I have some epic math goals and creating a study plan

[MathExplorer]

Summary:: Trying to become a warp drive pioneer. Yes, you read that right, lol.

Hello I’m a math student, and need advice on what pre-requisite math subjects I need to study AND in what order I need to study them in order to achieve my short term math goals. I’m also looking for the appropriate textbooks needed because I’m using the self taught method, and need textbooks that include examples, practice problems, quizzes/tests, etc.

My short term and long term math goals, and my questions about them are as follows:

Short term goals:

I need to bring my math skills to the level that it needs to be so I can have the ability to study the following math subjects:

• Division Algebras
• Jordan Algebras
• Clifford Algebras
• Quaternions
• Octonions

My Long term math goals are to then fully study the above five subjects.

Questions:

• Exactly in what order do I need to be studying these five math subjects?
• What specific pre-requisite math subjects do I need to study AND in what order do I need to study them, so that I can be educationally ready to begin studying the above five math subjects?
• Regarding the list of pre-requisite math subjects that I need to write down, are there any specific textbooks that you guys recommend that include examples, practice problems, quizzes and tests? Note: I’m using the self taught method instead of going to college to learn all or most of the math that I want to learn.

Note: if it helps answer my questions, I attached pictures of the most recent textbook that I finished studying so that you know where my math skills are at. As of this moment, I’m going over this textbook a second time to re-learn concepts that I’m already starting to forget while searching for the ‘next’ textbook to study.

If it helps answer the above questions, here is some information about myself regarding my current math skills, how I developed them, and why I’m interested in the five math subjects that I mentioned:

My learning path:

• I used youtube and google to teach myself basic math
• I attended a technical university to learn pre-algebra. But the way the school was designed, I ended up in a situation where I basically taught myself pre-algebra because the teacher didn’t teach us anything. He only tutored and graded the paperwork.
• I transferred to an online college to study a textbook labeled “elementary and intermediate algebra”. …And once again I found myself in a situation where I was paying lots of money(in student loans) to pay for the privilege of “teaching myself math”.
• By the time I finished the Elementary and Intermediate Algebra class, I came to the conclusion that after experimenting with attending college on campus, and then online, that either way I’m going to have to teach myself math. So if I’m going to have to teach myself, I might as well do it for free, especially since I just successfully went from basic math, all the way to intermediate algebra. So I’m continuing the self taught method for now, and just knocking out as much math as I can for free(saving myself thousands of dollars) until it’s time for me to enroll in a college again to major in physics.

What’s motivating me:

Three years ago I was inspired to get back into college and continue my education, despite finishing welding school back in 2007 and being a welder ever sense. I was inspired by two discoveries. The first discovery was learning about SpaceX and their plans to build fleets of rocket propelled starships that’ll be used for manned exploration missions throughout the solar system.

The second discovery was that I discovered that Warp Drive Theory, a subject I always viewed as nothing more than science fiction, is now an actual scientific theory that is being worked on by scientists around the world. The first warp drive theory scientific paper was published in 1994 by Dr. Miguel Alcubierre, and the work has been continued by other scientists such as Dr. Sonny White of NASA(who now works at Limitless Space Institute in Texas and is in charge of Warp Drive R&D), Dr. Froning, Dr. Musha, and a few others I discovered. They represent humanity’s first generation of FTL(faster than light) propulsion pioneers. So I originally started continuing my education due to breakthroughs in rocket science at SpaceX, but I’m now staying on my path of continuing my education to focus on FTL propulsion research and development. So right now, my main focus is developing the education I need to have the ability to research and develop FTL propulsion theories, mostly Warp Drive Theory, which is actually a thing now and not just a scifi prop on star trek.

I had the privilege of speaking with Dr. Miguel Alcubierre over email, and speaking with Dr. Sonny White of Limited Space Institute and got really good advice on what to study in order to bring myself up to their level, which obviously is going to take years lol. I basically have to fully study General Relativity, and Quantum Mechanics, and I need to educationally prepare myself to be able to study a unification theory that may come out in the near future(that according to Dr. Sonny White, scientists around the world are working on it). This unification theory will bridge the gap between General Relativity and Quantum Mechanics and partially re-write both subjects as well. It’s basically the final piece of the puzzle needed for Warp Drive Theory to mature the point of finally becoming an engineering problem to be solved, and not a physics problem to be solved.

This finally brings me to the five math subjects I’ve mentioned above. Through my own research, whatever unification theory that comes out that’ll help Warp Drive Theory take the next step in R&D is most likely going to use those five kinds of math subjects. There is a debate in the scientific community on whether or not that is even true, but the only way I’m going to find out for myself is to use the scientific method, which is to F around and find out lol.

