/sci/ - Science & Math

Hello friend
I am stanik. I am learning linear algebra
What are your?

>>9800443
Wikipedia.
There's a reason why Tao puts a bunch of wikipedia links in his blog posts.

Why is Tao so ugly?

>>9801059
He was born that way.

>tfw trying to self study baby rudin

>>9801106
Hey I'm self-studying papa rudin. I practically memorized baby rudin cover to cover as an undergraduate. Don't give up. You already know this but go as slow as you need to to understand everything before you go on.

>>9801124
Did you "cheat" on the exercises or did you do them all by yourself?

>>9800443
FUCK TRUMP AND FUCK WHITE PEOPLE

>>9801131
In college I was a shitty math student to be honest, so I cheated on quite a few exercises. I can be honest about that now. Hell I got caught and even "retroactively" suspended for it so the school found out and I stopped doing it immediately after that. I still memorized the fucking book though because I took 2 classes with it and by the second one I had already been "suspended" so there was no more cheating, at all, and I started working my ass off at that point.

Now I'm 2 years out of college and just now going back and self-studying math again. Reread baby rudin and remembered most of it, reread Topology by Munkres and some of Algebra by Michael Artin, all 3 being my favorite undergraduate math texts that I worked a lot with. Now I'm going through Rudin's RCA very, very slowly and doing all the exercises without any "cheating" on my own with no class. Occasionally I ask for clarification and help from Stack Exchange but that's it. This is how I learn best, as opposed to lectures or any type of classroom environment really.

>used Lagrange differentiation notation for 3 semesters of calculus
>uni physics uses Leibniz

>>9801189
Doesnt physics use newton notation?

>>9801141
>graduated and still reading undergrad books

this is what happens to every single one of you book worshiping brainlets. You get trapped memorizing set theory and real analysis books for the rest of your life. Let this be a real life example for you to learn from to change your evil ways now.

>>9801059
we can't all be handsome like you anon

Are Tao's books any good? Considering reading Solving Mathematical Problems.

>>9801189
No mathematician should ever care about notation. The fact that you do shows that you learned the material by rote memorisation. If you actually understood what is going on you would be able to freely jump from one notation style to another. Lagrange, Euler, Newton, your own -- doesn't matter.

>>9801696
I tried reading his random matrix theory book, but his writing style is boring and confusing. Maybe I'll try it again.

>>9801024
I am Chris , I'm also learning linear algebra.

Math is the latin of sciences

>>9801189
This isn't the brainlet general you moron.
You should be able to use them all interchangeably.

What do you guys think about the tangent space to a smooth manifold in terms of derivations?
It's a bit surprisingly how linearity and the Leibniz rule just make everything work.
They are so much cleaner than seeing things in terms of equivalence classes of curves or by doing everything in terms of coordinates, but something about them just feels off.

>>9801949
So some stupid language that is "renowned" for being old and doesn't provide any more utility than English?

>>9800443
Are there any general methods to determine whether a multivariate polynomial over a commutative ring is irreducible? The only one I know is Eisenstein's criterion, but that only applies to domains.

>>9802090
>What do you guys think about the tangent space to a smooth manifold in terms of derivations?
I don't care.

>>9802271
>but that only applies to domains.
Irreducibility isn't really a useful idea in rings with zero divisors. Even if you assumed your ring to keep unique factorization, polynomial degrees no longer add so you can do stupid shit like "factor" a polynomial into two factors that both have higher degree.

>>9802090
It feels off because it's abstract. When you think of derivations, the model example to keep in mind is directional derivatives at a point in Euclidean space. This connects nicely with the more geometric definition using equivalence classes of curves.

>>9802090
Nobody should care for the embedded tangent space. The only truth is m_p/m_p^2 and its dual.

>>9802438
>It feels off because it's abstract.
Only if you're some kind of subhuman engineer.

>>9802471
Where's that screenshot from?

>>9802494
It's a well-known anime.

>>9802486
>intuition isn't important

Is it possible to formulate ray optics in terms of a PDE?

>>9802471
Lecture video on analytic mechanics at my university (timestamp is crop).

This
>>9803246
was meant for
>>9802494

Dear people in this thread,
What can I do with more math in my life besides read more math books? I graduated with only a bachelors in math but have no real hope of going back to attain a higher degree. I have, however, been reading through plenty of math books since then and moving deeper into fields that interested me as an undergraduate.

What's the next step? Surely there's more to this than just reading more and more textbooks. Is it moving on to reading just math research papers, basically? And then eventually writing them? What do you guys think?

>>9801135
America belongs to Amerindians the same as europe belongs to europeans.

Time to get replaced, subhuman.

>>9803015
Yes, and it is done always. Eikonal equation

>>9801024
Hello faggot.
I am Chad. I am learning your girlfriend's pussy.
What are your?

>tfw trying to cure my brainletism doing every problem in Demidovichs Problems in Mathematical Analysis
Over 3000 problems, is this how the soviets cranked out so many good mathematicians

>>9801043
i also found wikipedia really good for getting an overview of concepts and ideas

Could someone mansplain me how to derive the isomorphism [math]\pi_5 (S^2) \cong \mathbb{Z}_2[/math] using the Serre spectral sequence and the Postnikov tower of [math]S^2[/math]?

>>9803290
Prove the Riemann Hypothesis

>>9803300
Cool, thanks!

>>9803866
What the hell, $\pi_n$ of a sphere can have torsion? This is even worse than I thought.

>>9804319
Yes. And if S^2 has that for n>3, then so does S^3 due to the Hopf fibration. Enjoy.

>>9802910
You needing something to be engineer-like to be intuitive just shows how you are a subhuman engineer.

How can you even compete with a math chad?

Is there any smart way to solve this for x (other than bruteforce)
[math](ax) mod p = b[/math],
where [math]p[/math] is prime?

>>9804519
multiply by a^{-1} mod p

>>9804519
... and what if we know a, p, b?

>>9804521
Oh, shit, seems like I'm getting it.

>>9804521
http://www.wolframalpha.com/input/?i=(1*x)(modulo+5)+%3D+2+solve+for+x

I change the x multiplier and see that sequence is changed, yet I can't find the pattern. It seems to be like prime * n + something.

>>9804519
Extended Euclidean Algorithm

>>9804519
What do you learn in high school those days?
You should know that this is just Bezout's identity, which means you can apply Euclidean division algorithm, which enough fast (you can also find the inverse to a and then multiply both sides of the equation by it, but this works just because p is prime).

