Layer 9: Graduate school This is where things branch off majorly in mathematics. Many advanced concepts and ideas end up here and even math majors may know little about what a graduate in economics uses for their mathematics. Concentration 1: Pure mathematics and the study of shapes There are two primary pure mathematics topics I'll discuss here, algebraic topology and algebraic geometry. These are typically reserved for graduate school because they use the tools of differential geometry, abstract algebra, and topology, and there's no guarantee that a math major had taken all of those. Algebraic topology: In this course students learn about translating topology into different mathematical languages. First, we have to talk about topology and invariances. Invariances are important because these are things that will not change. Differential geometry uses these invariances to separate out the things they are studying. Furthermore, only studying the properties of topological invariances allows someone to not care about the geometric properties of some shape. This exact idea of invariances is what algebraic topologists seek to classify. They use the properties of groups (and the study abstract algebra in general) to classify different topological spaces given certain unchanging things. This is why people say that a mug and a donut are the same thing. Change up that mug all you want (without breaking it), at some point, it will look like a donut. Why? Because both things have on thing that will not change. Their hole! This is a toplogical invariance. Algebraic topologists use very complicated methods (CW complexes, homology and cohomology, homotopy, esc, some of which may be introduced in an introductory topology course/seminar) to classify these spaces. You can then use these observations to make claims about the group in which this topological invariant lays. If you want an example, think of a larger group of infinitely many things with one hole in them. What can we say about them? Well, if we change their shape, they'll likely look like each other. Through some process we can change their shape, and voila! Once you figure out how we can change their shape (very abstract process, requires an explanation of linear maps and spaces then building up to category theory or measure theory, based on the type of course you take and what its main focus is on. Then an explanation of operators and how to really think about addition and multiplication, it's a lot!) Essentially, you can find out how to do mathematics on this special space and you can figure out how to turn it into another thing within this special space... (how do you turn a donut into a mug, and how do you describe the relationship between donuts and mugs and... pants and shirts?), it's very elegant and is super awesome, but it's hard to explain without getting a headache myself! Algebraic geometry: how do you solve equations the best way? Well, some people like to think of equations as shapes! One example is the equation a^2. If you think of it like a shape with sides a and a, with area a^2, it suddenly becomes interesting! It's an actual square! You can do this with many equations, like the quadratic formula, the equation for a vertex, and see how an equation like ax^2 +bx + c would relate to these things geometrically. ax^2 + bx + c... what does that even look like as a shape? Well... you have a square with a copies of itself, a line with b copies of itself... huh, thats not working. Perhaps I should turn it into a into a simpler equation (ax + d)(ax + e), you know, from the multiplies to c adds to b! Now, we can think of this as a rectangle with sides ax+d and ax+e! This is great because what are we doing? Converting an equation into a shape! Now, what algebraic geometers do is they literally study this for a living. They're steeped in this mindset and think of solutions of polynomial equations as shapes/geometries. Very complicated geometries, but geometries nonetheless! These are called algebraic varieties. Algebraic geometers will now use techniques from abstract algebra and toplogy to group many of these things together. Well, sadly I don't have the time nor knowledge to explain anything from the rest of layer 9, but I think this list of items should suffice! Measure theory (lebesgue integration and measures and paradoxical properties of sets) Stochastic calculus (calculus of randomness, brownian motion<- infinitely recursive random walks, ito calculus, esc) Tensor calculus (for theoretical physicists, the calculus of multilinear arrays, can be taught in differential geometry, is the most widely used model of general relativity. Why do I say most widely used model btw? Because there are two other formulations of gravity I've come across, one using vielbein fields and another using a specific lie algebra. I'll get to that in the next layer) Operator theory (what other things can integrals and differentials produce? Integral transforms (like laplace and fourier) come from this discipline, and physicists use many transforms to solve complicated calculus equations. Fun fact, old image processing used to use an integral transform with a dirac distribution as the kernel to figure out whether or not a line matches with a line in a photo) Spectral theory (advanced linear algebra with spectral decomposition and generalizing notions of eigenvalues and eigenvectors to much broader ideas of operators and mathematical spaces) Type theory (first developed as a logical alternative to set theory, it uses lambda calculus to formulate logical structure using these things called types. There are certain type theories, like Intuitionistic and Typed lambda calculus, but these are different than the one initially introduced by Bertrand Russell to avoid paradoxes in set theory.) Ramsey theory (uses combinatorics to study substructures of graphs given a structure of known size) (Continued...)