Sorry for this wall of text but I’ve already brought this subject up in a couple other math forums and they wanted to know why I’m interested in these topics so I decided to lay it out in this first post so we can focus this thread on answering my math questions.

Thanks in advance for any info/advice/textbook recommendations you guys can give me. Using the self taught route will be challenging, yes, but I’ll be saving myself thousands of dollars doing it this way. Plus there are tons of math textbooks I can get online for free or for very little money.

 

[fresh_42]

Summary:: Trying to become a warp drive pioneer. Yes, you read that right, lol.

Hello I’m a math student, and need advice on what pre-requisite math subjects I need to study AND in what order I need to study them in order to achieve my short term math goals. I’m also looking for the appropriate textbooks needed because I’m using the self taught method, and need textbooks that include examples, practice problems, quizzes/tests, etc.

My short term and long term math goals, and my questions about them are as follows:

Short term goals:

I need to bring my math skills to the level that it needs to be so I can have the ability to study the following math subjects:

• Division Algebras
• Jordan Algebras
• Clifford Algebras
• Quaternions
• Octonions

Linear Algebra; Abstract Algebra: Group Theory (basics), Field Theory (basics), Representation Theory; Homological Algebra (depending on how far you want to go); Commutative Algebra (basics); Algebraic Geometry (basics)

My Long term math goals are to then fully study the above five subjects.

Questions:

• Exactly in what order do I need to be studying these five math subjects?

• Quaternions
• Division Algebras
• Octonions
• Jordan Algebras / Clifford Algebras
• Clifford Algebras / Jordan Algebras
  

• What specific pre-requisite math subjects do I need to study AND in what order do I need to study them, so that I can be educationally ready to begin studying the above five math subjects?

(see above) LA, AA, RT, (HA), CA, AG

• Regarding the list of pre-requisite math subjects that I need to write down, are there any specific textbooks that you guys recommend that include examples, practice problems, quizzes and tests? Note: I’m using the self taught method instead of going to college to learn all or most of the math that I want to learn.

Note: if it helps answer my questions, I attached pictures of the most recent textbook that I finished studying so that you know where my math skills are at. As of this moment, I’m going over this textbook a second time to re-learn concepts that I’m already starting to forget while searching for the ‘next’ textbook to study.

I doubt that there is one book that fits all. Many are either good in the theoretical part, others have good exercises. I recommend
Lecture notes for Linear Algebra. You can find them online and see by yourself whether it fits your needs.
https://www.amazon.com/dp/0387901086/?tag=pfamazon01-20
https://www.amazon.com/dp/0387974954/?tag=pfamazon01-20
and to start from there to figure out what else do you need. Maybe they already include all basics you need that I mentioned from the other branches.

If it helps answer the above questions, here is some information about myself regarding my current math skills, how I developed them, and why I’m interested in the five math subjects that I mentioned:

My learning path:

• I used youtube and google to teach myself basic math
• I attended a technical university to learn pre-algebra. But the way the school was designed, I ended up in a situation where I basically taught myself pre-algebra because the teacher didn’t teach us anything. He only tutored and graded the paperwork.
• I transferred to an online college to study a textbook labeled “elementary and intermediate algebra”. …And once again I found myself in a situation where I was paying lots of money(in student loans) to pay for the privilege of “teaching myself math”.
• By the time I finished the Elementary and Intermediate Algebra class, I came to the conclusion that after experimenting with attending college on campus, and then online, that either way I’m going to have to teach myself math. So if I’m going to have to teach myself, I might as well do it for free, especially since I just successfully went from basic math, all the way to intermediate algebra. So I’m continuing the self taught method for now, and just knocking out as much math as I can for free(saving myself thousands of dollars) until it’s time for me to enroll in a college again to major in physics.

Physics is an entirely different matter. Your list has little in common with physics besides linear algebra and some fundamentals. Both meet again on an elaborated level of studies. What you need to know to major in physics and is on your list as well, is little more than what already can be found on Wikipedia (plus linear algebra). There are far more important branches of mathematics for physics: Calculus (real, complex, multivariate), Differential Equations (ordinary and partial), Differential Geometry, Lie Theory, or others depending on where you go in physics.

What’s motivating me:

Three years ago I was inspired to get back into college and continue my education, despite finishing welding school back in 2007 and being a welder ever sense. I was inspired by two discoveries. The first discovery was learning about SpaceX and their plans to build fleets of rocket propelled starships that’ll be used for manned exploration missions throughout the solar system.

