Math general

>>8748188
>implying that anyone in this board is older than 15 and have a decent grasp of mathematics

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>>8748199
>implying that anyone in this board is older than 15 and have a decent grasp of mathematics

https://en.wikipedia.org/wiki/Psychological_projection

>Psychological projection is a theory in psychology in which humans defend themselves against their own unconscious impulses or qualities (both positive and negative) by denying their existence in themselves while attributing them to others.[1] For example, a person who is habitually rude may constantly accuse other people of being rude. It incorporates blame shifting.

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>>8748188
>what are you researching?
Topological superconductors and CFT.
>what are you studying?
https://arxiv.org/abs/1506.05805
>any good problems?
See pic.
>book recommendations?
Turaev for TQFT and Bernevig for TSc/TI.
>cool theorems?
Verlinde formula :DDD

>>8748203
using the projection meme that was literally invented by gay millennial fags on /hsg/
fuck off back to your containment board

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I've been trying to find the significance of the new bases for matrices A and B in Strassen's multiplication algorithm.
I've been trying to find a genuinely rigorous proof without just multiplying the matrices to see that it holds true, so you could say I'm looking for a way to derive the theorem or algorithm or whatever.
>pic related, the base matrices

>>8748209

Can you explain to me what is topological superconductivity?

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>>8748221
i dont know what /hsg/ is but you're obviously a brainlet if you think projection is a meme invented by a board on 4chan

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>>8748260
Topological superconductivity is superconductivity characterized by topological invariants, such as the Chern number. In general superconductivity arises due to closures of the energy band gap, and for many-body second quantized Hamiltonains with specific symmetries (such as P, C, T or combinations thereof) the topology of the Brillouin zone becomes important when characterizing these possible band gap opening/closures. For instance for a 2-dimensional Hamiltonian with PT symmetry, [math]n[/math] bands above and [math]m[/math] bands below the gap, the Chern numbers are given by the elements of the homotopy group [math]c^1 = -\frac{1}{4\pi}\int_{BZ}d^2k \operatorname{tr}\left(gdg^{-1}\right)^3 \in \pi_2(BZ) = \{BZ,S^2\}[/math], which correspond to the Hall conductances (in units of the flux quantum) across the system at zero temperature. In addition, these topological excitations are protected from the bulk in the sense that the edge modes can remain topologically nontrivial while the bulk transitions to a topologically trivial state.
Studying how these topological excitations braid and fuse with each other can tell you about the topological orders that exist in the system, and this is where category theory becomes useful. This can be used to characterize all possible topological materials in the world.

>>8748203
This is the equivalent of saying "I know you are but what am I?"

a^2 + b^2 = c^2 always helped me out in physics. Highly useful theorem imo

>>8748303

What are some of the most important consequences of your research? Faster hard-drives that can store more bits? Smaller CPUs?

I know this is a lazy question because I could just figure it out with a couple minutes of googling but on the off chance you have a more interesting answer, shoot.

Lads, I need some help.

I'm applying to the honors math program at school and I need to research the research that a potential advisors is doing so that I am prepared to discuss their work and ask them to be my advisor.

Unfortunately, I have only take one quarter of analysis, and I'm expected to apply after I take another quarter of upper div math.

How am I supposed to understand modern math research? How should I go about researching current maths so that I can ask for an advisor? I'm just sort of lost on what I should be doing.

>>8748470

This was discussed in the previous thread.

>>8728665

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>>8748462
Topological superconductors are an example of strongly correlated many-body systems, and these are extremely difficult to study and characterize in general. However if there is a systematic way of studying topological superconductors and their behaviour in general using some mathematical machinery (such as braided fusion categories, link invariants, etc.) then we may be able to extend some of the methodology to other closely related strongly-coupled many-body systems, such as the fractional quantum Hall effect and out-of-equilibrium localisation problems. Holographic techniques via AdS/CFT are also being explored (by a faculty I work with closely, in fact) towards this goal.
In the more practical side topological superconductors can be used to produce high Tc superconductors since the topological excitations (vortices, skyrmions, chiral Cooper pairs, etc.) are believed to be topologically protected from thermal fluctuations. In addition they are also (in principle)m believed to be topologically protected from decoherence as well, so there has been work towards using topological excitations as qubits in quantum computers as well.

How do you guys study from textbooks? Do you just read them because you'll remember everything? Take notes? Make diagrams? Capture the main ideas in bullet points? Something else entirely?

What about exercises? Do you usually find them challenging? Do they take a long time even though they're easy?

And in general, what should I expect of myself? I know I should be the one answering that question, but I don't know how, so I want to see what others expect of themselves and other people in general.

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>>8748586
Make memes on the internet.

>>8748209
CFT as in conformal field theory? If so, could you elaborate a bit on what courses are necessary to take in late undergrad/early grad school to tackle quantum field theory later on? Topology, analysis? What are the prerequisites that lead to QFT and TQFT?
I'm very interested from what I've seen so far (my analysis I professor work in statistical field theory) and I'd like to get a headstart.

>>8748586
Take notes of everything. I'm pretty much rewriting huge chunks of the book as I go and try to do all the proofs.
It's the only way it can stay with me along the way. If I just try to browse through a book, I read too fast and forget what it's saying from one page to the next. Or I'll understand what's happening but forget it half an hour later.

>>8748586

Take diligent notes, trying to summarize key points. Also work out the proofs and the examples.

Then do the exercises with the notes as a reference.

>>8748625

And how does your note taking process work? What do you write in your notes and what do you leave out? I usually basically get the gist of things while being as complete as possible (so if the book gives a bit of history on a problem, some motivations and tries to tie things together wither other topics from the book, I just leave that out). But I feel like that's hardly the most effective way, couldn't tell you exactly why though.

>>8748477

All I found in that thread were a bunch of categorists circle jerking and posting anime

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>>8748621
Yes CFT is conformal field theory. The most important courses to take for QFT is classical field theory, and supplement with a cursory knowledge of the theory of distributions and Lie groups. Functional analysis and topology also help.
For the physical kind of TQFT (Chern-Simons, WZW, etc.) you will need knowledge of manifolds and differential geometry. For the categorical TQFT you need to know a bit of everything in abstract algebra.

>been with girl for several years
>one of these "im a scientist! science is lief" types
>found out yesterday she's taken only up to "calc 2"
couldnt sleep

>>8748675
Drop her anon, and search for the mystical math cutie3.14 girlfriend. Godspeed.

>>8748655
Something alone the lines of Landau/Lifschitz volume 2 perhaps? I don't think such a course exists at my university so I might have to make do with self-studying with the help of friends in physics.
Also, what book on QFT would be indicated for someone coming with a mathematics background as opposed to a physics student? The same books that are popular with physics students?

>>8748636
I write all the definitions in full and try to find examples and counterexamples if there are none in the book. For the theorems, I write the statements in full. If the proofs are routine, I just don't write them, but if they're complicated I try to work them out by myself on draft paper or read the proof, then write a neat and succinct summary in my notes.
Don't be afraid of writing too much.

Does anyone in this thread know of a good Latex editor/writing tool ? I'm going to have to write a paper and I'd rather make it as easy on myself as possible. The only Latex I know is from posting here.

>>8748694
texmaker

>>8748694
You could always use Overleaf to do the formatting of the paper.

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>>8748687
LL is great if you want to feel like a retard. But I learned classical field theory and QM from LL as well and I am a retard so you may be able to do it too.
There are a few standard texts on QFT (Schrednicki, Schroeder, Bjorken and Drell, etc.) for physics students, and they all have some level of handwaving. This is inevitable since they all need to cover some form of Feynman diagrams and the functional integrals in their generating functionals are far from rigorously defined. Those books are good if you're fine with that but if you're not then I'd suggest you start with them and then move on to books on non-perturbative approaches in QFT.

good book on lambda calculus ??

>>8748612

>the proof is trivial

>>8748709
Thanks for your help, do you have a recommendation for a non-pertubative approach to QFT?

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>>8748738
Strocchi, Swieca, Streater, Wightman.

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I was supposed to do an independent study of modular forms with a professor next quarter, but I've lost all drive and passion for mathematics. I feel like up until now I've given up the most important things in life in a pursuit of absolute truth. It's similar to the way Grothendieck abandoned math because he felt "spiritually impoverished". (Of course, I'm nowhere near as driven or intelligent as he was.) I'm wondering if anyone else knows this feel.

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>>8748770
A clear goal helps to remotivate. I've been close to giving up a lot of times, but always just having some concrete thing to achieve has always given me back my will to study more.

>>8748770
i know that feel

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>>8748761
Thank you so much, you helped me more than you probably think. Here's a cute Wakaba for you.

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>>8748798
What is this warm fuzzy feeling inside me?