A fucking tenth grader in high school took and apparently did extremely well in my Algebraic Topology class with me, at an Ivy League school no less. Made me feel like such a brainlet it hurt.
>He wasn't even Asian.

>>9801141
Goodness mate, you are still reading Rudin and you have graduated. I was finished with Rudin's books after my first year. And I am more into Algebra than Analysis.

>>9804967
This thread is making me feel terrible. I'm a good way through Rudin - RCA now. Is his Functional Analysis book also worth reading?
To put things in perspective, you can call me a brainlet all you want but I absorb this stuff way slower than most other students did, I think. I did rather poorly in math classes but great when I just started a book at the beginning, did every proof at my own pace, and slowly work everything out with a lot of stupid questions along the way. Seeing lectures did almost nothing for me. Right now I'm just trying to catch up to the point where I can start reading research papers; obviously I'm still a good ways away.

>>9804515

makes me remember of that university professor that did porn

>>9804972
>This thread is making me feel terrible.
Ignore the trolls, they learned HoTT & QFTNLLL when they were 5 years old, and now they are so ahead thay can't even see themself. Take your time and enjoy it.
The books you are reading are quite heavy. For example Munkres really has too much stuff you would never learn as an undergraduate, but if you like analysis it is known to have some good chapters.
I can't give you any opinion on Rudin's books, I'm not into analysis and I learned it without doing any exercise (if we don't take into account the exams). Probably the only thing I enjoyed were complex analysis, measure theory, some result on density of functions and lagrangian/hamiltonian mechanics (because it was clear and it was a revenge against shitty physics I mechanics).
Not to discourage you, but I can't imagine anyone getting into research by just reading few technical books. A friend of mine is doing his master thesis in analysis, and is solving some strange problem using different methods of those there were used in an articles (namely variational methods and things I don't understand). But it is very different from what you learn in books, and without the push and help of the relator he would't be able to do any of that.
What I am saying is that you might enjoy take things easier at the beginning, see other topics, and when you feel ready ask someone good to help you get into research.
And most of all, I hope you get some joy studing our beloved math.

>>9805164
>QFTNLLL
?

>>9801945
Nuce. Where topic are you??

Is baby rudin useful for engineers?

>>9805442
>Is baby rudin useful for engineers?
Rudin is a meme.

>>9805178
Something too advanced for us.

>>9805574
Relative CW homology has literally nothing to do with QFT, or whatever QFTNLL is, so why exactly did you post That pic from your notebook, sweetie?

>>9805578
Not a sweetie, not my notebook, not related to anything. It was a random photo.

>>9805587
>Not a sweetie
Then you have to leave.

>>9805578
>replying to obvious shitposts so you can announce to everyone that you know what homology is

I'm learning operator theory right now and that shit is nuts math but I am loving it

>>9805774
Her post doesn't deal with homology, it only mentions relative CW homology.

>>9803866
Answer this you useless fucks.

>>9806033
Don't know, have you tried RTFM?

>>9805164
Thanks. As for research I'll start reading published papers and then ask somebody to help me get into it when I'm ready for it.

>>9806043
Fuck you.

>>9806033
>Answer this you useless fucks.
Do you need to swear?

https://totallydisconnected.wordpress.com/2018/05/09/the-latest-hot-abc-news

how much longer until this is acknowledged as a nothing burger?

>>9801476
Only in mechanics if at all

>>9800443
I just audited a summer precalc 2 class not sure if I'm a brainlet or the professor was ass but I'm thinking it was a mixture of both

I'm at the end of my undergraduate, I want to do Random Matrix Theory
What honours (aka post-grad) subjects do I take

>>9801945
Hi I'm Jordan. I learned Linear algebra in 2003. I hope you enjoy it.

He's next.

>>9806572
I wish I was named Freeman

>>9806699
What about Dyson Freeman? 2nd coolest name in the world behind Freeman Dyson.

>>9801725
No speaker should ever care about language, if you actually understood what was going on you'd be able to freely junp from one language to another. French, English, russian, your own - doesn't matter

>>9801725
This is bullshit, good notation makes working with stuff a lot easier. If what you said were true mathematicians would never invent new notation for stuff, because they certainly know what's going on.

>>9806713
A difference in math notation is more like a difference in programming language.

I finally got 2 math papers published this year.

It makes a hell lot of difference to babble about IUTT on 4chan and actually publish something

>>9806572
>abstract cancer
What did he mean by this?

>>9806761
maybe next you can learn English

>>9806742
>programming language
Refer to the >>>/g/hetto/.

What is the new function [math]g'(x) = f^{-1}(g(f(x))[/math] called?

>>9804966
>Algebraic Topology class
this is how i know both of you are brainlets

>>9807290
i've seen this referred to as the 'meme' function in literature

>>9806779
What's wrong with the post?

>>9807296
>this is how i know both of you are brainlets
Mathematicians use "we", not "I".

I'm interested in the expected values of the function

b * x + (1-b) * y

where the triple (b,x,y) is uniformly distributed. I sampled a bunch of points and I'd like to know what the distribution of the values are.
The plot reminded me of this entropy kind of bump, but that's just a blind guess and it. Another guess it Dirichlet distribution for simple exponents, but I don't have tools to check those.

Can someone quickly remind me how to compute the joint distribution from three uniform ones? What's that bump curve?

Hey lads starting phd this fall how do I prep for qual exams

>>9806699
Me too. Then people would call me "Freeman" as though I were Gordon Freeman

>>9807867
*tips fedora*

Scholze & Manolescu rumored for Fields medal.

>>9808709
No shit Sherlock, Scholze will get a fields medal. He's the most important mathematician alive.

>>9809010
well they could have waited until next time.

>>9805055
male or female

>>9800443
Math with bad drawings is the best

Why do so few people know about the generalized Fourier transform? It's a much more powerful technique than most people realize desu.

[math]| \{ x \in \mathbb{R}^{d} \, | \, 0 \leq x_{1} \leq x_{2} \leq ... \leq x_{d} \leq 1 \} | = \, ?[/math]
You should be able to calculate this.
Tip: No integration

>>9809010
>He's the most important mathematician alive.
But I'm not Scholze.

>>9809090
Which generalization are you talking about? There are many ways to look at Fourier.