Layer 10: PhD in Theoretical STEM Many PhDs focus on research, but if you have a PhD in a more theoretical discipline you might learn about one of these areas of mathematics! Lie Theory (Connection between lie algebra and lie group using the exponential map. Lie theory can be taught to particle physicists, as it's useful in reconstructing the conformal spacetime group and connecting Einstein's equations and Hilbert's equations.) Noncommutative geometry (String theory, used to describe open bosonic strings) Homotopy theory (A discipline of algebraic toplogy which studies homotopy. Aka, how a shape changes over time, or more precisely, how many paths can be drawn within a region? Not sure if you can get a PhD in this, though...) K theory (an advanced version operator theory and linear algebra in general, it studies rings generated by vector bundles over a topological space or a scheme. Basically, it studies vectors (arrows) without geometric properties and with geometric properties on certain spaces/shapes) Category theory (you use this a lot in abstract mathematics, but studying it specifically is a very different thing and leads to many other theories. Basically, you are given an amount of objects and a morphism. What you study in category theory will be the morphisms between objects, their relationships, not the objects themselves. This is why it's useful! You can study the relationships between things. Before, they studied the objects themselves. Now, with category theory, you study the relationship of these objects? Why is this cool? Well, now, instead of thinking of objects as topological spaces or functions... think of them as categories themselves! How do you relate toplogy spaces to functions?! Boom! Functors. Functors are morphisms between categories. This is why category theory is called abstract nonsense. And we're not even done yet. I saw a video on category theory in neurology, which gave a reason as to why we can't see color differently. However, I never took notes. Computer science also uses category theory. How? I have no idea. Perhaps proof assistants come to mind, but the other ways they use category theory is beyond me...) Most of these are for more theoretical disciplines and there is a lot missing, but from what I've heard, this is the types of mathematics for theoretical disciplines! Layer 11: PhD in mathematics/Active areas of research This is the point where if you try to explain it without experience, you will be wrong and barely anyone can correct you unless they have a degree, so I'll do my best, but know that I can only speak in terms of wikipedia and my own personal reading and not experience! Also, I will speak in terms of general fields of mathematics that are still active and not on specific subdisciplines. Complex geometry (Complex variables but with algebraic geometry if you care, this includes hodge theory, symplectic geometry, kahler manifolds, and can encompass real and arithmetic geometry. Since complex geometry is basically just advanced algebraic geometry, I included it here) The langlands program (An active area of mathematical research that seeks to unify all of mathematics. This is not the only theory studying the fundamentals of mathematics, however. The langlands program first arose as a connection between number theory and algebraic geometry. It connected galois groups of fields to certain general linear groups.) Chromatic homotopy theory (and any subdiscipline of homotopy theory thereafter. This starts with the redshift conjecture, which says that the K theory of an E infinity ring R is a single chromatic level higher than R. There is also stable and unstable homotopy theory, stable homotopy theory is concerned with structure remaining after sufficiently many suspension functors, and unstable homotopy theory is just when they don't remain, I guess? There's also a ton more homotopy theory subdisciplines, like motivic and fractional, but I haven't read about any of them!) Homotopy type theory vs Univalent Foundations! (another type theory that uses the methods of higher category theory to formulate the study of logical structures. It arose that intensional type theory had a model in the category of groupoids (generalizations of groups to category theory)! Well, they needed higher dimensional models of type theory (as groupods are homotopically 1-dimensional, with sets being 0-dimensiomal), so they did exactly that through something called the univalence axiom, which they called "universe extensionality". This development of higher dimensional type theory allowed for the development of computerized assistants that can help with mathematical proofs. A proof assistant! They're still developing one called HOL4, but it looks like the current most useful one is probably Lean? Wikipedia says it can work with all forms of mathematical logic. In essence, HoTT'ers think of logical statements (a=b) as saying something about the relationship between two shapes that represent those symbols. In this case, the shape of a and the shape of b are toplogically the same. Note how this is different from algebraic geometry. For one, they study the properties of shapes, and two, they need equations. Equations being the same means they can now study the shapes themselves. However, in HoTT, you must derive the logic behind why these shapes are the same. It's not about the shapes, but about the fundamentals question of why they are related in such a way. Well, why did I put univalent foundations on here? Well, because HoTT is closely related to it! They're built off of the same structure, type theory, but they say something different. Univalent Foundations is more fundamental than HoTT and says that types and topoi are equivalent. I will go over that next! There are subdisciplines of HoTT, such as Modal HoTT, which aims to equip HoTT with modalities, but oh well. Haven't read about it. Onward!) Topos theory (A certain type of higher category is called a topos if it behaves like the category of sheaves of sets on a topological space. I.e. tracking the data of a set. Topos theory can also be described as... the theory of different universes in mathematics! Some universes of math will have different axioms and relationships than others, and topos theory wants to study exactly this. This means there are infinitely many mathematical theories attached to a single topos!) TQFT (Studies the topological invariants of quantum fields using category theory! There is also Homotopy QFT, which studies QFT's on these things called cobordisms, which is an equivalence relation between compact manifolds of the same dimension. There are also interesting books on something slightly related by Cecilia Flori on Topos Quantum Theory, but I've yet to read it and no one has done any research on it, so it seems like it failed to garner interest :,), but there are a non-zero amount of people currently studying it! I found two lecturers, Cecilia and William Donovan, as well as a third author, Benjamin Eva.) Level 12: newest directions in fundamental mathematics Olivia Caramello's Topos' as bridges. Using the concept of topoi, Olivia Caramello believes that Topoi can be used to spread knowledge and methods between areas of mathematics. This is called Olivia Caramello's unifying theory. She claims that topoi can connect infinitely many pairs of objects (theories), not just one! This is ripped straight from her website. IUTT (Shinichi Mochizuki's controversial theory regarding the abc conjecture. He uses approaches of anabelian geometry and teichmuller theory to construct an "arithmetic teichmuller theory for number fields equipped with an elliptic curve", Inter-universal Teichmuller Theory, the theory wouldn't be controversial if Shinichi Mochizuki wasn't a bitch! A few number thoerists decoded his absolutely insane mathematics (there was a joke someone's professor once told a few years before Mochizuki's papers in IUTT came out that he was a smart mathematician, but he was bad at communicating ideas) and found that corolarry 3.12 was left unproven by the paper. Recently, another mathematician, Kirti Joshi, tried to help prove his theory, but Mochizuki basically said he was also stupid and didn't understand his theory.) If I missed anything, feel free to let me know! And also, corrections about interpretations or easier ways to explain certain areas of mathematics is super useful! (Everything I've found is from personal rabbit holes mixed with my college math classes, so these hot takes should be taken with a massive grain of salt!)