>>8748522
>strongly correlated many-body systems
>protected from thermal fluctuations
>(believed to be) protected from decoherence as well

Thank you for taking the time to respond. I'm sure you have better things to do than answer our trivial questions. Much appreciated.

>>8748694

Texmaker, like >>8748699 said

I just started using LaTeX and these were also helpful to get started.

https://www.youtube.com/watch?v=rT5kIQ-JHhw
https://tex.stackexchange.com/

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>>8748694
I personally use TeXworks. It's flexible but more bare-boned than other editors. I'm also looking into using emacs for typesetting.
>>8748856
>I'm sure you have better things to do than answer our trivial questions
I wrote that up within minutes because I've worked so closely with it. It's no sweat.

Math Major Physics Minor here,

Is this decision a terrible mistake?

Is there any supplementary reading I should be taking in order to learn faster? I'm just taking Calc 1 now and feel like I get all the material and want to learn more.

I am reading pic related.

No one has ever read it, the most anyone reads of ancient geometry is Elements. :(

>>8749062
>reading non-existent books
next level

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>>8749062
Pic

numberphile's new video

very humble guy, i like him :) no homo

https://www.youtube.com/watch?v=MXJ-zpJeY3E&feature=youtu.be

Currently reading elementary number theory by David Burton, really enjoying it so far...

>>8749047
Just make sure you get your integrations and derivations down for now. Calc 2 and 3 all rely on you really knowing this shit. Is there a reason you're taking physics as a minor? I'd consider maybe cs instead if you're not completely sure.

>>8749047
can second >>8749103- i took thermophysics for giggles and it'd have majorly helped to be fluent in manipulating derivatives, integrals, doing derivations of that sort

>>8748365
All I ever needed in my MIT physics class was y=mx+b, where'd you learn that one?

>>8749103
Because my school doesn't offer a Physics major and I do not have the capability to transfer. (For my own sanity, finances, and due to my gpa.)

I just really enjoy physics.

>>8749113
I think I have to take a few CSCI classes for my math degree, actually. Should be good

>>8749113
Yeah a few friends of mine actually ended up dropping thermo because of that.
>>8749127
If you enjoy it then go for it.

>>8748470
You're not going to understand their research. They're not going to expect you to.

>>8749047
>Math major
>Taking calc 1
Isn't it a bit early to have settled on a major?

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could use a bita help with M_{e <- B} (T) in part A. was asleep that lecture.

>>8749164
I'm in my second year and had to declare something

>>8749175
How does someone make it to second year before taking calc 1 and still get enrolled into a math major and physics minor? What did you do in first year?

>>8749176
General education and precalculus

>>8749168
Your pic reminded me just how little I remember from linear algebra.

>>8749178
I seriously hope you had a fucking wild year of partying.

>>8749181
mostly LAN parties

>>8749184
oof

>>8749178
holy shit,nice time get

>>8749199
You made me get all introspective and realize I basically did nothing but play video games my freshman year. Wew lad im depressed now

>>8749168
Find [math]U[/math] such that [math]U^{T}\mathcal{D}U = \mathcal{B}[/math] then calculate [math]U^{T}TU[/math].

>>8749208
Its good to reflect man

>>8748188
>>8747606
Il faut faire des maths si tu veux faire des maths pures, donc inscris toi au cours par correspondance de l3 de math de l'upmc...

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How do I learn Inter-Universal Teichmuller Theory?

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>>8749485
https://www.maths.nottingham.ac.uk/personal/ibf/notesoniut.pdf
>3. STUDYING IUT AND RELATED ASPECTS

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>Teenage math autists bow to Mochizuki
>In contact with frobenoids
>Rumoured to possess mathematical abilities
>Solved the ABC conjecture with an iron, but indecipherable proof
>Direct descendant of Minoru Mochizuki
>Will bankroll the first cities in between universes (Mochizukugrad will be be the first city)
>Owns basically every IUT research facility on Earth
>First designer babies will be Mochizuki Babies
>Said to have 200+ IQ
>Ancient Indian scriptures tell of one angel who will descend upon the Earth and will bring an era of enlightenment and unprecedented technological progress with them
>This is Mochizuki
>Owns Femtobot R&D labs around the world
>You likely have Shinichibots inside you right now
>In regular communication with the Archangel Grothendieck, forwarding the proofs of God to the book
>Learned fluent Algebraic Geometry in under a week
>Invented Inter-Universal Teichmuller Theory, a complex field of mathematics which only he can comprehend fully
>This ingenious development proved to be the last piece in the puzzle of time travel
>This lead to the design and construction of the first time machine through a joint effort between himself, Lockheed and the CIA
>Nation states entrust their proof reserves with the twins. There are no proofs in Ft. Knox, only Ft. Mochizuki

Oh fuggg....It's true...the prophecy's come true...

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>>8749570
Fucking kek

>>8749156
imo, coming in to a non-introductory physics class from a pure math perspective leaves you scratching your head at the derivations: for every 1 new symbol that's mentioned, there's like 5 ways it can be written in terms of 10 others. having your head on with the dimensional meaning of everything you're doing is important and if that guy enjoys physics and has interest/background in it then he'll steer clear of most of those pitfalls. but otherwise, without some physical perspective, doing something like deriving say the density of energy function can seem like totally arbitrary manipulations of symbols-- almost like writing the theorem after writing its proof.

Okay /sci/, Math plebe here but I feel like I just got jipped on my Calculus 1 Midterm. I got asked

"Define, using limits, what it means for a function 'g' to be continuous at 'a'."

So I read define and saw it as "Give me a verbal answer as to how to define continuity at g." So I wrote something along the lines of "The limits as you approach 'a' from either side are the same and g(a) is defined." But then I read "using limits" and so I wrote out the formulaic version of lim x->a- g(x) = lim x->a+ g(x) and where g(x) is defined.

Am I stupidly dense or does this feel like the question is worded weird?

I can do the maths just fine for calculating continuity and derivations and all that jazz, but it's hard to answer a question when I don't know what the question is asking. Does that make any sense? I couldn't tell if he wanted a verbal answer or a formulaic answer and I'm not sure if it's because of my stupidity or the way the question was written.

I am not smart enough to know how stupid I am so I don't need to be railed on about "omg this problem is so simple." Thanks.

>>8750407
It sounds like you just can't handle reading, dude. That was a perfectly reasonably worded question and then you started adding more words to it.

>>8750422
Okay I'm probably just autistic, thanks!

>>8750424
Since I feel like actually being helpful, a problem with undergrads (which you'll grow out of) is this weird discomfort with notation. The notation is totally irrelevant. The equations say the exact same thing you could say in words, and oftentimes people will use words. We like equations because they fit those long sentences into a compact form.

>>8750430
I didn't particularly mind the notation, mind. I just literally didn't understand which of the two he was asking for.

But in general, I should just go with equations because they're concise? Cool deal, thanks anon.

>>8749168
>the vectors have arrows on top of them
is this high school LA?

>>8750545
what the fuck do you do? bold font them?
sounds like more effort than drawing a line on paper

>>8750580
You use letters. Like this:
Let v be in R^n (or whatever you say in English for it is an element of it) Then v is a vector in your sense

>>8749158

I guess I'll just do my best to understand the general idea of what they're trying to do?

I just don't want to look like an unprepared doofus when I ask someone to be my advisor

>>8750007
I mean at least from my personal experience my calculus profs all referenced what we were doing to physics.

>>8750595
Honestly, that's probably out of reach. They won't think you're an idiot. You should talk about what you've learned and what excited you most, and then they can describe some facet of what they like that relates.

>researching
Nil and void, like my soul
>studying
Abstract Algebra from a Category Theoretical perspective
>book recommendations
Algebra: Chapter 0, understandable to undergrads who aren't brainlets yet contains lots of interesting material

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[math] \int_0^\infty x^{S-1}e^{-A\,x}\dfrac{1-(Z\,e^{-x})^M}{1-(Z\,e^{-x})}{\mathrm d}x = \Gamma(S)\sum_{n=0}^{M-1}\, \dfrac{Z^n}{(n+A)^s} [/math]

>>8750945

That's a cool identity but check this bad boy out.

[math]3\sum_{n=1}^\infty \frac{1}{4^n}=1[/math]

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Slow week.
Anybody else smokes?