>wrote my first one page proof
>it's shit

>>9809530
What did you prove?

>>9809662
I cant be arsed to type it

>>9809010
>He's the most important mathematician alive.
*ahem*

>>9809677
Is it the union of all sets in F?

>>9809821
Correct

>maths

>>9800443
Daily reminder that applied math is best math

>>9810338
>applied math
This is not well-defined.

https://arxiv.org/pdf/1806.05538.pdf
>A marriage of category theory and set theory: a finitely axiomatized nonclassical first-order theory implying ZF
>Marcoen J.T.F. Cabbolet
>(Submitted on 11 Jun 2018)
>The main purpose of this paper is to introduce a finitely axiomatized theory that might be applicable as a foundational theory for mathematics. For that matter, some twenty axioms in a formal language are introduced, which are to hold in a universe consisting of a class of objects, each of which is a set, and a class of arrows, each of which is a function on a set. One of the axioms is nonclassical: it states that, given a family of ur-functions - i.e. functions on a singleton - with disjunct domains, there exists a uniquely determined sum function on the union of these domains. This 'sum function axiom' is so powerful that it allows to derive ZF from a finite axiom scheme. In addition, it is shown that the Loewenheim-Skolem theorem does not hold for the present theory, which therefore can be considered stronger than ZF. Furthermore, the axioms of category theory are proven to hold: the present universe may therefore serve as an ontological basis for category theory. However, it has not been investigated whether any of the soundness and completeness properties hold for the present theory: the inevitable conclusion is therefore that only further research can establish whether the present results indeed constitute an advancement in the foundations of mathematics.

trying to implement subtraction using functions, with peano's axioms. taking a - b = c as an example, one can find the value of c by counting from b to a. i havent found a notation to represent this, here is an attempt, but i feel it is wrong, code in C below to show my reasoning, unsigned ints are natural numbers:

-(a,b) = (a = b + s^c(1))


unsigned int subtract (unsigned int a, unsigned int b)
{
unsigned int c = 0;

while(b + c <= a) {
c = c + 1;
}
printf("%d - %d = %d\n", a, b, c); /* format: a - b = c */
}


>>9801059
Because if he was handsome he’d have drowned in pussy many years ago and the human race would have missed his genious.

>>9811287
Why are you using <=? Once you find c such that b+c = a you want to stop, not increment again.

Redpill me on math, where does a HS dropout begin?

>>9811595
From the hospital KYS.

>>9811633
Kentucky Yeast Service Hospital

>>9811636
Secret.

>>9811582
That is standard notation for "coimplies", brainlet.

>>9809122
SPOILER

Symmetry arguement, multiplying it by d! gives us the volume of the unit cube which is 1

>>9806570
Did your colleagues turn back to you and grin when you were learning about the Jordan canonical form?

So who's going?

>>9811737
What's Como?

>>9811668
That isn't how these things work when you have infinite sets.

>>9811633
KYKS my dude.

>>9811738

https://www.youtube.com/watch?v=yQ_PibZ1Q8M

this place.

>>9811746
So, it's beautiful but too hot. I'm out.

>>9803290
If you wanr a challenging but interesting problem try this:
show that
sigma(n) <= Hn + e^(Hn)*ln(Hn)
where sigma(n) is the sum of the dibisors of n
and Hn is the nth harmonic number

>be first yearath student
>everyone seems to be signing up for these 2nd semester optional seminars
>cant miss out on that
>"representation theory of finite groups"
>do the first presentation nothing more than a few definitions, get a dexent grade
>3rd week is tensors
>defined with the universal property for n spaces
>slowly stop following the presentations and just do other things during the seminar
Now I sit there every week and feel terrible, I didnt learn anything.
In addition I now have shitty associations with algebra and seminars (I dont know shit about either)
I should have never signed up for this

>>9811737
Are you italian?

>>9811595
What did you learn in HS?
What do you want to learn? Do you have any goals?
Why do you want to learn math?

>>9803494
No. They just had lots of jews.

>>9806572
he's gonna dyson

ded general ded subject

>>9812543
>>9811595
this should probably be it's own thread desu, but here:
http://4chan-science.wikia.com/wiki/Math_Textbook_Recommendations
http://4chan-science.wikia.com/wiki/Mathematics

>>9811595
>Redpill me on math, where does a HS dropout begin?
High School:
• Euclidean geometry, complex numbers, scalar multiplication, Cauchy-Bunyakovskii inequality. Introduction to quantum mechanics (Kostrikin-Manin). Groups of transformations of a plane and space. Derivation of trigonometric identities. Geometry on the upper half-plane (Lobachevsky). Properties of inversion. The action of fractional-linear transformations.
• Rings, fields. Linear algebra, finite groups, Galois theory. Proof of Abel's theorem. Basis, rank, determinants, classical Lie groups. Dedekind cuts. Construction of real and complex numbers. Definition of the tensor product of vector spaces.
• Set theory. Zorn's lemma. Completely ordered sets. Cauchy-Hamel basis. Cantor-Bernstein theorem.
• Metric spaces. Set-theoretic topology (definition of continuous mappings, compactness, proper mappings). Definition of compactness in terms of convergent sequences for spaces with a countable base. Homotopy, fundamental group, homotopy equivalence.
• p-adic numbers, Ostrovsky's theorem, multiplication and division of p-adic numbers by hand.
• Differentiation, integration, Newton-Leibniz formula. Delta-epsilon formalism.

>>9813223
Freshman:
• Analysis in R^n. Differential of a mapping. Contraction mapping lemma. Implicit function theorem. The Riemann-Lebesgue integral. ("Analysis" by Laurent Schwartz, "Analysis" by Zorich, "Theorems and Problems in Functional Analysis" by Kirillov-Gvishiani)
• Hilbert spaces, Banach spaces (definition). The existence of a basis in a Hilbert space. Continuous and discontinuous linear operators. Continuity criteria. Examples of compact operators. ("Analysis" by Laurent Schwartz, "Analysis" by Zorich, "Theorems and Problems in Functional Analysis" by Kirillov-Gvishiani)
• Smooth manifolds, submersions, immersions, Sard's theorem. The partition of unity. Differential topology (Milnor-Wallace). Transversality. Degree of mapping as a topological invariant.
• Differential forms, the de Rham operator, the Stokes theorem, the Maxwell equation of the electromagnetic field. The Gauss-Ostrogradsky theorem as a particular example.
• Complex analysis of one variable (according to the book of Henri Cartan or the first volume of Shabat). Contour integrals, Cauchy's formula, Riemann's theorem on mappings from any simply-connected subset C to a circle, the extension theorem, Little Picard Theorem. Multivalued functions (for example, the logarithm).
• The theory of categories, definition, functors, equivalences, adjoint functors (Mac Lane, Categories for the working mathematician, Gelfand-Manin, first chapter).
• Groups and Lie algebras. Lie groups. Lie algebras as their linearizations. Universal enveloping algebra, Poincaré-Birkhoff-Witt theorem. Free Lie algebras. The Campbell-Hausdorff series and the construction of a Lie group by its algebra (yellow Serre, first half).