The second discovery was that I discovered that Warp Drive Theory, a subject I always viewed as nothing more than science fiction, is now an actual scientific theory that is being worked on by scientists around the world. The first warp drive theory scientific paper was published in 1994 by Dr. Miguel Alcubierre, and the work has been continued by other scientists such as Dr. Sonny White of NASA(who now works at Limitless Space Institute in Texas and is in charge of Warp Drive R&D), Dr. Froning, Dr. Musha, and a few others I discovered. They represent humanity’s first generation of FTL(faster than light) propulsion pioneers. So I originally started continuing my education due to breakthroughs in rocket science at SpaceX, but I’m now staying on my path of continuing my education to focus on FTL propulsion research and development. So right now, my main focus is developing the education I need to have the ability to research and develop FTL propulsion theories, mostly Warp Drive Theory, which is actually a thing now and not just a scifi prop on star trek.

I had the privilege of speaking with Dr. Miguel Alcubierre over email, and speaking with Dr. Sonny White of Limited Space Institute and got really good advice on what to study in order to bring myself up to their level, which obviously is going to take years lol. I basically have to fully study General Relativity, and Quantum Mechanics, and I need to educationally prepare myself to be able to study a unification theory that may come out in the near future(that according to Dr. Sonny White, scientists around the world are working on it). This unification theory will bridge the gap between General Relativity and Quantum Mechanics and partially re-write both subjects as well. It’s basically the final piece of the puzzle needed for Warp Drive Theory to mature the point of finally becoming an engineering problem to be solved, and not a physics problem to be solved.

This finally brings me to the five math subjects I’ve mentioned above. Through my own research, whatever unification theory that comes out that’ll help Warp Drive Theory take the next step in R&D is most likely going to use those five kinds of math subjects. There is a debate in the scientific community on whether or not that is even true, but the only way I’m going to find out for myself is to use the scientific method, which is to F around and find out lol.

Sorry for this wall of text but I’ve already brought this subject up in a couple other math forums and they wanted to know why I’m interested in these topics so I decided to lay it out in this first post so we can focus this thread on answering my math questions.

Thanks in advance for any info/advice/textbook recommendations you guys can give me. Using the self taught route will be challenging, yes, but I’ll be saving myself thousands of dollars doing it this way. Plus there are tons of math textbooks I can get online for free or for very little money.

Definitely differential geometry!

 

[MathExplorer]

Linear Algebra; Abstract Algebra: Group Theory (basics), Field Theory (basics), Representation Theory; Homological Algebra (depending on how far you want to go); Commutative Algebra (basics); Algebraic Geometry (basics)

• Quaternions
• Division Algebras
• Octonions
• Jordan Algebras / Clifford Algebras
• Clifford Algebras / Jordan Algebras

(see above) LA, AA, RT, (HA), CA, AG

I doubt that there is one book that fits all. Many are either good in the theoretical part, others have good exercises. I recommend
Lecture notes for Linear Algebra. You can find them online and see by yourself whether it fits your needs.
https://www.amazon.com/dp/0387901086/?tag=pfamazon01-20
https://www.amazon.com/dp/0387974954/?tag=pfamazon01-20
and to start from there to figure out what else do you need. Maybe they already include all basics you need that I mentioned from the other branches.Physics is an entirely different matter. Your list has little in common with physics besides linear algebra and some fundamentals. Both meet again on an elaborated level of studies. What you need to know to major in physics and is on your list as well, is little more than what already can be found on Wikipedia (plus linear algebra). There are far more important branches of mathematics for physics: Calculus (real, complex, multivariate), Differential Equations (ordinary and partial), Differential Geometry, Lie Theory, or others depending on where you go in physics.
Definitely differential geometry!

Awesome! Thank you sooo much! It's been a real pain in the neck trying to find the answers to my questions. You know how the internet can be sometimes. Sometimes the internet will provide everything accept for what you're looking for, rofl.

After writing down a whole page of notes from what I learned from your reply, I only have two questions:

1. Where would Differential Geometry fit in with my study plan?

So far, this is what I gathered:

First study these and in this specific order:

• Linear Algebra
• Abstract Algebra
• Group Theory (basics)
• Field Theory (basics)
• Representation Theory
• Homological Algebra (depending on how far you want to go)
• Commutative Algebra (basics)
• Algebraic Geometry (basics)

Then study these in this specific order:

• Quaternions
• Division Algebras
• Octonions
• Jordan Algebras / Clifford Algebras
• Clifford Algebras / Jordan Algebras

2. Am I correct to assume that because of how you listed Jordan Algebras and Clifford Algebras, that you are implying that by the time I get to the point of studying those two subjects that it doesn't matter which of the two I study first?