>>8751034

i'm a retard

[math](n-1)\sum_{k=1}^\infty \frac{1}{n^k}=1[/math]

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>>8751034

[math] \dfrac{1}{1-q} = \sum_{k=0}^{\infty} q^k [/math]

eh?
Yes, I'm jelly.
You made me jelly. Mad jelly.
I'm gonna show you

[math] [n]_q := \sum_{k=0}^{n-1} q^k [/math]

[math] [n]_1 := \sum_{k=0}^{n-1} 1 = n [/math]

[math] \dfrac{1}{1-q} \int f(x) d_q x := x \sum_{k=0}^{\infty} q^k f(q^k x) [/math]

[math] \int x^m d_q x = \dfrac{x^{m+1}}{[m]_q} [/math]

[math] \left(\dfrac{d}{dx}\right)_q f(x) := \dfrac{f(qx)-f(x)}{qx-x} [/math]

[math] \left(\dfrac{d}{dx}\right)_q \, x^m = [m]_q\, x^{m-1} [/math]

[math] e_q(x) = \sum_{n=0}^\infty \dfrac{x^n}{[n]_q!} [/math]

[math] \left(\dfrac{d}{dx}\right)_q \, e_q(x) = e_q(x) [/math]

>>8750751

Alright, thanks. I'll do my best. Hopefully I get an interesting project and a cool advisor

>>8750838
I fairly intensely disliked Aluffi when I tried to read it
I don't know if people here genuinely like it or if it's just memed because muh categories

You're not learning "Abstract Algebra from a Category Theoretical perspective", you're learning foundational algebra with occasional breaks to rephrase everything in very (very) primitive category language which serves no purpose other than ramping up the pretension for 3/4 of the book

Not to mention his writing style is aids

>>8751119

*pushes glasses up*

Nice try kid...

Did you know that [math]\sum_{n=1}^\infty \frac{1}{n^x}[/math] converges when [math]x \geq \sqrt2[/math]?

...that's what I thought.

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>>8751187
*unzips notebook*

For all functions [math] f [math],

[math] \int_{- \sqrt{2} }^\sqrt{2} f(x^2) \dfrac{1}{1 + {\mathrm e}^{x^2\sin(x)}}\,{\mathrm d}x = \int_0^\sqrt{2} f(x^2) \,{\mathrm d}x [/math]

For all functions [math] f [/math],

[math] \int_{- \sqrt{2} }^\sqrt{2} f(x^2) \dfrac{1}{1 + {\mathrm e}^{x^2\sin(x)}}\,{\mathrm d}x = \int_0^\sqrt{2} f(x^2) \,{\mathrm d}x [/math]

I need a good elementary algebra book, but with more elaborated set of exercises.
Schaum and Khan Academy are easy

Brainlet here
I was exploring different things you can do with absolute values when I found out this problem

Does the following evaluate to 1,-1, both, or undefined?

[eqn]\lim_{k \to \infty} f(x) = \frac{|sin(kx)|}{sin(kx)}[/eqn]

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>>8751279
How hard do you want?

Andreescu has at least one problem book for everything you could do within the bounds of high school math to know and they're mostly very well chosen. But they're tough.

If you just want not baby tier it's probably enough to just work through that Lang meme book

>>8751279
Go through Gelfand or something similar.

>>8751281
It's not defined. A limit cannot evaluate to "both"; limits are always unique. There's either one or none.

Roughly speaking the problem is that no matter how big k gets, it doesn't help the values of f get any closer to each other; f never stops oscillating between -1 and 1. Since your values aren't approaching any one number, there's no limit.

>>8751289
I mean this elaborated

[math]\frac{(81^2)^\frac{1}{4} \cdot \sqrt[5]{32^{2}} \cdot 125^\frac{2}{3}}{\sqrt[3]{27} \cdot \left (\frac{2}{3} \right )^{-3} \cdot \left ( \frac{9}{4} \right )^{-2}}[/math]

powers, roots, equations, inequalities only

>>8751335

That looks like a Brilliant.org question. It might be what you're looking for.

bogpill me on algebraic geometry

>>8751581
We'd like to solve polynomials. Turns it it's pretty hard.

>>8751635
What are the applications of solving these polynomials?

>>8751684
Have you never wanted to know when a function is zero? It comes up literally all of the time in every technical field. Not to mention there are reasons for studying things beyond immediate practical applications.

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>alg geo is about solving polynomials
>scientists use alg geo to find roots of polynomials
>this fucking retarded

>>8751187
it already converges for x > 1

Who /topos/ here?

>>8748203
>>>www.reddit.com

>>8751123
I got accepted to one of the best grad programs in my country and I just emailed profs at the school saying I'd like to study something combining analysis, algebra, and topology, and said their research looked cool while attaching a list of courses Ive taken. I generally received warm responses.

>>8751937
Fuck off Grodendieck

>>8751684
One good application of abstract algebra (and also algebraic geometry) is in symbolic computation. The algorithms that allows maple to quickly multiply integers, factor polynomials and so fourth depend on algebraic/ number theory notions. Anything more complicated than that depends on even more sophisticated math.

It also sounds like you've missed the point of math. The point isn't to produce machinery that other people will use, it's to come to conclusions and connect things together coherently. It just so happens that the occasional "useful" theorem will be sound.

>>8751259
that woman is literal pleb

>>8750945
Bitch looks like a fucking alien lfmoa

Im want to go back to uni after 8 years but my mid and highschool math knowledge has rusted.

So I just finished serge lang basic mathematics. What should I do next? Is there anything missing that i need to know?

>>8751980
>The point isn't to produce machinery that other people will use
>it's to come to conclusions and connect things together coherently

This is a myth and you're regurgitating grothendieck. All great mathematicians for all of history worked as astronomers doing tables, or for the government doing statistics, taxes and military logistics or for banks doing finance.

No one actually does math for math's sake unless they're independently wealthy (fermat, pascal) or part of the priestly class (the pythagoreans) and can afford to do it for fun. People being able to write cute little treatises on this and that just doesn't make up the bulk of math and you're lying to people when you say "that's not what math is *really* about." You know who employs the largest number of mathematicians in the country? The NSA. That's military money, military research. Besides, there isn't a single "pure" math problem that can't be traced to a "real world" motivation. I mean, do you really think category theory is about "concepts" and "relations"? No! It's transitivity checking! You're finding invariants! We've been doing that since Planck started trying to make light bulbs cheaper.

I mean really, what you're doing when you say things like "the point isn't to produce machinery that other people can use" is covering your eyes and saying "The world is truly a dark place! You just don't understand!"

>>8752212
>Besides, there isn't a single "pure" math problem that can't be traced to a "real world" motivation.
Can you trace the Collatz conjecture to a real world motivation?

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https://en.wikipedia.org/wiki/Cumrun_Vafa
>this dude's name is literally cum run

>>8752220
airthmetic is useful for counting things

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>>8752355
Very vague response.

What does the Collatz conjecture count exactly?

>>8752358
>traced

WHERE DO I GET A BOATLOAD OF PROBLEM SETS TO WORK ON IN ONE PLACE?

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>>8752362
Feel free to trace it back to something then.

>>8752371
all the operations used in it can be used for counting things
no one claimed the problem itself is useful

>>8752365

this is fun to work through

http://www.isinj.com/mt-usamo/250%20Problems%20in%20Elementary%20Number%20Theory%20-%20Sierpinski%20(1970).pdf

>>8752380
>addition, multiplication and division are useful for counting things
wow you sure convinced me all pure math can be 'traced back' to 'real life' applications

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>>8752392
cheers, nigger!

>>8752393
I'm glad

>>8752396
hi glad im dad

>>8752393
>>8752380
>>8752371
>>8752358
>>8752355

This is not >>8752212

You're talking to a different anon. Just to make that clear.

I'm not going try and prove that the Collatz conjecture was motivated by a real world problem. Obviously that's a wild goose chase. It's not even clear when it was originally introduced to the mathematics community. Wikipedia's claim of 1937 has no evidence to support it. Anyway- I have no idea where or not it was. That doesn't change the point I was making originally. There are a lot of toy problems that are made because they're fun to play with, hard to solve, or unsolvable. Obviously puzzles play a role in some of mathematics. Who doesn't have fond memories of solving puzzles as a child? Or showing friends new puzzles?

My point remains that this post >>8751980 doesn't paint a historically accurate picture of mathematics and instead favors a narrow, recent *trendy* point of view which, like I said in the previous post is only a point of view people have if you can afford to have it. It's an ivory tower perspective. It's not a universal view of mathematics, despite often claiming to be. "Math is about relations and analogy" often goes hand in hand with "Math is about freedom", and so on. These people often think they're literally doing you a service but end up restricting your thinking to their own formalism. "No, you really need to understand it from the categorical perspective, then you'll really understand it" is one I keep seeing popping up.

There is such an enormous amount to be learned from what is look down on and trivialized as "applied math" or "conceptually simple" and this attitude encourages that kind of unwarranted and narrow-minded snobbery that probably discourages people from talking about something other than the literary side of mathematics. I know this might be hard to believe but some mathematicians actually like numbers and shapes.

>>8752395

You're welcome my friend.