>>9813224
Sophomore:
• Algebraic topology (Fuchs-Fomenko). Cohomology (simplicial, singular, de Rham), their equivalence, Poincaré duality, homotopy groups. Dimension. Fibrations (in the sense of Serre), spectral sequences (Mishchenko, "Vector bundles ...").
• Computation of the cohomology of classical Lie groups and projective spaces.
• Vector bundles, connectivity, Gauss-Bonnet formula, Euler, Chern, Pontryagin, Stiefel-Whitney classes. Multiplicativity of Chern characteristic. Classifying spaces ("Characteristic Classes", Milnor and Stasheff).
• Differential geometry. The Levi-Civita connection, curvature, algebraic and differential identities of Bianchi. Killing fields. Gaussian curvature of a two-dimensional Riemannian manifold. Cellular decomposition of loop space in terms of geodesics. The Morse theory on loop space (Milnor's Morse Theory and Arthur Besse's Einstein Manifolds). Principal bundles and connections on them.
• Commutative algebra (Atiyah-MacDonald). Noetherian rings, Krull dimension, Nakayama lemma, adic completion, integrally closed, discrete valuation rings. Flat modules, local criterion of flatness.
• The Beginning of Algebraic Geometry. (The first chapter of Hartshorne or Shafarevich or green Mumford). Affine varieties, projective varieties, projective morphisms, the image of a projective variety is projective (via resultants). Sheaves. Zariski topology. Algebraic manifold as a ringed space. Hilbert's Nullstellensatz. Spectrum of a ring.
• Introduction to homological algebra. Ext, Tor groups for modules over a ring, resolvents, projective and injective modules (Atiyah-MacDonald). Construction of injective modules. Grothendieck Duality (from the book Springer Lecture Notes in Math, Grothendieck Duality, numbers 21 and 40).
• Number theory; Local and global fields, discriminant, norm, group of ideal classes (blue book of Cassels and Frohlich).

>>9811595
Why would you want to?

>>9813225
Sophomore (cont):
• Reductive groups, root systems, representations of semisimple groups, weights, Killing form. Groups generated by reflections, their classification. Cohomology of Lie algebras. Computing cohomology in terms of invariant forms. Singular cohomology of a compact Lie group and the cohomology of its algebra. Invariants of classical Lie groups. (Yellow Serre, the second half, Hermann Weyl, "The Classical Groups: Their Invariants and Representations"). Constructions of special Lie groups. Hopf algebras. Quantum groups (definition).

Junior:
• K-theory as a cohomology functor, Bott periodicity, Clifford algebras. Spinors (Atiyah's book "K-Theory" or AS Mishchenko "Vector bundles and their applications"). Spectra. Eilenberg-MacLane Spaces. Infinite loop spaces (according to the book of Switzer or the yellow book of Adams or Adams "Lectures on generalized cohomology", 1972).
• Differential operators, pseudodifferential operators, symbol, elliptic operators. Properties of the Laplace operator. Self-adjoint operators with discrete spectrum. The Green's operator and applications to the Hodge theory on Riemannian manifolds. Quantum mechanics. (R. Wells's book on analysis or Mishchenko "Vector bundles and their application").
• The index formula (Atiyah-Bott-Patodi, Mishchenko), the Riemann-Roch formula. The zeta function of an operator with a discrete spectrum and its asymptotics.
• Homological algebra (Gel'fand-Manin, all chapters except the last chapter). Cohomology of sheaves, derived categories, triangulated categories, derived functor, spectral sequence of a double complex. The composition of triangulated functors and the corresponding spectral sequence. Verdier's duality. The formalism of the six functors and the perverse sheaves.

>>9813227
Junior (cont):
• Algebraic geometry of schemes, schemes over a ring, projective spectra, derivatives of a function, Serre duality, coherent sheaves, base change. Proper and separable schemes, a valuation criterion for properness and separability (Hartshorne). Functors, representability, moduli spaces. Direct and inverse images of sheaves, higher direct images. With proper mapping, higher direct images are coherent.
• Cohomological methods in algebraic geometry, semicontinuity of cohomology, Zariski's connectedness theorem, Stein factorization.
• Kähler manifolds, Lefschetz's theorem, Hodge theory, Kodaira's relations, properties of the Laplace operator (chapter zero of Griffiths-Harris, is clearly presented in the book by André Weil, "Kähler manifolds"). Hermitian bundles. Line bundles and their curvature. Line bundles with positive curvature. Kodaira-Nakano's theorem on the vanishing of cohomology (Griffiths-Harris).
• Holonomy, the Ambrose-Singer theorem, special holonomies, the classification of holonomies, Calabi-Yau manifolds, Hyperkähler manifolds, the Calabi-Yau theorem.
• Spinors on manifolds, Dirac operator, Ricci curvature, Weizenbeck-Lichnerovich formula, Bochner's theorem. Bogomolov's theorem on the decomposition of manifolds with zero canonical class (Arthur Besse, "Einstein varieties").
• Tate cohomology and class field theory (Cassels-Fröhlich, blue book). Calculation of the quotient group of a Galois group of a number field by the commutator. The Brauer Group and its applications.
• Ergodic theory. Ergodicity of billiards.
• Complex curves, pseudoconformal mappings, Teichmüller spaces, Ahlfors-Bers theory (according to Ahlfors's thin book).