 

[fresh_42]

Awesome! Thank you sooo much! It's been a real pain in the neck trying to find the answers to my questions. You know how the internet can be sometimes. Sometimes the internet will provide everything accept for what you're looking for, rofl.

After writing down a whole page of notes from what I learned from your reply, I only have two questions:

1. Where would Differential Geometry fit in with my study plan?

So far, this is what I gathered:

First study these and in this specific order:

• Linear Algebra
• Abstract Algebra
• Group Theory (basics)
• Field Theory (basics)
• Representation Theory
• Homological Algebra (depending on how far you want to go)
• Commutative Algebra (basics)
• Algebraic Geometry (basics)

Then study these in this specific order:

• Quaternions
• Division Algebras
• Octonions
• Jordan Algebras / Clifford Algebras
• Clifford Algebras / Jordan Algebras

2. Am I correct to assume that because of how you listed Jordan Algebras and Clifford Algebras, that you are implying that by the time I get to the point of studying those two subjects that it doesn't matter which of the two I study first?

Your study plan is rather algebraic. This is nice and exciting. But as much as mathematics needs algebra, as much does physics need calculus in all its variants, and there are many. They don't really match because they require a different mindset. And a third one for the physical part. It is far more about language than one might expect from STEM fields.

One could say for short, that algebra - and your subjects on the list are all in algebra - is the science of structures. Calculus, however, is the science of functions. And physics needs a third perspective, namely a coordinate orientated view.

Physics measures things, therefore we need coordinates. It is all about coordinates since physics describes quantities and motions. The motions are represented by functions in coordinate systems and differential equation systems. E.g. differential geometry provides the coordinate systems for general relativity, functional analysis the framework for quantum mechanics. These goals require an approach that is not very algebraic. In fact, an ordinary algebraic person will probably hate coordinates - they disguise the view on the structures.

If you really want to study physics, then calculus is in my opinion the only valid starting point. If you will follow your plan, then you will likely end up in pure mathematics. There are touching points like Lie algebras, but even they stem from the study of Lie groups which are - even literally by definition - analytic, i.e. a matter of calculus. Other touching points are at the frontlines of research, e.g. the shape of our universe, or string theory. But this is a long way to go and there are no shortcuts.

Study your algebras if you like! They require a lot less knowledge than physics does, and there is a lot of research to do. But to find a way back from pure mathematics to physics usually does not work. Maybe you should have a look at Virasoro algebras ...And, yes, linear algebra is needed everywhere, in physics and in mathematics.

 

[MathExplorer]

oh ok. Based on what you said, then I should plan on heavily studying calculus as well. The point of all these math goals of mine are:

1. Develop the mathematical abilities needed to fully study General Relativity. When I talked to Dr. Miguel Alcubierre, he made it very clear that I need to fully study GR if I want to R&D Warp Drive Theory...and he warned me that this is no small feat and that most physicists don't fully study out that subject while they are in college.

2. The reason for the interest in the 5 math subjects I mentioned was to prepare me for a scientific theory that doesn't exist yet, but Dr. Sonny White told me that there are scientists around the world who are working on it. There is a third Warp Drive R&D expert who introduced me to those five math subjects via some information he shared to me that suggests that the five math subjects will be used a lot in the future unification theories, and thus my interest in them.

So I guess I have three categories of math goals now instead of two:
1. The short term goals I shared
2. The long term goals I shared
3. Whatever math I'll be studying while fully studying General Relativity and then Quantum Mechanics...that...according to what I just read...is Calculus? If the answer is yes, then am I correct to assume that I'll likely be studying calculus 1-4? and that learning differential equations is just a part of learning calculus?

 

[MathExplorer]

Just to confirm something I read either in this thread or heard in other places I've asked the above questions, am I correct to assume that I'll learn "differential equations" when I'm going through the process of learning Pre-Calculus, Calculus 1,2,3 and 4?

 

[Mondayman]

Just to confirm something I read either in this thread or heard in other places I've asked the above questions, am I correct to assume that I'll learn "differential equations" when I'm going through the process of learning Pre-Calculus, Calculus 1,2,3 and 4?