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>>8752421
I don't know why you went on such a rant again, all I was pointing out was that you're essentially doing the same thing you're criticizing just in the opposite direction. You have such a personal backlash against the abstract, birds-eye-view motivation that you just throw everything towards the applied instead
> Besides, there isn't a single "pure" math problem that can't be traced to a "real world" motivation

That's all I was pointing out

Also
>No one actually does math for math's sake unless they're independently wealthy (fermat, pascal) or part of the priestly class (the pythagoreans) and can afford to do it for fun.
is patently false, whether you look at all the amateurs who tried (and still try) to solve famous problems like Fermat's Last Theorem in their spare time, or even Yitang Zhang who before becoming a lecturer spent his spare time reading arithmetic geometry journals when he wasn't working at Subway, or any of the other many examples...

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Okay guys, I'm doing my master's thesis on the Mitchell embedding theorem, and I'd like to show how it can be used to generalize diagram lemmas. If I choose some of these results, say the five lemma, is it ok to just say it can be proved by diagram chasing, or should I explain what the idea of diagram chasing is, or should I have the snake lemma accompany it and prove one and tell the reader to chase the other?

>>8752437
>>No one actually does math for math's sake unless they're independently wealthy (fermat, pascal) or part of the priestly class (the pythagoreans) and can afford to do it for fun.
is patently false

Ooooh, "patently" false. Well now I know for certain that it's false. It wasn't that it was hyperbole that was enough or that anyone with brain cells would have been able to come up with at least one example that could have contradicted it. Good thing you added patently. Now we all know for sure.

See, the big stinky problem with your reply is that you think I am against abstraction and a bird's eye view when really I am against narrow-mindedness and trend following, which you conveniently didn't read, even though I explicitly stated in the post that you are directly replying to but not quoting. Abstraction and bird's eye view have their place. I'm not against them. Everything has it's place. No harm was done when asking what the applications of polynomials might be. No one's going to get hurt if they approach algebraic geometry with numerical questions instead of wondering about predicates and their relations. For all you know I could be studying category theory! Abstraction could be my bias (spoilers: it is)! I could actually despise applied math (more envy it). Did you ever consider that? My point would still be the same. Mathematics is much larger than what your professor and and that reallllly smart guy whose using a lot of advanced sounding terminology thinks. Math should not be a social event where you fall prey to peer pressure (watch this get misquoted as "math should not be a social event"). Don't follow trends. Think for yourself.

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>>8752564
>Think for yourself.

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>>8752564
there's a difference between hyperbole and making blatantly incorrect blanket statements

this is why its ironic when you say you're against narrow-mindedness

the rest of your post is a bunch of rambling, with you hilariously still making the same mistake that you're criticizing

>>8752600

I don't know how to talk to you if you're not going to read my posts, take the very lame and uninteresting cop-out of, "well he's not using well-formed formulas so I can just pretend to not understand him" and dismiss it as rambling. I guess we're going to have to end this conversation. Too bad.

>>8752617
>I guess we're going to have to end this conversation
if you insist

feel free to come back when you understand the difference between hyperbole and incorrect statements, and can manage to make a single post where you don't contradict yourself

>>8752497
Why not just prove one via diagram chasing and say the other proofs use the same method?

>>8752638

I'm pretty sure he's being facetious.

>>8752643
Heh, you never know.

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>>8752638
>or should I have the snake lemma accompany it and prove one and tell the reader to chase the other
That's what I meant. I'd prove the five lemma and say the same method is to be used for the other results. This is probably the best way.

Anyone work with linear algebraic groups?

What would be a decent book for someone that has to learn highschool math?

>>8752839
What are you looking to learn it for? Just for your highschool classes? Or in hopes to do higher math?

>>8748612
Why should I solve the problem when it's not kurisu?

Eventually ready for computer science math.

>>8752212
Historically this is true, yes. The context of my comment is within your comment about algebraic geometry and in general pure mathematics. If you're going to paint the picture of mathematics as a whole, then really the point of math at the end of is whatever it means to you. To me, it's a means to itself. I acknowledge that other people expect application. Other people expect several things from me but I only care to the extent that I am not harmed in any way.

>Besides, there isn't a single "pure" math problem that can't be traced to a "real world" motivation.

Are you telling me that all of the theorems and lemmas that go into solving pure math problems are intended to be applied directly to the real world? No. All of that machinery gives the problem a sound logical context. That's the point of the mathematician.

My thesis advisor works in harmonic analysis and specifically has been proving density results for the support of certain functions. His work can be traced back to Fourier analysis which in turn is based off of the vibrating string. At no point is his work being directly applied to the real world. He's just showing when these functions attain certain properties in a mathematical context.

>>8752421
I didn't say anything about applied math. I was talking specifically about pure math. Again, I thought the context would've made that clear.

>what are you researching?
https://psychicapparatuses.wordpress.com/2017/03/16/topoi-over-algebraic-bases/

(...Among other things, including a pretty sharp method of knot tabulation and some work on dynamical systems using synthetic differential geometry.)

>what are you studying?
Mostly higher Chern-Weil theory.

>any good problems?
I couldn't intentionally solve a problem if my life depended on it, but a knot calculus I developed helped my resolve the problem of the additivity of crossing number under knot sum.

>book recommendations?
I don't know, probably Lurie's DAG stuff. All of it is on arXiv, so you don't even have to pirate it.

>cool theorems?
de Rham cohomology is literally infinitesimal simplicial cohomology in synthetic differential geometry, and the proof of Stoke's theorem almost becomes a one-liner.

>>8753054
>>8752421
Perhaps another point I would like to add, is that applied mathematics is a lot less applied than applied mathematicians sell themselves to be.

>>8752991
>>8753021
whoops this was for you.

yeast

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Why not just have supercomputers put mathematical symbols (and words too for proofs) randomly together following some loose syntax to exclude the most obviously impossible and then automatically evaluate for logical validity and numerical correctness using verification and computation?

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>>8752497
Learn what your audience is. Who's on the defense committee? If there are non-experts on what you're doing then you better write out everything.

>>8753498
for what end?

>>8751054
trying to quit

littledragonbitchcrab.gif

>>8752497
If they are familiar with the fact that diagram chasing works in the classical case (in RMod), then just use the fact that Freyd-Mitchell guarantees a fully faithful functor. All of the universal constructions are thus reflected by the inclusion, so if they accept that diagram chasing works for modules, they must accept that it works in general. As >>8753513 said, you'll have to gauge the audience to determine if an explanation for diagram chasing (in the classical sense) is necessary. Good luck Animenon, be sure to report back when things go well!

>>8753498
a neural net to surpass memeuchzuki?

How to start understanding Lie algebra? What are we operating on and what does the lie bracket operation even mean??

Ive been staring at the first page of a lie alg book for a month

>>8753551
its literally just an algebraic structure of a vector space with a product satisfying certain identities

what are you confused by?

>>8753551
Lie algebras are just the tangent space of Lie groups at the identity lmao it's very simple

>>8753551
If it helps, look into enveloping algebras. See why the commutator of an associative algebra yields a Lie bracket.

A Lie algebra is just the tangent space at the identity in a Lie group, and the Lie bracket is what you get when you start differentiating vector fields (and tensor fields in general).

>>8751275
How the fuck did you prove that?

>>8751275
cute

let g(x)=exp(x^2sin(x)). then g(-x)=1/g(x).

[math] \int_{-\sqrt{2}}^0{\frac {f(x^2)}{1+g(x)}}dx= \int_{0}^{\sqrt{2}}{\frac {f(x^2)}{1+1/g(x)}}dx=[/math]

and [math] {\frac 1 {1+1/g(x)}}={\frac {g(x)}{1+g(x)}}=1-{\frac 1 {1+g(x)}}[/math]

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Redirecting into here because I didn't know this was a thing

>Taking discrete
>Literally every other word out of the profs mouth is "But we can also use this symbol instead"
>There are straight up 4 symbols for every operation and the prof insists on using her own made up shit
>She calls cross-products "mutlipladd" because you multiply and then add instead of just calling it a fucking cross product
Can somebody PLEASE for the love of fucking god recommend a book that covers this shit in a way that isnt completely retarded?
I'm going to lose my goddamn mind

>>8753054

Thank you for being civil with me. Looking back I was hysterical. Sorry for that. You're not as narrow minded as I accused you of being originally.

>his work can be traced back to Fourier analysis which in turn is based off of the vibrating string

That's how it's introduced in many textbooks and lecture notes, but what I remember is that Fourier analysis was motivated by an attempt to unify functions as his claim was that any function (what was considered a function in the 19th century) can be approximated with a combination of sine and cosine functions, which were known before Fourier, just not whether they could be used to approximate any function. I don't know what proportion of this was motivated by aesthetics, commercial or military interests, to advance mathematics as a whole, or some mixture of all of the above.