>>9813230
Senior:
• Rational and profinite homotopy type. The nerve of the etale covering of the cellular space is homotopically equivalent to its profinite type. Topological definition of etale cohomology. Action of the Galois group on the profinite homotopy type (Sullivan, "Geometric topology").
• Etale cohomology in algebraic geometry, comparison functor, Henselian rings, geometric points. Base change. Any smooth manifold over a field locally in the etale topology is isomorphic to A^n. The etale fundamental group (Milne, Danilov's review from VINITI and SGA 4 1/2, Deligne's first article).
• Elliptic curves, j-invariant, automorphic forms, Taniyama-Weil conjecture and its applications to number theory (Fermat's theorem).
• Rational homotopies (according to the last chapter of Gel'fand-Manin's book or Griffiths-Morgan-Long-Sullivan's article). Massey operations and rational homotopy type. Vanishing Massey operations on a Kahler manifold.
• Chevalley groups, their generators and relations (according to Steinberg's book). Calculation of the group K_2 from the field (Milnor, Algebraic K-Theory).
• Quillen's algebraic K-theory, BGL^+ and Q-construction (Suslin's review in the 25th volume of VINITI, Quillen's lectures - Lecture Notes in Math. 341).
• Complex analytic manifolds, coherent sheaves, Oka's coherence theorem, Hilbert's nullstellensatz for ideals in a sheaf of holomorphic functions. Noetherian ring of germs of holomorphic functions, Weierstrass's theorem on division, Weierstrass's preparation theorem. The Branched Cover Theorem. The Grauert-Remmert theorem (the image of a compact analytic space under a holomorphic morphism is analytic). Hartogs' theorem on the extension of an analytic function. The multidimensional Cauchy formula and its applications (the uniform limit of holomorphic functions is holomorphic).

>>9813233
Specialist: (Fifth year of College):
• The Kodaira-Spencer theory. Deformations of the manifold and solutions of the Maurer-Cartan equation. Maurer-Cartan solvability and Massey operations on the DG-Lie algebra of the cohomology of vector fields. The moduli spaces and their finite dimensionality (see Kontsevich's lectures, or Kodaira's collected works). Bogomolov-Tian-Todorov theorem on deformations of Calabi-Yau.
• Symplectic reduction. The momentum map. The Kempf-Ness theorem.
• Deformations of coherent sheaves and fiber bundles in algebraic geometry. Geometric theory of invariants. The moduli space of bundles on a curve. Stability. The compactifications of Uhlenbeck, Gieseker and Maruyama. The geometric theory of invariants is symplectic reduction (the third edition of Mumford's Geometric Invariant Theory, applications of Francis Kirwan).
• Instantons in four-dimensional geometry. Donaldson's theory. Donaldson's Invariants. Instantons on Kähler surfaces.
• Geometry of complex surfaces. Classification of Kodaira, Kähler and non-Kähler surfaces, Hilbert scheme of points on a surface. The criterion of Castelnuovo-Enriques, the Riemann-Roch formula, the Bogomolov-Miyaoka-Yau inequality. Relations between the numerical invariants of the surface. Elliptic surfaces, Kummer surface, surfaces of type K3 and Enriques.
• Elements of the Mori program: the Kawamata-Viehweg vanishing theorem, theorems on base point freeness, Mori's Cone Theorem (Clemens-Kollar-Mori, "Higher dimensional complex geometry" plus the not translated Kollar-Mori and Kawamata-Matsuki-Masuda).
• Stable bundles as instantons. Yang-Mills equation on a Kahler manifold. The Donaldson-Uhlenbeck-Yau theorem on Yang-Mills metrics on a stable bundle. Its interpretation in terms of symplectic reduction. Stable bundles and instantons on hyper-Kähler manifolds; An explicit solution of the Maurer-Cartan equation in terms of the Green operator.

>>9813235
Specialist (cont):
• Pseudoholomorphic curves on a symplectic manifold. Gromov-Witten invariants. Quantum cohomology. Mirror hypothesis and its interpretation. The structure of the symplectomorphism group (according to the article of Kontsevich-Manin, Polterovich's book "Symplectic geometry", the green book on pseudoholomorphic curves and lecture notes by McDuff and Salamon)
• Complex spinors, the Seiberg-Witten equation, Seiberg-Witten invariants. Why the Seiberg-Witten invariants are equal to the Gromov-Witten invariants.
• Hyperkähler reduction. Flat bundles and the Yang-Mills equation. Hyperkähler structure on the moduli space of flat bundles (Hitchin-Simpson).
• Mixed Hodge structures. Mixed Hodge structures on the cohomology of an algebraic variety. Mixed Hodge structures on the Maltsev completion of the fundamental group. Variations of mixed Hodge structures. The nilpotent orbit theorem. The SL(2)-orbit theorem. Closed and vanishing cycles. The exact sequence of Clemens-Schmid (Griffiths red book "Transcendental methods in algebraic geometry").
• Non-Abelian Hodge theory. Variations of Hodge structures as fixed points of C^*-actions on the moduli space of Higgs bundles (Simpson's thesis).
• Weil conjectures and their proof. l-adic sheaves, perverse sheaves, Frobenius automorphism, weights, the purity theorem (Beilinson, Bernstein, Deligne, plus Deligne, Weil conjectures II)
• The quantitative algebraic topology of Gromov, (Gromov "Metric structures for Riemannian and non-Riemannian spaces"). Gromov-Hausdorff metric, the precompactness of a set of metric spaces, hyperbolic manifolds and hyperbolic groups, harmonic mappings into hyperbolic spaces, the proof of Mostow's rigidity theorem (two compact Kählerian manifolds covered by the same symmetric space X of negative curvature are isometric if their fundamental groups are isomorphic, and dim X> 1).
• Varieties of general type, Kobayashi and Bergman metrics, analytic rigidity (Siu)

>>9813225
>(in the sense of Serre)
I don't know why but I always lose it here

>>9811737
Sounds cool but no. Say hi to my girl olivia though.

based rigorous curriculum poster

>>9813775
>rigorous
What do you mean?

What books should I read and in what order if I want to get into several complex variables?

>>9800443
https://graphicallinearalgebra.net/

I really enjoyed this blog :3

>>9813856
Read complex analysis in one variable and then use induction to several variables

>>9813878
>then use induction to several variables
>induction
You can't induct on learning anon.

>>9813223
>>9813224
>>9813225
>>9813227
>>9813230
>>9813233
>>9813235
>>9813236
Not the other anon, I just wanted to say thanks for posting.

Here's a fun problem for you, /mg/:
Find all [math] n \in \mathbb{N} [/math] such that [math] \binom{n}{2} [/math] is a perfect square. No cheating allowed.