Layer 10: PhD in Theoretical STEM Many PhDs focus on research, but if you have a PhD in a more theoretical discipline you might learn about one of these areas of mathematics! Lie Theory (Connection between lie algebra and lie group using the exponential map. Lie theory can be taught to particle physicists, as it's useful in reconstructing the conformal spacetime group and connecting Einstein's equations and Hilbert's equations.) Noncommutative geometry (String theory, used to describe open bosonic strings) Homotopy theory (A discipline of algebraic toplogy which studies homotopy. Aka, how a shape changes over time, or more precisely, how many paths can be drawn within a region? Not sure if you can get a PhD in this, though...) K theory (an advanced version operator theory and linear algebra in general, it studies rings generated by vector bundles over a topological space or a scheme. Basically, it studies vectors (arrows) without geometric properties and with geometric properties on certain spaces/shapes) Category theory (you use this a lot in abstract mathematics, but studying it specifically is a very different thing and leads to many other theories. Basically, you are given an amount of objects and a morphism. What you study in category theory will be the morphisms between objects, their relationships, not the objects themselves. This is why it's useful! You can study the relationships between things. Before, they studied the objects themselves. Now, with category theory, you study the relationship of these objects? Why is this cool? Well, now, instead of thinking of objects as topological spaces or functions... think of them as categories themselves! How do you relate toplogy spaces to functions?! Boom! Functors. Functors are morphisms between categories. This is why category theory is called abstract nonsense. And we're not even done yet. I saw a video on category theory in neurology, which gave a reason as to why we can't see color differently. However, I never took notes. Computer science also uses category theory. How? I have no idea. Perhaps proof assistants come to mind, but the other ways they use category theory is beyond me...) Most of these are for more theoretical disciplines and there is a lot missing, but from what I've heard, this is the types of mathematics for theoretical disciplines! Layer 11: PhD in mathematics/Active areas of research This is the point where if you try to explain it without experience, you will be wrong and barely anyone can correct you unless they have a degree, so I'll do my best, but know that I can only speak in terms of wikipedia and my own personal reading and not experience! Also, I will speak in terms of general fields of mathematics that are still active and not on specific subdisciplines. Complex geometry (Complex variables but with algebraic geometry if you care, this includes hodge theory, symplectic geometry, kahler manifolds, and can encompass real and arithmetic geometry. Since complex geometry is basically just advanced algebraic geometry, I included it here) The langlands program (An active area of mathematical research that seeks to unify all of mathematics. This is not the only theory studying the fundamentals of mathematics, however. The langlands program first arose as a connection between number theory and algebraic geometry. It connected galois groups of fields to certain general linear groups.) Chromatic homotopy theory (and any subdiscipline of homotopy theory thereafter. This starts with the redshift conjecture, which says that the K theory of an E infinity ring R is a single chromatic level higher than R. There is also stable and unstable homotopy theory, stable homotopy theory is concerned with structure remaining after sufficiently many suspension functors, and unstable homotopy theory is just when they don't remain, I guess? There's also a ton more homotopy theory subdisciplines, like motivic and fractional, but I haven't read about any of them!) Homotopy type theory vs Univalent Foundations! (another type theory that uses the methods of higher category theory to formulate the study of logical structures. It arose that intensional type theory had a model in the category of groupoids (generalizations of groups to category theory)! Well, they needed higher dimensional models of type theory (as groupods are homotopically 1-dimensional, with sets being 0-dimensiomal), so they did exactly that through something called the univalence axiom, which they called "universe extensionality". This development of higher dimensional type theory allowed for the development of computerized assistants that can help with mathematical proofs. A proof assistant! They're still developing one called HOL4, but it looks like the current most useful one is probably Lean? Wikipedia says it can work with all forms of mathematical logic. In