I'd like to repeat that I appreciate you being so civil in your response. I will be more level-headed before I post next time and make more of an effort to be specific and not crude. I am sorry if I offended you. I regret my rudeness even more now that I see you are actually reasonable and open minded.

>>8753654

Mutlipadd is just plain retarded, maybe check if your prof has autism. But afaik math has a lot of jargon for specific shits.

>>8753654
>be russian
>study math in germany
>most material is in english
>have to memorize 4x times the terminology and ambiguities
>trying not to kys myself very hard

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>>8753513
>an algebraic topologist
>a homotopy theorist
>anyone who downloads it while browsing the archives
The first two will check it, the others are people who, if lucky, have a course or two in algebraic topology, and would then be familiar with the method. Otherwise, they will be like "nigga what the fuck?" if I skip them all. Every passing moment I'm even more confident about having a proof for one of them to show what to do, and then get the others done by saying the proof is a diagram chase.

>>8753546
The only two certain readers are. The easiest way to generalize them would be to have the diagram of the lemma at hand, then express the claim in an equivalent diagrammatic form, for example that if the second and fourth vertical arrow are epimorphisms and the fifth is a monomorphism, then I'd have the exact sequence

[math]0 \to \text{ker\ }f_i \to A_i \to B_i \to 0[/math] for [math]i=2, 4[/math], and one more for the fifth, and this would be the diagrammatic statement [math]P[/math]. That I'd have the same sequence exact for [math]i=3[/math] would then be [math]Q[/math]. The full metatheorem, as Freyd calls it, then guarantees that [math]P \rightarrow Q[/math] is true in all "fully abelian" categories if it is true in all module categories, which it is. By the embedding theorem, all abelian categories are fully abelian, so it holds for all. A similar reasoning would then not only prove the monomorphism claim, but actually work for all such theorems/lemmas. Or am I doing something wrong?

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>>8753006
Here.

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>>8753823
And the formatting of my post sucks, sry.

[math]0 \to \text{ker } f_i \to A_i \to B_i \to 0[/math], [math]i =2, 4[/math]
and
[math]0 \to A_5 \to B_5 \to \text{coker } f_5 \to 0[/math]

If these are exact ([math]P[/math]), then ([math]Q[/math]) the sequence

[math]0 \to \text{ker }f_3 \to A_3 \to B_3 \to 0[/math]

is exact.

>>8753823
Hmm, it sounds right to me at least. I'm not used to these sorts of theorems in category theory, so I'm not familiar with this sort of proof.

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>>8753853
Yes, I think that should do it too. Here's a pic of how Freyd defines stuff. How's life, by the way?

>>8753801
Yeah I understand that math is defined locally, but she is fucking worthless as a teacher.

i like Linear Algebra

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>>8753936
This is a really cute post

>>8753916
I didn't realize that theorem was that powerful! Wow, that's awesome.

Yeah, I think what you have will do quite nicely. And, life is pretty good! I'm in contact with a small group of students learning category theory and homotopy theory in Chicago, and we've had some cool conversations. My research as a visiting scholar is going very well, and we should be able to produce some concrete results in the next month or two. I've gotten my friend hooked on knots theory and some combinatorial problems, and we are developing a formula giving the number of knots with a given crossing number (it employees Alexander's theorem, reduces things to a simpler case by collapsing all crossings into a meta-crossing, and then the problem comes down to counting certain elements in braid groups modulo a commutation relation).

Yeah, life is good. How are you?

>>8753950
(Pardon the typos, I'm pretty baked.)

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>>8753916
>very Abelian categories
>Abelian categories
>Ab-categories
>yfw they're all different objects

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>>8753950
That sounds pretty cool, actually! My life's pretty much just studying, sometimes watching meme videos, walking around and shitposting. Not that I'd complain about it, though.

>>8753984
Fear not! The first embedding theorem says:
>every small abelian category is isomorphic to an exact full subcategory of [math]\textbf{Ab}[/math]
Equivalently:
>for every small abelian category [math]\textbf{A}[/math], there is an exact embedding [math]\textbf{A}\to\textbf{Ab}[/math]
Equivalently:
>every abelian category is very abelian

>>8753984
Anon, think of it this way: abelian categories are the "topoi" of Ab-categories (which also include deloopings of abelian groups, groupoids corresponding to the 2-truncation of simply connected homotopy types, et cetera). All abelian categories are fully abelian, so they don't show up in the literature.

>>8754022
I misspoke, I meant to say 2-truncations of fundamental [math]\infty[/math]-groupoids of simply connected spaces.

>>8753679
Not the anon you're replying to, but I'm pretty sure that Fourier series developed out of Fourier's solution to the heat equation (a partial differential equation).

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Maths is safe space of coherence created by normies once they whine that what they call reality is not enough coherent to them.
Math is the step beyond the one done by normies with their little legal rules, justifying them from what they experience, by some rationality, common sense, necessity and other spooks like progress (and whine when they see that most people do not care about their little rules even if the first people manage to enforce them)
Then they get butthurt when some guy not spooked about all these spooks recall them that all these people is projecting lots of feelings and fantasies or just create other spooky formal languages after the same intention.
Of course, these normies create an idea of ''accuracy of my formal language with what I experience'' because deep down people know that their little inferences are just the result of their imagination, so they crave some spook called ''non-human objective third party '' which would make everybody agree on anything while shitting on empiricism, because ''my senses dupe me since the straw bends in water, but not in the air, thank יהוה for giving the faculty of reason I can totally see the real reality now''.

>>8754942
nice meme

i stopped reading after
>''my senses dupe me since the straw bends in water, but not in the air, thank יהוה for giving the faculty of reason I can totally see the real reality now''.

>>8754942
logic is a spook, i know this because i thought logically

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The (compactly supported) Alexander-Spanier cohomology is constructed quite nicely, very intuitively.

The set [math]\Phi^k(X; G)=\text{Hom}_{\textbf{Sets}}(X^{k+1}, G)[/math] of [math]k[/math]-functions into an abelian group [math]G[/math], which itself is an abelian group. The groups are defined to be trivial for negative indices.

Using the [math]k[/math]-diagonal [math]\Delta^k_X=\{ (x, \dots, x)\in X^{k+1}\ |\ x\in X\}[/math], one can define local equivalence for the [math]k[/math]-functions the following way: [math]\varphi, \psi \colon X^{k+1} \to G[/math] are locally equivalent if there is an open [math]W \subset \Delta^k_X[/math] such that [math]\varphi |_W=\psi |_W[/math]. A trivially local [math]k[/math]-function is a [math]k[/math]-function locally equivalent to [math](x_1, \dots, x_{k+1}) \mapsto 0[/math]. One then defines [math]\Phi^k_0(X; G)[/math] to be the normal subgroup consisting of locally trivial [math]k[/math]-functions.

One can also define [math]\Phi^k_c(X; G)=\{ \varphi \in \Phi^k(X; G)\ |\ \text{supp}(\varphi) \text{ is compact}\}[/math] to get a subgroup consisting of all compactly supported [math]k[/math]-functions. Since [math]\varphi, \psi[/math] locally equivalent implies [math]\text{supp}(\varphi)=\text{supp}(\psi)[/math] and the empty set is compact, one has [math]\Phi^k_0(X; G) \subset \Phi^k_c(X; G)[/math]. This allows defining [math]C^k(X; G)=\Phi^k(X; G)/\Phi^k_0(X; G)[/math] and, for the compactly supported case, [math]C^k_c(X; G)=\Phi^k_c(X; G)/\Phi^k_0(X; G)[/math].

One defines the coboundary map [math]\delta^k \colon \Phi^k(X; G) \to \Phi^{k+1}(X; G)[/math] by [math]\delta^k\varphi (x_1, \dots , x_{k+2})=\sum\limits_{n=1}^{k+2} (-1)^{n+1}\varphi (x_1, \dots , x_{n-1}, x_{n+1}, \dots , x_{k+2})[/math]. This gives rise two to two cochain complexes:

[math]0 \to \Phi^0(X; G) \to \Phi^1(X; G) \to \cdots[/math],

and, restricting to [math]\Phi^k_c[/math],

[math]0 \to \Phi^0_c(X; G) \to \Phi^1_c(X; G) \to \cdots[/math].

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>>8755642
This, then, induces cochain complexes

[math]0 \to C^0(X; G) \to C^1(X; G) \to \cdots[/math]

and

[math]0 \to C^0_k(X; G) \to C^1_c(X; G) \to \cdots[/math].

Defining things the usual way, one gets [math]H^k(X; G)=H^k(C^*)[/math] and [math]H^k_c(X; G)=H^k(C^*_c)[/math].

That this works for the case with compact supports follows from the fact that [math]\text{supp}(\delta^k\varphi)\subset \text{supp}(\varphi)[/math].