>>9813403
>I don't know why but I always lose it here
Why?

brainlet CSfag here. How do I derive the inverse hessian BFGS update formula (2nd equation) from the hessian update formula (1st equation)?
I know you're supposed to use the Shannon-Morrison-Woodbury formula, but I can't seem to get the correct result

>>9814328
>brainlet CSfag
No need to say "brainlet". It's redundant.
>How do I derive the inverse hessian BFGS update formula (2nd equation) from the hessian update formula (1st equation)?
Ask in >>>/sci/sqt/.

>>9813775
>>9813910
it's a meme...
>>9813223
sad

>>9815199
>it's a meme...
What do you mean?

>>9814476
>that whole post
I should've know /sci/ doesn't have any actual academics. It's just filled with undergrad wannabes and IQ shitposters

What is a good and modern intro to topological K-theory?

>>9815236
hatcher's vector bundles and topological k-theory
if you hate hatcher, then probably karoubi although it's a big step up in difficulty

I found atiyah's book to be pretty good however, although opinions may differ.

Where can I learn about modeling phenomena using stochastic processes? I recall seeing a paper that used some process to model the vibrations on the web of a spider as the spider moved and I remember thinking that was cool. Want to learn more. I've studied stochastic processes a fair amount, but haven't really done much in the way of applying it.

>>9815315
ignore the troll

>>9815295
Thanks!

Why do you think I'm trolling

>>9815385
if you're the person i responded to, i just preemptively told you before someone posted about how this is not math and to go to /toy/ or some shit

>>9813228
>>9812543
flunked out of 10th grade due to medical problems.. been bugged for years that I never learned beyond algebra, I'm applying myself to learn again and attend college.

>>9813223
>>9813224
>>9813225
>>9813227
>>9813230
>>9813233
>>9813235
>>9813236
Jiggin Jighaboos holy fuq, didn't expect such a thorough list of material. Godbless you anon.. ill study what you've given vigorously.

>>9815609
Make him proud.

>>9801476
hey, physics grad student here
Basically never. In undergrad, I used Newton's notation more often in my ODE's classes than in physics classes. Baby's first classical mechanics used Newton's notation, that was it.>>9801476

>>9814108
Well I found a Fibonacci style formula how to find x such that x^2 = n(n-1)/2, and a much more convoluted one for n, but I don't know how they work or how to prove it. Is this supposed to be hard for someone who has no experience with number theory or modern maths in general?

>>9812827
i wonder who is behind this post

Is it over for me if I go to a state school (America) where complex analysis is a fucking graduate course....

>>9815657
I wish. Then I wouldn't be a brainlet.

>>9815689
there are state schools where complex analysis is an undergrad class
I go to one of them

>>9815772
congrats, i don't
a uc i'm guessing?

>>9812827
interesting bit of trivia - to exclude jews from higher education, what the soviets did was write problems with simple but difficult-to-derive solutions so that the jews who didn't pass couldn't complain about having unfairly difficult entrance examinations.

https://arxiv.org/abs/1110.1556

I'm poor, is studying from pdf textbooks meme??

>>9815498
Gotcha. Yeah that guy who calls anyone who posts about analysis or probability an "engineer" is pretty annoying.

>>9815804
Why would it be? Some studies say that retention isn't as good from an e-book, but I think as long as you work exercises and prove the Theorems, you'll be fine.

>>9815811
>Yeah that guy who calls anyone who posts about analysis or probability an "engineer" is pretty annoying.
I'm not a "guy".

Guys, is there any comprehensive book on trigonometry that goes from basics all the way to Integrals? My professor is skipping everything involving trig, so I need to study it on my own.

>>9815804
yeah, you should learn it from a hard copy (maybe from the library) and then save the pdf for reference. pdf's are totally fine for doing homework and whatnot, but they suck for learning something new.

>>9815852
Serge Lang's Basic Mathematics will have everything you need to know on trig and more. Also a great reference for when you forget trig identities the day after finishing calc 2.

>>9815857
>Serge Lang's Basic Mathematics
Thank you, anon. I'll attack it next week after exams.

How do you know if it's worth trying to become a research mathematician?

>>9816034
>How do you know if it's worth trying to become a research mathematician?
Ask yourself whether the work of a research mathematician is worth the average salary of a research mathematician

>>9816034
if you are a research mathematician, you probably don't have to try to be at that level

Give an example that illustrates the connection between triangular numbers and factorial

if you can't do this, you're too busy huffing your own farts to truly understand mathematics.

>>9816034
>doesn't define a worth metric
son, you've already failed

https://imaginary.org/background-material/school-taskbook-from-5-to-15

>The taskbook for school students by renowned Russian mathematician Prof. Vladimir Igorevich Arnold.
Surely you can solve all the problems aimed at children 5 to 15 years old, /mg/.

>>9816093
>Give an example that illustrates the connection between triangular numbers and factorial
prod_{k=1}^n 2*T_{2k+1} = (2n+2)!

>>9816101
bourbaki was a mistake
What you said is extremely simple

>>9816099

>>9816102
>What you said is extremely simple
give me one (1) good reason to look for anything less simple

>>9816110
I don't need to address your failure

>>9816112
>I don't need to address your failure
and I don't need to do your homework for you

>>9816113
As long as you're comfortable with my feeling that you have failed, I am comfortable with you thinking this is homework

i'm about to drop out from mathematics after 4 years but am still taking some random electives to explore my options
what buzzwords should i have learned in math to impress chicks

>>9816114
>As long as you're comfortable with my feeling that you have failed
I don't feel anything regarding your feelings

>>9816115
>i'm about to drop out from mathematics after 4 years
why would you drop out when you have enough credits to graduate?

>>9816115
just put on some 3blue1brown and aesop rock

>>9816121
Try your hand at VI Arnold's problems for 5-15 year olds to cement your humiliation

>>9816122
i don't, i'd have at least another year of work to do
and it'll be a fuckton of work as a consistent C student who didn't understand anything

>>9816123
gay

>>9816123
>Try your hand at VI Arnold's problems for 5-15 year olds to cement your humiliation
no thanks, I've already felt humiliation from his books for adults

>symbol pushers

If you can't explain yourself, then you probably don't understand things as well as you think you do. This is the secret wisdom of teaching, which often gets shat on.
What goes on in your spergy head is of no consequence to anyone. If you can't relate it from any possible point of inquiry (even if you must trace a path of prerequisites) then you simply don't understand it as well as you think you do.

