Terence Tao :What is his weakest area in mathematics ?  

A.G.I Will be man's last invention

Well in general, algebra and topology become really wild, abstract and mechanic and there is no trickery like there is in number theory or analysis. Ramanujan could never have pulled off in algebra what he did in number theory and analysis, because in algebra, by its reason for existence, one wants abstract representation, classification and canonical forms to simplify and determine solutions. Same for topology. Studying invariants does not leave that freedom to randomly concat concepts and bring forth shiny new identities. That is never an algebraic research subject of itself, rather it arises from the abstract invariant and classification study and is only emphasized upon when necessary to complete that higher task.

@@the_unknown.chronicler Just a guess by what I've heard from profs and students. Meaningful results in algebra come by just that much harder, the task of forming insightful structures from operations quantifies over much more than, say, going through an estimation, transformation or equation. In analysis courses 70-80% throughout assignments was usual majority of time, algebra, well, about onw third

@@fragileomniscience7647 I'd completely disagree with this sentiment. There is just an entirely different route to coming up with sweeping statements inside of these field, often times it comes from having good intuition and translating between different frameworks, understanding how to create a new framework and make it do what you want it to do. Id say that the analogue to terry tao for algebra and topology, is peter Scholze, and jacob lurie. Maybe the analogue of ramanujan here is alexander grothendieck

I doubt that topology is parity sensitive, it can't even tell the difference between donuts and coffee mugs! :)

​@@julianbruns7459 indeed. If the more recent developments in number theory are any indication, algebraic geometry would be the more likely candidate. Which, to be fair, is "culturally" quite close to algebraic topology

@@martiensventer9191 maybe knot, there seems to be an interesting correspondence between primes, class groups and knots and link groups, statistical evidence shows a really deep correspondence is going on, factorizations of numbers ~ "unravellings"

@@PerfectoidJosh Could you be more specific about what you mean by "unravelings" and "statistical evidence"? I'm currently learning about quantum groups, tensor categories and their relationship to low-dimensional topology, so if you have any references related to connections to number theory I would love to go check them out.

@@martiensventer9191 what I mean by "statistical evidence" is pretty much explained inside of Akshay Venkatesh's IAS talk on the subject, but large scale information about the factorizations of primes are related to linking. At a more general level though It's based off the fact that there are many properties of Spec(\mathbb{Z}) (who's closed points are primes) that make it act like its of dimension 3, for example Spec(\mathbb{Z}) \cup (\infty) has etale cohomological dimension 3, also that the etale fundamental group of that space is trivial, which makes one think about it as though its a sphere, and the idea is to also think about the embedding of Spec(\mathbb{F}_p) into Spec(\mathbb{Z}) as embeddings of a knot into the 3-sphere. and I've heard (though I'm not sure) that Peter Scholze has made this correspondence precise through his analytic stacks framework, where Spec(Z) is realized as some sort of 3-dimensional stack. For some references id recommend looking at the book "Arithmetic topology". For what its worth the statement that topology maybe useful for solving the twin prime conjecture was mostly made in jest, im not entirely such much number theoretic theorems could be proven through this correspondence. maybe not until there is theorems behind this

same. did that in elementary and highschool.

I mean yeah, it's pretty reasonable to translate geometric problems to algebraic problems and vice versa at that level.

I feel ya. If you are like me, your visualization ability is really bad... much easier to turn it into statements instead of visuals.

You would love modern geometry, then. Open a journal on algebraic geometry and you'll struggle to find a single picture

I've read that people who struggle with visualization or have aphantasia typically also struggle with geometry when they ask to rotate or the like. I have aphantasia and geometry is my weakest area as well!

It's easy

topology¡?

Both are good.

why you say perelman is smarter than Terence?

to come at such a conclusion, one must be able to read both their papers and hence be at an extra ordinarily high level intelligence him/herself.

No such classification is possible. There is no total ordering in Rⁿ with n >= 2, because you can always fix one component to order, while the other can be arbitrary. As such it doesn't make sense to compare in two different subjects. And randomly a proof may either be complex or simple, independent of ones intelligence, because mathematics is just that chaotic.

@@fragileomniscience7647 R2 can be ordered lexicographically. Mathematicians can also be ordered, perhaps not like well ordered (ordering of natural numbers) but more like pre ordering (ordering with equivalent classes) based on ingenuity of their ideas. Perelman Tao it's impossible for a mere mortal like me to distinguish between their intellect but definitely they are in a different space than say the average Mathematician.

dude your're in just grade 8 now, wait and see how the turntables

@@notbased8158 😂

Do you know what a Derham Cohomology is because that's the kind of algebra and topology he means 😂

He is talking about ultimate algebra which consider much symbols than normal

Innocent kid 🤣