I think this is cool, and more intuitive than the construction of for example singular cohomology.

>>8755664
[math]C^0_k[/math] should be [math]C^0_c[/math]

>>8755642
whats this shit and what do you use it for? You can talk to me like I know Analysis, lin algebra, combinatorics and proof theory

>>8755694
Don't indulge the categorists' ego, or they'll keep circlejerking the math general to death with their anime and the abstractions only they understand.

>>8755642
Why are you masturbating in public instead of trying to have any kind of disussion?

>>8755705
>with their anime
They do this show show their sexual orientation, its a "code" in the gay community.

>>8755705
Are you serious? All kinds of mathematicians have used some sort of cohomology theory to great effect.

>>8755705
>the abstractions only they understand

If only you knew what they were "really" talking about you wouldn't see it as abstract. Don't conflate abstraction with your own ignorance. They're just talking about physics.

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>>8755733

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>>8755694
That's algebraic topology. You take a topological space, you try to do something using the tools of topology, realize it's too hard, and you have two choices: give up and get depressed, or try solving your problem using a bit of algebraic structures.

Assuming you chose not to give give up, you would then have a nice arsenal of different methods at your disposal. Among these you have homology and cohomology theories that share the core idea but differ otherwise. In a homology theory, you would construct an algebraic structure (groups, modules, vector spaces) for every non-negative integer n such that your construction somehow resembles the notion of dimension, or n-dimensional holes. This allows you to sort of decompose the space into a chain of algebraic objects. You will also have a fixed map from the object of dimension n to the object of dimension n-1 such that the composite of any two subsequent maps takes everyting to 0. This sounds complicated but makes proving stuff a lot easier.

Cohomology is the same in the sense that you, again, construct an algebraic structure for every non-negative n, but this time the associated maps go from n to n+1, like with Alexander-Spanier. This is more complicated than homology, but also yields more powerful results.

I hope this clarifies things a bit. It's hard (for me) to explain this properly when you don't know topology or algebra.

>>8755705
That you don't want to discuss it doesn't mean no one would.

>>8755829
What's a cool introduction to category theory for someone that only knows basic undergrad algebra/topology? MacLane is a bit difficult right now but I still want to do what it's all about. I want to learn algebraic topology and algebraic geometry in the future so I thought this might help.

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>>8755921
I used Awodey's "Category Theory" and Freyd's "Abelian Categories", but I've heard Borceux's "Handbook of Categorical Algebra" is good, too. You should ask someone else, probably. That OHP guy would surely know. Mac Lane is probably very good if you want to focus on algebraic topology and geometry.

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>>8755972
Thanks for your recommendation anon, I'll ask him when he's around. Here's a cute Yotsuba for you.

>>8755989
I heard conceptual mathematics is a good first book

>>8756043
I thought it was a project that grew out of nand2tetris? Do you have a link or something anon?

>>8755642
>>8755664
Can this be made into a sheaf cohomology theory like singular,derham,etc. cohomology?

>>8756066
https://www.amazon.com/Conceptual-Mathematics-First-Introduction-Categories/dp/052171916X

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>>8756071
https://books.google.fi/books?id=JFXSBwAAQBAJ&pg=PA185&lpg=PA185&dq=alexander+spanier+cohomology+sheaf&source=bl&ots=TAVTABqSND&sig=O34wqkKBtZwL8S3v7GNPekuj6AI&hl=fi&sa=X&ved=0ahUKEwj-vsOGtd7SAhWFjSwKHcsGAt0Q6AEIbDAI#v=onepage&q=alexander%20spanier%20cohomology%20sheaf&f=false
It seems so. I just started reading about it today, so I don't know too much yet.

>>8755921
I don't really think category theory is a thing that can be learned "beforehand" so to speak, that is before you practically need to.

You need two things to decently grasp why categories are useful; a reservoir of examples from all over mathematics (a decent undergrad degree's worth is enough) and a subject (say, algebraic topology) where you really need the language to do shit properly.

It's one of those things where once you understand it you're going to think "all that would have been SO much easier if somebody told me this first" but in reality your reaction would have been "why is my teacher overcomplicating everything with all this wank".

This isn't a consensus or anything, but anybody who disagrees with me is wrong.

Mac Lane is still _the_ category theory book all these years later; if you can't understand it, come back later.

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>>8755642
Interesting.
A wavefunction [math]\psi[/math] in quantum theory can be considered as a Lie algebra-valued section on the Hermitian G-line principal bundle [math]G \rightarrow P \rightarrow M[/math], and therefore [math]\psi \in \Psi^{k}(X,\operatorname{Lie}G)[/math] if we can write [math]M = X^k[/math], such as the case for the [math]k[/math]-body Euclidean space [math]M = (\mathbb{R}^n)^k[/math] or the torus [math]M = (S^1 \times S^1)^k[/math]. The diagonals in these manifold can be considered as the points at which each of the [math]k[/math] particles have identical positions, which are forbidden by Fermi statistics if we're interested in electrons. This means that either we have to exclude the diagonals [math]\Delta[/math] from [math]M[/math] or identify any (Fermi) wavefunctions that agree on the diagonal. Is there a way to extend Alexander-Spanier cohomology to global equivalence, where I define my equivalence classes of [math]k[/math]-functions if the representative coincide on the entire [math]\Delta[/math] instead of just an open subset? How would that change the cohomology structure?
If we are able to do this then we may be able to extrapolate [math](k+1)[/math]-body quantum properties from what we know about [math]k[/math]-body systems with exact sequences.

>>8756379

gorilla has a good question

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>>8756400
Could this answer your questions? https://www.google.fi/url?sa=t&rct=j&q=&esrc=s&source=web&cd=4&ved=0ahUKEwierczi2N7SAhXIdCwKHd6NCpgQFgg8MAM&url=http%3A%2F%2Fwww.mscand.dk%2Farticle%2Fdownload%2F12317%2F10333&usg=AFQjCNG9aY-gvlJEt3zujkFVAVqXE0zbsQ&cad=rja
(sorry for the link, it's from a google search)

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>>8756444
>Hannu Honkasalo
Th-thanks

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What are the prerequisites to post-anabelian froeboid geometrics?

Why are all the math threads eventually dominated by all of you anime autist category diarrheaists? It's either undergrads asking about what fucking book to use for your babby analysis or it's you homos babbling about topos-valued functors from the category of vectorizable negligent post-quasicoherent assholes to the space of left-exactoid toposes over the category of you moms dick.

fuck you all, you're just mediocre shitheads who think generalizing trivial algebraic constructions to the point that they become unrecognizable is deep mathematics. Stop trying to ape Grothendieck and go calculate something, or at least give a hint as to why anyone should care about your endless generalizing without just coming up with paragraphs of more self-reference jargon. Fuck. Yes I mad

>>8757029
>eventually
this is the only place where you're incorrect anon

these threads are made by and for circlejerking faggots

>>8752720
all day every day

Anyone have recommendations for an introductory measure theory text?

>>8757046
Stein Shakarchi
Zygmund Wheeden
Folland
Rudin
Literally all are good. Just pick one and read it.

>>8757029
>Stop trying to ape Grothendieck
None of the discussion is even about geometry.

>>8757066
Thank you friend :)

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>>8757066
>Rudin
>good

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@8757029
The funny thing is that neither of us are studying purely category theory. I'm not even a mathematcian ffs lmfao.
No (You) for you because the bait is weakshit.

Heyo /sci/, I'm tripping balls right now. I had to share some ideas I had today looking at the existence of adjoints to the canonical embedding of an enriching category into enriched categories of presheaves (you can take it all to be enriched over sets if you want).

The bare minimum of information needed to do enriched category theory is basically a monoidal category V, called a cosmos or something similar. The idea is that the object of morphisms from A to B in some V-category C is an object of V, and functors work by supplying morphisms in V between hom objects.

So, say you have V. Now, you take some V-category C*, and then you look at [C*,V] the "V-presheaves" on C. The enriched Yoneda lemma, just proven in full generality last year, says that C sits inside of [C*,V] fully and faithfully, even if all we have is a monoidal category to enrich over (not necessarily closed or symmetric!).

But notice that V sits inside of [C*,V] by sending objects to the constant functor on them. I was looking at homs into enriched constant presheafs, and I proved that natural transformations into the constant functor on v are just lifts along the forgetful functor V/v->V. Now, if you are familiar with the process of internalization in category theory, a lift along a forgetful functor encodes the structure that is forgotten onto the presheaf being lifted.

This is mostly useless until you realize that v could be some kind of moduli object. In this case, V/v is the category of structures that v modulates, and a lift along this is an internalization of this structure into another category. So, enriched presheaves constant on moduli objects are themselves global moduli objects!! This is profound to me.