>>9816163
no shit

>>9815626
Yes, I would consider it as a hard problem if you're not familiar with number theory. In fact, I posted it because I found a very similar homework question in a number theory text I found online.

>>9816123
>problems for 5-15 year olds
>the basel problem

>>9815835
Yes you are, fucking monster.

>>9806713
>math is a language
No.

>>9808709
>gypsy is rumored to get a fields medal

>>9816127
>algebraists

>>9800443
The answer to the OP is definitely Tim Gowers' blog. The mathematical content aside, he occasionally bangs out a great essay on current political events in the UK (the EU referendum and Alternative Vote referendum come to mind).

Obviously he political insights aren't anywhere near as advanced as his mathematical ones, but I find it interesting reading the political thoughts of somebody who is a genius in another field.

https://gowers.wordpress.com/2016/06/02/6172/

>>9817490
>not Baez

>>9817496
>No Fields Medal
>Counting it as a math blog

>>9816163
Yes, isn't that common knowledge that understanding something means that you can explain it?

>>9817490
>politics

>>9818101
selfies belong on /soc/

My uni is offering a course on Lebesque Integration in the fall. I've taken real analysis I/II but it's been a while since I've look at my notes. What should I brush up on before considering registering?

>>9818604
Set theory and the fundamentals of topology.

>>9818679
How far into set theory should I go? Just the practical stuff?

>>9818604
Just go read Rudin's RCA first 2 chapters and Chapter 11 from Rudin's PMA. If you've taken real analysis already this should be a breeze.

>>9818683
I'll take a look, thanks.

>>9818604
What >>9818679
said should have really been taught in your real analysis course. Lebesgue integration is part of analysis; it's just a more general way to integrate compared to the Riemann integral. Once you get it, you start to realize that there was no real reason to learn the Riemann integral except as a kind of historical exercise or stepping stone to be replaced with Lebesgue integration.

>>9805055

SAUCE

>>9818702
Tell me a single instance where you had to integrate something that wasn't continuous besides a set of measure 0. The lebegue integral is pretty meh. Measure theory is the important stuff.

>>9818759
Fair enough, but the importance of formally defining and exploring the properties of Riemann integrals are still greatly exaggerated in analysis classes. Calculus student, obviously, need to know how to integrate. But a student learning analysis may as well know the pre-requisite measure theory and be able to define a Lebesgue integral, measure, sigma algebras or sigma rings depending on which one they're taught to work with, Lebesgue measure and all that and have the Riemann integrals downgraded to a historical proof or exercises.

>>9818765
I know this point is exaggerated, but I'm in a literally who spic country uni, and we learn Lebegue in our analysis class. Riemann is considered calculus stuff.

nLab is a gift from heaven

>>9818927
The formatting on that site is shit.

>>9816163
>this is what the average undergraduate /sci/ poster believes is good advice to spread to his studious peers
why the fuck do i interact with you retards

>>9808709
Who're the other 2?

>>9819236
Mochizuki is definitely one of them.

>>9819248

>>9819248
He's 2 old son.

>>9806180
>May 9

I need to pick one from the following for next semester

>applied (ew) linear algebra
>cryptography and coding theory
>complex analysis
>dynamical systems
>graph theory

which one and why /mg/?

>>9820083
what are your interests?

>>9820106
combinatorics/cryptography/algebra

I've already filtered the courses which I find boring

>>9820117
flip a coin heads for cryptography tails for graphs

nigger

>>9820083
>one
why not take all of them

What's a good introductory text to topology?
Taking algebraic topology next sem and I thought I should probably learn the basics first.

>>9820387
Dugundji.

>>9820387
Simplicial Homotopy Theory by Goerrs and Jardine.

>>9820303
already had other slots filled with more important couses

What's the fastest suicide method?

>>9820117
>combinatorics/cryptography/algebra
>>>/g/

Any good books to learn quantum mechanics for someone with a math background (never seen it before)?

>>9820813
Dividing by zero

>>9820935
Von Neumann

I'm losing weight and learning spectral sequences.

>>9820945
>I'm losing weight
bulk up pussy

>>9820948
I must become a trap.

>>9820941
thank you

what do you guys do about the topics you never mastered in high school, For eg i dont know much of statistics, probability and complex no and so if i go to college, will i get fucked?? Or will learning high level stuff in college will automatically fill in the gap left by my poor high school knowledge?

I hate this general.

>>9821144
>I hate this general.
Why?

>>9818759
the point of Lebesgue integration is that it gives you nicer function spaces.

>>9821144
sorry, senpai.

>>9821142
I'm grinding through college and doing stupid shit like >>9820813
Be sure your foundation is solid, otherwise you'll eventually out yourself as a fraud just like I did.

>>9820935
Try asking in >>>/toy/ or >>>/x/.

>>9820387
may's concise course

Where can I find this in a format that's compatible with my kindle? It won't process the PDF for some reason.

https://www.seas.upenn.edu/~cis515/linalg.pdf

>>9821640
You can convert a pdf to just images (or some other format just bundling images), maybe a kindle can display that?
Or look on libgen.io, but I doubt another version exists.
What even can a kindle read?

>>9800443
>what's the second best math blog?

using Mathstodon.xyz as a social network has taken the place of blogs for me.

My prospective PhD advisor in number theory/algebraic geometry has just sent me an email asking me to ask him any questions. What are some good things to ask to make it seem like I'm interested but not badly informed ?

I just don't really have any pressing questions, since everything I'd think to ask I'd already researched a bit, but I don't wanna look like I'm a fucking mushroom

>>9822250
>asking a bunch of college drop-outs for academic advice
This isn't only against the rules, it's also stupid.

>>>/adv/

>>9822327
I want to ask a targeted question about the field, I don't want to ask pleb shit about "how's the student life"

niggas whats a good abstract algebra textbook that I should be using for self study?
Dummit seems to be really nice from the ratings and from a cursory look but the exercises don't have solutions and the ones on the internet got taken down.
Pinter's book seems to be pretty good too but people are saying that he rushed through the later chapters.
Herstein's seems like it's the most digestible but it doesn't cover nearly as much it seems.

It would also be nice if i could get a full solutions manual for any of them.

