on Topology in undergraduate school. But never did cover much of the other topics here. It is amazing how much more Math there is out there, considering all the time I spent on it ! thanks
Hello I have a question, if you can answer I would appreciate.
What is the branch of mathematics that deal with divergence, grad, curl, laplacian of curvilinear coordinates (spherical, cylindrical etc.)? Is it differential geometry? Algebraic geometry?? Topology? I am electrical engineer, we used spherical coordinates in electromagnetic and robotics classes but I always tried to memorize the formulas which is hard (they usually give them in exams for cylindrical coordinates though). I wanna know which branch /math class deals with this. Is it shown in undergraduate to all math students?
Thanks in advance
I'm definitely buying the Garrity book, thanks.
0=180 ? forgot where i learned it book? possibly auslanders or trig lock no variables remain solution
Do you have a paid comment section where we can ask questions?
I don't, but you can ask here or email me. My email is on the about page.
@@TheMathSorcerer Thank, I will.
i studied abstract algebra a lot in physics (grad level tho)
oh wow very nice!!
I worked out some math. Without ever being good at it. I made a homemade Mandalorian board game. Worked out how to apply twenty cards to work out on three out of the for games that correspond to twenty double sided tokens half are one type the other half are a different type. To work for tokens on two different games. Between two to four players. With one game being a cross between memory and poker.
What? No Algebraic Curves or Knot Theory?!
Is there any graduate math program which doesn't require abstract algebra? Few may get to Galois Theory, but groups, rings, fields and polynomials are as fundamental as general topology.
FWIW I came across the Nonparametrc Statistical Methods text in an undergrad course, but then again I also came across Box and Jenkins in an undergrad course. And to get picky, you haven't really done differential geometry and manifolds if you lack Spivak's multiple volumes. That is unless you enjoy the 19th century approach immersed in differential forms.
This guy has information addiction
Temperature was one and had negative measurements but so was acceleration when it stood singular from both acceleration and the steady state velocities of special relativity
Each was also another one reigning like separate yet equal infinities ( Google wrote very very small like an Incredible Spring Band ballad but the geometry had sunk two cubic square points into one as if they were pubic as if hiding the points of the square. Perspective would flatten that line of information requiring some movement or deviation from absolute zero which was still a quantum problem but they were getting closer but it would evoke florene
How do I problem solve? I try to solve as much problems as I can, but when I face a completely new idea I have never seen before , it takes me more time to solve it. Practicing alot makes me solve problems similar to the ones I have practiced much faster , but when I see a problem I have never seen before ,then either it takes me a really long time , or I end up not being able to solve it. How do I train my self to solve new problems?
I am surprised you didn't show spherical trigonometry.:)
find books when MIT abandoned fluid dynamics as a study 1898 1893 also or so was last structured curriculum
Maybe its cause I specialise in theoretical computer science but I have had plenty of abstract algebra in my CS education so I'm not really sure if it holds that computer scientists never touch on abstract algebra.
I would have killed to have a class on the topology on manifolds... I'm having to learn so much on my own to fill my knowledge gaps...
As someone going to apply for grad school in astrophysics l feel like I’m missing out not getting to study most of these fields.
I’m a physics and electrical engineering undergrad student, I have a few of these books.
I'm currently in the final year of my network engineering degree and planning to go to grad school for telecommunications engineering. Can you suggest a good book for learning information theory and math related to signal processing? I see these two topics come up a lot in most telecom engineering syllabuses.
For Signal processing you probably want to read real analysis and Fourier analysis. I haven't studied Fourier anaysis so I can't recommend anything, but some good intro texts for real analysis would be Understanding Analysis by Abbott, or Real analysis by Gaughan. I think Terence Tao has a two book series on analysis which coveres some fourier anaysis, but from the bit I've seen it is not as introductory as the other books i mentioned.
Currently using the linear analysis and dummit/Foote algebra books
What is even crazier. The entirety of theoretical physics relies on these abstract concepts, they're not just some made up bogus or freetime activity for bored mathematicians. This crazy abstraction level is indispensable as it clearly seems to be fundamentally embedded in our world
Books from Springer used to be absolute hardcore, very condense and advanced.
Thx for sharing!
Show us the forbidden math, Sorcerer. 🧙♂
Man, I studied mechanical engineering in Spain and we all had to study abstract algebra on the first year...
@@maalikserebryakov Maybe not, but we all had to study it anyway, and I actually found it very useful. You do need to be able to solve differential equations as an engineer (e.g., material mechanics, fluid dynamics, etc.) and it is very useful to know that the solutions to certain linear differential equations form a vector space and that you only need to find a base (i.e., a certain set of functions) of such vector space and then apply the initial conditions in order to find the particular solution (a certain vector) to your problem.
I read the thumbnail as “math-people never see”! I guess I’m blind now
Any book that you can recommend me to learn series and successions? Thank you
Where can I find all these
I put links in the description.
i remember a touch of auslander ,krysig
Students interested in mechanics ( the normal force et al ) classes should comence in the fitness gymnasium arena with the springs and pullys obeying Sir Issac Newton. Mathematics reminds me of hand sewing.
I wish I could find a copy of Kreyszig at a decent price.
i was allowed to plAY a little in manifolds WANTED MORE