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I've also been looking into Clifford algebras and their connection with symmetry groups lately, and I've learned that they can be characterized by [math]\left|t - s\right| \in \mathbb{Z}_8[/math] as matrix algebras.

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>>8755921

>>8757807
the point of those is to do geometry less retardedly than with space vectors

a great subject for a phd is to define them constructively, which is not easy because A∆B = (A∪ B)\(A∩ B) is not constructive on first thought, because of lack of decidable equality for the reals.

Bouma, T., Dorst, L., & Pijls, H. (2001). Geometric Algebra for Subspace Operations.

Hestenes, D. (2003). Spacetime physics with geometric algebra. American Journal of Physics, 71(7), 691.

Hestenes, D. (2005). Gauge Theory Gravity with Geometric Calculus. Foundations of Physics, 35(6), 903–970.

>>8758312
Hestenes, D. (2010). New Tools for Computational Geometry and Rejuvenation of Screw Theory. Geometric Algebra Computing.
Hestenes, D. (2008). Gauge Gravity and Electroweak Theory.
Hestenes, D. (2010). Modeling Theory for Math and Science Education. Modeling Students’ Mathematical Modeling Competencies.
Hestenes, D. (1968). Multivector calculus. Journal of Mathematical Analysis and Applications, 24(2), 313–325.
Hestenes, D. (1974). Proper particle mechanics. Journal of Mathematical Physics, 15(10), 1768.
Hiley, B., & Callaghan, R. (2010). The Clifford Algebra approach to Quantum Mechanics A: The Schroedinger and Pauli Particles, 29.
Macdonald, A., & College, L. (2009). A Survey of Geometric Algebra and Geometric Calculus.
Ablamowicz, R. (1982). Clifford algebra approach to twistors. Journal of Mathematical Physics, 23(2), 231.
Baylis, W., & Jones, G. (1989). The Pauli-Algebra approach to special relativity. Nuclear Physics B - Proceedings Supplements, 6, 129–131.
Pavšič, M. (2003). Clifford Space as the Arena for Physics. Foundations, 33(9), 1277–1306.
Hestenes, D., & Ziegler, R. (1991). Projective geometry with Clifford algebra. Acta Applicandae Mathematicae, 23(1), 25–63.
Hestenes, D. (1986). Clifford Algebra and the interpretation of quantum mechanics. Clifford Algebras and Their Applications in Mathematical Physics, 321.

what's a good book for intro for computer science math?

/tg/ here, this is probably the best place to ask:
I've been trying to figure out a way to calculate the probability of throwing exactly N of the same kind on K dice. Getting a solution for a certain face hasn't been a problem, but I want to see how likely it is to get four/five/six of any kind.

>>8758372
d(k choose n)(1/d)^n ((d-1)/d)^(k-n)

why is baiting professional mathematicians on mathoverflow so enjoyable? posting anything related to mochizuki triggers responses so quickly

Why did you guys decide to major in math instead of engineering?

h-hello sceintists,

how do i get more dubs???

>>8758509
more interesting

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I only learned "college" algebra in college.
Where can I continue to study math for free?

>>8758528
the library
http://edx.org/
http://coursera.org/
http://ocw.mit.edu/index.htm
http://gen.lib.rus.ec/

>>8758410
what's d here?

>>8758533
What do I study next after college algebra? Is there an order?

>>8758515
Engineers learn most of the same math though

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>>8758504
Baiting people is pretty much always enjoyable, but it's not the result that matters. It's the thrill of the chase. That's why you should bait them using a clever plot instead of using something that always works. An analogous situation would be playing the Genghis Khan campaign in AoE2, and instead of assassinating the shah of Persia and then conquering his lands and Russia at the same time, you would just send "black death" and win instantly.

>>8758509
I tried it, and it wasn't for me. I just didn't enjoy it.

>>8757253
These things without knowing categories are like analysis without knowing topology.

>>8757379
Nice. You speak of lifts. Are they somehow related to homotopy?

>>8758548
please post this engineering curriculum

there weren't any engineers in any of my classes after multivariable calc in my 3rd semester, so i'm not sure where you get that idea from

>>8758551
How was the sauna?

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>>8758562
I missed my bus and had to wait an hour for the next one. But it will relax my body so that my mind can run free!

>>8758509
Because I don't want to be an engineer? Seems like a silly question.

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>>8758574
It happens. We should take a sauna together some time.

>>8758552
Engineers take a lot of classes on PDEs and ODEs

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>>8758611

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>>8755642
These typos... there should be
[math]\Delta^k_X \subset W[/math] instead of
[math]W \subset \Delta^k_X[/math].

>>8758590
That could be fun. Now I hopped off the bus at a wrong stop. Feels dumb man.

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>>8758646
>[math]\Delta^k_X \subset W[/math]
What? So [math]W[/math] actually contains the entire diagonal? How can it be open then?

>>8758552
I did EE at UCLA and we were required to take a complex variables and PDE course, but these were taught by the ME department not the MATH department. However in general you are correct; most engineers stop after the regular math courses that all STEM people take, so for the vast majority of engineers ODE is the last math class they will take taught by the math department.

181A. Complex Analysis and Integral Transforms. (4) Lecture, four hours; outside study, eight hours. Enforced requisite: course 82. Complex variables, analytic functions, conformal mapping, contour integrals, singularities, residues, Cauchy integrals; Laplace transform: properties, convolution, inversion; Fourier transform: properties, convolution, FFT, applications in dynamics, vibrations, structures, and heat conduction. Letter grading.

182B. Mathematics of Engineering. (4) Lecture, four hours; discussion, one hour; outside study, seven hours. Enforced requisite: course 82. Analytical methods for solving partial differential equations arising in engineering. Separation of variables, eigenvalue problems, Sturm/Liouville theory. Development and use of special functions. Representation by means of orthonormal functions; Galerkin method. Use of Green’s function and transform methods. Letter grading

http://catalog.registrar.ucla.edu/ucla-cat2016-541.html

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I need some dank maths memes

>>8758538
Depends exactly what "college algebra" means. Probably try a Calculus book next. If you feel you don't have sufficient background in something when trying that, just go and read up on whatever you're not comfortable with and then continue.

>>8758678
nvm i just had a nigger moment

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>>8758538
http://www.freebookcentre.net/SpecialCat/Free-Mathematics-Books-Download.html

I'm assuming college algebra included trig. Next you would take calculus I & II(differential and integral). Then you can take ODE's or multi variable calc then PDEs after you've taken those. You can also take linear algebra and complex variables. At this point. At this point your at engineer level.

To understand serious mathematics you need to take a course on proofs(although sometimes you can learn proofs through higher level undergrad courses). This would be a sequence on abstract algebra, a sequence analysis and 2nd course on linear algebra(mit's 18.700 for example). Then most degrees would let you choose the final few credits, although you would want one course on topology, one course a number theory, and on course on geometry.

After that your pretty much at grad student level and can choose to study the topics you want.
So congratulations you are not longer a brainlet

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>>8758720

anyone ever done bayesian statistics?

>>8758548
Engineers might take some classes specific to their needs, like solving differential equations. I, as a math grad student, would have never touched this stuff because it's totally disjoint to what I do. It's not a matter of learning as much math as math majors, which is patently false; this also is not a slight on engineering. Of course engineering students are not going to spend as much time with varying fields of math as math students. Why would they? You need to pull your head out of your ass and stop comparing fields.

Does anyone know an illustrative proof of the fact that the geometric walk [math](0, 1, 1 + 2, 1 + 2 + 3, ..., 1 + 2 + ... + 2^N-1)[/math] is injective in [math]\mathbb{Z}/(2^N\mathbb{Z})[/math]?

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Do we do homework help here? yes? please ;_;

or does anyone have any resources that would help with Aluffi's Chapter 0 book? I'm probably retarded but there is quite the jump from the chapter material to the questions

What are some good advanced complex analysis textbooks?
I've already worked through "An Introduction to Complex Function Theory" by palka and "Topics in Complex Analysis" by andersson

>>8758966
Huybrechts "Complex Geometry" if you are interested in multivariables.

>>8758965
maybe get Leinster's 'basic category theory' if you're having trouble with the categorical notions, if you're having trouble with the group theory maybe armstrong's 'groups and symmetry'

3.8 is easy as long as you understand the definition of a coproduct, what have you done so far?

>>8758747
>After that your pretty much at grad student level
I hope not, because the curriculum you gave is pretty shitty and incomplete.

I'm an undergrad who was asked to lecture for a grad course. I'm just wondering if you guys have any lecturing tips for someone who's only given one lecture before. Topic is cobordism theory.

>>8758965
Do you understand the universal property of coproducts? That should help you construct the map for 3.7.

>>8759106
I know who you are lol

>>8758747
>To understand serious mathematics you need to take a course on proofs

While this is true, we do not know if he does not already use the techniques taught in proof classes implicitly when solving problems.