>>9822336
jacobson

>>9822250
ask him if root 2 exists

>>9822250
Maybe ask him for references on some stuff you're interested in, or connections between something specific you would like to study and what he does or things like this.
Don't worry about sounding badly informed really, you are an undergrad. No matter how interested you are, you cannot have the same depth as someone who spent three years nonstop studying a specific subject (unless you are a very exceptional case and did it during your undergrad, but I doubt that you would be asking about this if it were the case).

>>9806742
well in that case, why dont you go ahead and show me some sorting algorithms in machine code

>>9822336
>niggas whats a good abstract algebra textbook that I should be using for self study?
catch-all algebra books are a literal waste of ink, get a proper book on group theory, a proper book on homological algebra, etc.

>>9822334
See
>>9822327

>>9800443
Hello mathematicians. I'm not often on this board but I have a question I have no idea how to solve. How do I add 2 circles together into 1 large circle? I have the diameter of 1 circle, what would the diameter be if I had a circle with exactly double the area, not asking for anyone to work it out but just tell me the equation to work it out myself

>>9822822

>>9822843
I make polythene film for a living, machines wind it into frames. I want to know if I can run a job onto 1 frame or if have to split it over 2...1 roll will have a diameter of 1.7 meters, the max size I can fit in a frame is 2.3 metres

>>9822888
Doubling the amount of film on a roll doesn't double the rolls diameter as each Revolution takes slightly more film to complete ... I want to know how to work out the size of a roll by doubling the meterage of film

Hello comrades, I’m currently studying Abstract Algebra. Just wondering if there’s a good Abstract Algebra problem/exercise book complete with solutions for independent study? Like ‘Burn, A pathway into number theory’ for Number theory?

Any recommendation?

>>9802285
So go fuck yourself

>>9814328
What a strange notation. Please enlighten me with your CSfaggotry, I feel interested in this mystical way of writing things.

So much arrogance and emotion in what should be a rational and sane thread. Be ashamed, autists.

>>9823114
>So much arrogance and emotion in what should be a rational and sane thread.
What do you mean?

>math grad
>not worked on anything in a month
kill me i'm going back to work tomorrow

>>9823187
>kill me i'm going back to work tomorrow
why not today?

>>9823195
dont want to

>>9823187
just drop out

>>9823226
kill yourself faggot

>>9822336
https://www.ocf.berkeley.edu/~abhishek/chicmath.htm#i:general-abstract-algebra
http://4chan-science.wikia.com/wiki/Mathematics
Maybe this will help you

>>9823228
>faggot
Why the homophobia?

>>9823234
because i'm homophobic
is that meme still spreading here?

>>9823234
Homophobia is a good thing since gay is wrong. It’s proven that gay sex have an extraordinarily high chance of giving you Aids and HIV.

>>9823007
There is a website with solutions to D&F (really too many exercises), and there should be a pdf with solutions to Kleshchev Algebra. There are also some books such as Dixon on group theory with exercises/solutions, but I would not recommend it. Instead try doing the exercises from any book by yourself, think, and if you really can't or you picked up Lang search on stackexchange, probably someone already asked there.

What is the best book to learn about metric spaces? I'd prefer a book with little to no topology. I want the metric-dependent definitions (such as the definitions of neighbourhoods and open sets)

>>9822822
The area of a disk of radius r is proportional to r^2, hence to double the area, you have to double r^2, ie. multiply the radius by sqrt(2)

>>9823249
Thanks buddy.

>There is a website with solutions to D&F
By any chance, do you have the link to it?

>>9823335
This. I'm also the other guy that asked. I would really appreciate it as well.

I'm a complete and utter dumbshit and I'm not too proud to start right from the start

>>9823445
keep at it
math makes the world a better place

>and here are all the contributions mathematics has made to the 20th century

>>9823573
0/10

>>9823577
we understand each other perfectly.

>>9823249

Last I checked that site was taken down. Its been at least a year since I looked though.

>>9823445

Starting right from the start is the best way to address any deficiencies from the get go.

Building a strong and complete foundation.

>>9823570
>>9823594
Thanks boys. I tried starting at year 7/8 but I was missing some basic things so I decided to just swallow and go to the extreme basics. Of course, I didn't learn anything from grade 2 and below, but it has eliminated any doubt

>>9823649

You might find yourself surprised by what comes easily to you later on, compared to how you might have gone with it in school. I certainly did.

>>9823573
Thank god I don't have to contribute to Pr*gress if I am doing math.

>>9823582
Can you at least give me the link? Maybe the wayback machine archive it or something.

>>9824204
That's unfortunate, I didn't know it was taken down. If I'm not wrong it was a wordpress(?) blog named crazyproject. It really was crazy, so many exercises. Anyway >>9823007 should be able to do most of them alone (but it will take a huge amount of time), and for harder one check stackexchange.

yo /mg/, not sure if this is the right place to ask a PDE question, but here I go anyways:
So I wanted to test my finite element code on some test problems with known solutions, but I'm struggling with this one:

I want to find eigenfunctions of the laplacian on the unit disk with homogeneous neumann conditions, i.e.
Find [math] u \in H^1(D) [/math], s.t.
[math] \Delta u - \lambda u = 0 [/math] on [math] D [/math] and
[math] \frac{\partial u}{\partial n} = 0 [/math] on [math] \partial D [/math]
where [math] n [/math] is the outward normal at the boundary.
Let's hope the latex didn't fuck up. preview doesn't seem to work for me

Now this is not a well posed problem since there multiple solutions, including [math] u = 0 [/math] and combinations of bessel and trigonometric functions, leaving an infinite amount of solutions, since rotating a solution is still a solution.
Now my question is: What kind of additional constraints would lead to a nontrivial unique solution?

>>9823445
You're doing the right thing. I should be doing it too.

>>9823578
Well played, even though you're wrong.

Hi guys just got an f on my linear algebra exam. I've never been this suicidal

>>9824893
Just do it again next semester.

>>9824922
I want to talk to the professor to see if we can cut out a plan first

Guys, am I a complete brainlet for not being able to understand inverse Laplace transformations? I understand the concept and can do all the exercises, it's the ones with IVPs that trip me up. I have a shit professor though, so I have an excuse to ease my consciousness.

>>9823320
Thanks for the response. I don't understand it but thanks anyway.

hey guys, is there a /sci/-approved introductory statistics book?