essence, HoTT'ers think of logical statements (a=b) as saying something about the relationship between two shapes that represent those symbols. In this case, the shape of a and the shape of b are toplogically the same. Note how this is different from algebraic geometry. For one, they study the properties of shapes, and two, they need equations. Equations being the same means they can now study the shapes themselves. However, in HoTT, you must derive the logic behind why these shapes are the same. It's not about the shapes, but about the fundamentals question of why they are related in such a way. Well, why did I put univalent foundations on here? Well, because HoTT is closely related to it! They're built off of the same structure, type theory, but they say something different. Univalent Foundations is more fundamental than HoTT and says that types and topoi are equivalent. I will go over that next! There are subdisciplines of HoTT, such as Modal HoTT, which aims to equip HoTT with modalities, but oh well. Haven't read about it. Onward!) Topos theory (A certain type of higher category is called a topos if it behaves like the category of sheaves of sets on a topological space. I.e. tracking the data of a set. Topos theory can also be described as... the theory of different universes in mathematics! Some universes of math will have different axioms and relationships than others, and topos theory wants to study exactly this. This means there are infinitely many mathematical theories attached to a single topos!) TQFT (Studies the topological invariants of quantum fields using category theory! There is also Homotopy QFT, which studies QFT's on these things called cobordisms, which is an equivalence relation between compact manifolds of the same dimension. There are also interesting books on something slightly related by Cecilia Flori on Topos Quantum Theory, but I've yet to read it and no one has done any research on it, so it seems like it failed to garner interest :,), but there are a non-zero amount of people currently studying it! I found two lecturers, Cecilia and William Donovan, as well as a third author, Benjamin Eva.) Level 12: newest directions in fundamental mathematics Olivia Caramello's Topos' as bridges. Using the concept of topoi, Olivia Caramello believes that Topoi can be used to spread knowledge and methods between areas of mathematics. This is called Olivia Caramello's unifying theory. She claims that topoi can connect infinitely many pairs of objects (theories), not just one! This is ripped straight from her website. IUTT (Shinichi Mochizuki's controversial theory regarding the abc conjecture. He uses approaches of anabelian geometry and teichmuller theory to construct an "arithmetic teichmuller theory for number fields equipped with an elliptic curve", Inter-universal Teichmuller Theory, the theory wouldn't be controversial if Shinichi Mochizuki wasn't a bitch! A few number thoerists decoded his absolutely insane mathematics (there was a joke someone's professor once told a few years before Mochizuki's papers in IUTT came out that he was a smart mathematician, but he was bad at communicating ideas) and found that corolarry 3.12 was left unproven by the paper. Recently, another mathematician, Kirti Joshi, tried to help prove his theory, but Mochizuki basically said he was also stupid and didn't understand his theory.) If I missed anything, feel free to let me know! And also, corrections about interpretations or easier ways to explain certain areas of mathematics is super useful! (Everything I've found is from personal rabbit holes mixed with my college math classes, so these hot takes should be taken with a massive grain of salt!)

Well my brain hurts and my replies aren't showing so I guess I just wasted my time. ;~;

Thats not a proof. A proof doesnt just show something holds for a few cases it shows it is true for ALL cases. If I did find a counter example to that it would disprove it. But if I found examples no matter how many I did, it would not prove it.

What is bro trying to cook 🔥

That's just how adults are

Fr 😭

Yeah like where is the analysis at, where are the numbers at, where is counting and probability, where is stat, where is the monster (my favorite)?

You‘re so right. Stopping at linear algebra and calculus is like end of highschool level.

I think the most advanced is by far number theory. Whereas the others have practical use, number theory is very rigid in its use - the only example that comes to my mind is cryptography.

You did abstract your first yr of undergrad? That'd usually a 3rd/4th yr undergrad course

There is no way you learned rational expressions in 3rd grade

@@paulspielvogle290 im not joking💀💀

@@Midnightenlight HUH?

Maybe he just has a different school system. Otherwise he's lying

​@@Midnightenlight you're not joking, you're lying.