Proof classes should only give you the standard notation for the techniques you are already using, not the techniques themselves... otherwise mathematics probably is not right for him and no class on proofs will fix that.

>>8759120
Lel, I've been exposed. Then you're either a friend of mine, or someone else in the course.

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>>8759106
>undergrade
>asked to lecture
That's the first I've ever heard about something like this. Just for one class?

>>8759372
Yeah, professor is absent for a week and asked me to fill in.

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>>8759390
Do the students know that you're undergrad? I'd say the main thing is to really know your shit, including some related topics that you aren't going to cover in case they ask difficult questions.

>>8759431
Yeah. I just hope they don't give me a tough time. But at the same time I'm half afraid they just won't bother to show up because I'm an undergrad.

>>8759444
Or you could just spread your boipucci and let them lay into you

>>8759106
undergrads belong no where near a grad course.

>>8759508
kek. but grad courses are so much easier.

>>8759444
>I'm half afraid they just won't bother to show up

desu i would do this if that was my class

who here has a gf

>>8759742
I've had an ex-bf does that count?

btw I being nofap today. last time I went for a full week.

>>8751123
Advisors want competent, interested students. Like the other poster said, there's no expectation for you to immediately understand their research. If you're doing a thesis program, that understanding will come over time in the form of a whole lot of reading and background work.

That being said, It's not a bad idea to brush up on the work of anyone you plan to have meetings with. Not with the intention of bringing it up out of the blue, but as a possible conversation point if their research area intersects with some interest of yours. I'm sure most researchers would prefer a student they know would be driven in their work.

>>8753679
It's all good, it took me some thinking to see where you were coming from. You're main point is that application has a tendency (maybe it should, maybe it shouldn't) to drive mathematics. That I won't disagree with. It's just what the mathematician literally does is pretty far from having application to the real world, even if he is working on a problem with direct applications.

>>8760137
REEE ROASTIEEEEEE

>>8760308
Nope, just gay.

>>8759742
math here w/ bio gf

Current year 1 undergrad here. About to finish my second semester

Feeling like I've lost passion in mathematics. I say this because I have so much trouble trying to understand anything. Proving anything is insanely difficult- most of the time I have absolutely no idea how to start, much less connect the dots to the course material (specifically analysis and adv linear algebra). Im tired. I really wanna give up. Can't believe just 3 months ago Im excited to learn some 'real' math, compared to calculus and introductory linear algebra.

What do? Sudoku is an option

I'm a brainlet, taking a basic calculus course.

>>8760503
if you're in year 1 you either likely never had a passion in math to begin with or were deluding yourself

study harder brainlet

>>8760503
try cat theory since it is different maths>>8758293

leave bourbaki to brainlets

>>8760137

depends on if you take it in the ass or not, if you yes if you both do then yes

What is the best website to Learn/Practice High School+ level mathematics? I figured it should be about time that I started properly appreciating the mathematical world.

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>>8760503
Try making some sort of flow chart how things lead to one another, one for each studied chapter. For example:

(Compactness if you are familiar with this ->) closed interval [math][a, b][/math] -> for every continuous function [math]f \colon [a, b] \to \mathbb{R}[/math], there are [math]c, d\in [a, b][/math] such [math]f(c), f(d)[/math] are the minimum and maximum of the image set.

If you could do this, you'd maybe see the connections better. Proving is, basically, just a chain of implications, and the chart would consist of precisely those.

>>8758352
Are you talking about Discrete Math?

>>8760586
I think deluding is a pretty strong word.. I had read up on algebra and analysis books before start of uni

>>8760786
Cat theory needs a rather solid foundation in math, no? And I got no foundation to speak of


>>8760860
Sounds like a great idea! Will try it

Anyone here for doing actual math research at a university? Can you tell all about your research in a few sentences ?

>>8760891
>>Cat theory needs a rather solid foundation in math, no?
no , on the contrary and with the universal properties in definition, the proofs are different form usual math.
before looking a the proofs, try to do them yourself. go at least to the definition of complete category and limits in the red book, Theorem 2.8.1 . it is about 60 pages.

>>8760891
It's almost pointless to learn category theory without a foundation in math. Sure, you can follow it, but it will be dry, unmotivated, and devoid of interesting examples.

>>8760975
>Sure, you can follow it, but it will be dry, unmotivated, and devoid of interesting examples.
yeah this is what engineers say about pure math. they need little applied illustrations other they get bored.
The good news for him is that he never has done maths, so he does not even understand the examples, interesting or not.
Plus the proofs are simpler in CT

>>8760990
Holy shit, why is this board so obsessed with category theory? It's not even particularly interesting.

>>8760503
Are you American? I can't take classes like that until 3rd yr

>>8760997
CT is more interesting than arithmetic

>>8760990
>>8760972

I wouldnt say that i havent done maths, ive understood proofs on cayley hamilton, jordan and rational canonical forms, and other fun facts like embedding of rationals in any ordered field.

But im nowhere near 'real' maths i guess so your point still stands

I find cat. theory pretty technical (from the first 10 or so pages), will see how it goes

Thanks for the suggestion mate

>>8761111
Please don't listen to that retard. This is the problem with /sci/: anyone can shoot his big fucking mouth and the people who don't know better take it seriously.

abandon the sinking ship, guys
>>8761147

>>8760948
How about you read the thread?

>>8759742
Math here with Comp Sci gf.

>>8759508
Why not, if they have demonstrated themselves competent?

please tell me the next statement is always true and why is isnt a corollary in books. I dont think it is obvious obvious.


If S is an upper bound of S_1 and S belong to S_1, then S must be the supremum

>>8761798
its always true and its obvious

>>8760997
Because it's very abstract and people on this board seem to think that doing abstract and pedantic math will make them sound smarter

>>8761798
>water is wet

>>8761869
>>8761824
>state it s obvious
>it s an actual exercise in many respectable real analysis books
brainlets

>>8762045
you're the brainlet if that 'exercise' takes you longer than 5 seconds

>>8762046
took me one second, it s an important result that speeds things up.
>brainlet that hasnt learned multilinear algebra on his freshman year calling others brainles
kys brainlet

What is the difference between median and mean?

Is median always superior?

>>8762055
its not 'important', its trivial

if a maximum exists it's the supremum

>>8762063
brainlet

>>8762071
it s not true for other fields though
>he works only on R1
brainlet

>>8762072

answer me then genius. or fuck off this is stiupid question thread

>>8762063
Mean is the average of the values, i.e. sum over number of values in the sample.

median is "middle" value of the sample when ordered by size.

No, median is not always larger. Consider {0,0,1}. Median value is 0. Mean is 1/3.

>>8762077
This is remarkably pathetic, dude.

>>8762089

no no, i meant in practical way. Because afaik, given a set of data {-100000, 100, 101, 102}

it is better to use median because the data is skewed far far to the left.

So does that mean that median is always better than mean? Is there a specific type of data where mean is better? maybe mean is easier computationally?

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>>8762144
>So does that mean that median is always better than mean? Is there a specific type of data where mean is better? maybe mean is easier computationally?
Consider a country where 99% of people make 100 dollars a year and 1% makes 10000000 dollars a year. Which one gives more realistic information about the wealth distribution in that country? If the differences are enormous, then go with the median.

>>8762178

alright, i can see how its a subjective matter what you want to extract from the set.

thanks cute anime girl

>>8755921
http://www.logicmatters.net/resources/pdfs/GentleIntro.pdf

WTF Real analysis
WHY CANT I FUCKIN STUDY FOR A TEST 5 DAYS PRIOR AND ACE IT? WHY THE FUCK DO I HAVE TO PAY ATTENTION TO LECTURE AND DO HW AND NOT MASTER THE Material 5 days prior? MATH IS A SHIT MAJOR SHITTTT

>>8748209
turaev is god-tier

>>8748188
I'm a grad student working on (mathematical) Gauge theory and low dimensional topology. Means I also have side interests in knot theory and symplectic geometry. Currently trying to understand Manolescu's work on the triangulation conjecture using Seiberg-Witten Floer homology.

Anyone else in the area? :O will probs see/might have already seen you at conferences this year

>>8755972
Emily Riehl also has a new book out which is quite nice. Idk if it really offers anything above what's available, though.

>>8758966
Depends on what you want to work on. There's things like several complex variables (I would look at hormander's book if you really want more analytic flavour than a typical complex geometry flavour). A lot of advanced complex analysis also goes into stuff like quasiconformal maps (see ahlfors), or complex dyanmics(see milnor or carlesson's books). After basic complex analysis, it gets "advanced" in many different directions.

anyone able to help me plotting lyapunov exponents in mathematica?

>>8748209
Take your pedophile cartoons back to >>>/a/