Some textbooks include a short introduction, but not a full on course. As I understand it where I took it,

Calculus I : Differentiation
Calculus II : Integration
Calculus III : Multivariate Calculus
Calculus IV : Vector Calculus

Differential Equations is its own two courses where I went.

 

[FactChecker]

To get where you want to go, General Relativity and Quantum Mechanics, I think you should look at the books "Linear and Geometric Algebra" and "Vector and Geometric Calculus" by Alan Macdonald. They might give you a different perspective on many of the other subjects that you have listed. You can branch out from there.

 

[PAllen]

I think many of your math choices are off track. What I suggest is going to the course schedule of major university and tracking back the prerequisites for the graduate GR course, the QM courses, and the QFT courses. These prerequisites will include both physics and mathematics. As far as I know, they will not look at all like your list in the OP. Then, you can see what textbooks are being used for GR, QM, and QFT. For GR itself, at this point in time, I would suggest Carroll, then Wald. Note, the online version of Carroll is missing many important topics compared to the textbook.

With pure GR, all warp drive solutions require large amounts of exotic matter. If anyone has told you otherwise, they are flat out wrong. Exotic matter violates the dominant energy condition, and has the feature that such matter, by itself, as a macroscopic body, can move faster than light. No classical relativistic theory of matter or energy corresponding to known matter admits such a possibility. Thus, from a purely classical perspective, warp drives cannot exist. Thus, the importance of QM and QFT and the effort at unification of these with GR. QM does allow small scale exotic matter states within existing models. Conceivably, some unifications of GR and QM might allow larger scale exotic matter. Thus, you need to master QM, QFT, and GR, and then start to explore unifications favorable to your goal. An epic program indeed!

 

[mpresic3]

It looks as if your current level of math is roughly ninth grade as it is taught in the US with some eleventh grade topics thrown in. It used to be the usual curriculum in the US was:

ninth grade: elelmentary algebra
tenth grade: geometry
eleventh grade: intermediate algebra and trigonometry
twelfth grade: more advanced trig and calculus(1/2 term) or calculus (full term) for advanced students

So it seems what it missing so far is your effort in geometry. This really should not be slighted because of your long term interest in general relativity. You have set before you an ambitious set of goals. However, given many years, it is might be possible to progress to this understanding. Who knows, everyone started out in math at a understanding that was lower than your current one, albeit many learned geometry at an earlier age.

Anyway, my advice if it is to work through plane geometry first. Make sure to retain those algebra skills that you have practiced up to now. Then study precalculus math (intermediate algebra, some probability, trigonometry). Then you may be ready to take calculus.

For students in HS this alone, takes 3-4 years. You may have half of this already.

This brings us to college math. Those topics you request are (only) touched upon (by talented students) after at least 3-4 years of college math, and investigated much later. (So you see what you are in for)

There is a good chance you will never have an understanding of these topics of division algebras, octonions etc. Moreover, if you plunge further and further deep, and research abstraction as much as possible, you will miss a lot of the beauty of the study. Think of the journey, not the destination

By this I mean, for me personally, I will never forget:

1. My first college physics lecture
2. The time they told me in class the charm-anticharm particle was discovered
3. The lecture where Maxwell's equations were presented.
4. My first lectures in quantum mechanics

and a few others, although these stick out in my mind mostly. The bottom line is feel good about the small successes in the journey. They will motivate the difficult roads in the path ahead

 

[PeroK]

If you want to build a warp drive engine, I'd ditch the linear algebra books and build a science and engineering lab in the basement instead.

 

[FactChecker]

Here is the reason that I recommended two books on Geometric Algebra and Geometric Calculus. In a lot of physics, there comes a point where mysterious coordinate systems are invented to make the calculations work. They are all a little different and yet all eerily similar. Are they part of advanced physics? What is going on?
In Geometric Algebra, it becomes clear that they often are the result of dirt-simple geometry, along with (tedious) bookkeeping. It is a good, methodical way of addressing complex analysis in multiple dimensions, quaternions, Pauli matrices, and other mathematics subjects needed in physics. It is the same as Clifford Algebra but the emphasis is on the geometric basis of it. It gives you good, concise mathematics to express a lot of physics. For instance, using Geometric Algebra, Maxwells' equations are just one simple equation. The algebra provides the geometry and bookkeeping that consolidates the Maxwell equations into one.
In addition to the two books I mentioned in post #8, you might look at "Geometric Algebra for Physicists" by Chris Doran and Anthony Lasenby.

That being said, there is a learning curve in Geometric Algebra that does not really pay off until later. I am not sure that your current background is the right time to go in that direction. Just be aware of it.