/sci/ - Science & Math

>>9695018
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).

>>9695019
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).

>>9695020
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.

>>9695022
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).

>>9695024
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).

>>9695026
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.

>>9695028
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)

bump

But is this a real path? I mean, is it realistic to understand this in order, or is this bait for people unfamiliar with these subjects?

no, it's some deluded russian guy's list that gets posted here as a meme all the time

>>9695686
>deluded russian guy's list
How you know he is russian?

>>9695686
WHO?! Where does this fucking meme come from??!!!

>>9695759
>WHO?!
https://en.wikipedia.org/wiki/Misha_Verbitsky

>Where does this fucking meme come from??!!!
http://imperium.lenin.ru/~verbit/MATH/programma.html

>>9695686
>nazbol
>deluded
How about no? Verbitsky is a good man.

>>9695686
>no, it's some deluded russian guy's list that gets posted here as a meme all the time
But he's an actual working mathematician unlike the people who make memelists on /sci/

>>9696031
He literally made that memelist

>>9696083
He's talking about other lists that people post here faggot.

>>9695018
A lot of it isn't even related to math. It's acceptable if you can filter stuff well, which probably isn't the case if you're just starting out.

>>9696152
>A lot of it isn't even related to math.
What do you mean?

>>9695576
>is it realistic to understand this in order
Yes.

have any other professors made ambitious lists like this?

>>9696998
>ambitious
In what way?

>>9697008
the timespan

>>9696998
math 55 is the closest familiar thing but has nothing to do with this list

>>9696998
see >>9696242

>>9697436
Very nice.

Anybody got a math curricula for me starting from Calculus, along with books and other resources? Currently working through Spivak's Calculus

>>9698005
the FAQ has genuinely good info running from Calculus 1 (it recommends Stewart then Spivak) through more or less a full BS in math. all you need is time, a computer, some paper, and the ability to find quality books on libgen. there's also a huge torrent out there that has every book used in course 18 (math major) at MIT

>>9697414
math 55 is linear and abstract algebra. the list ITT recommends basically doing the first semester (math 55a) in high school and not really touching the curriculum of 55b

you're not wrong just providing more context for people not familiar with the course

>>9698022
actually correction, linear and abstract algebra are covered in 55a, 55b is real & complex analysis and would be covered in the "freshman" section of OP's list

>>9698020
Holy shit thanks. Do you mind telling me where I can find this torrent? I searched Kickass.to and thepiratebay

>>9698028
yeah I got it from thepiratebay.
search "MIT math books" and it should be the first result
The books are generally good scans, but check libgen.io if you ever need better quality. Sometimes you'll have to download a couple and check them out, but that is going to be a fraction of the time you spend on the book so no biggie. Still 100x faster than even going to a book store.

I also recommend getting a good reader app that can read .djvu files. There are several that are free and open source. Djvu is considered a "superior" format to pdf for various reasons (almost always smaller file sizes and the readability is great), although this is less the case now than it was in 2006. But because of that a lot of math profs and math students that have created these scans made them as djvus instead of pdfs. Every once in a while you'll get a prof who has uploaded an advance copy pdf that they were given to review and the quality is absolutely based straight from the publisher.

Fuck publishers man

>>9698028
also MIT has been one of the best schools about making their classes open, especially math, so this is another good resource for you:

https://ocw.mit.edu/courses/find-by-department/

If you want some structure as you go through a book, access to course materials etc, it's there

>>9698038
Just started downloading, loads of books! Is probability worth learning? Endgame is topology, so could I get away with skiping probablity?

>>9696226
Retard.

>>9698209
>Retard.
What do you mean?

>>9698210
Can you keep up?

>>9698216
>Can you keep up?
Can you isn't even related to math?

>>9698221
>Can you isn't even related to math?
What do you mean?
Can you speak English?

>>9698224
>What do you mean?
Can you keep up?

>>9698236
>Can you keep up?
What do you mean?
Are you following this conversation? Am I speaking to an ESL monkey?

>>9698241
>What do you mean?
Can you isn't even related to math?

>>9698244
Are you okay?

>>9698249
>Are you okay?
Can you keep up?

>>9698253
Please. Have you considered seeking help?
Are you genuinely not of sound mind?
Can you understand?

>>9698266
>Have you considered seeking help?
Can you isn't even related to math?

algebra is "mathematics" for people who don't like mathematics

>>9698366
I don't know what algebra you're referring to.

Terry Tao says it's the mathematical subject that he finds most challenging and he often has to put algebra in terms of other fields to understand it the way other mathematicians understand it. Any babby math in any field is babby math, and any field can be made complex until you press the boundaries of what mathematicians have currently proven and congratulations you probably have done enough work in the field to merit a PhD if you get to that point.

Your post was low effort/low IQ

>>9698203
I don't know topology so I can't say. But if you read the preface Munkres wrote he will probably tell you what you need to know. Since you're self-learning, prefaces are always good to at least skim, and they will substitute somewhat for the direction of a professor

>>9698620
>Terry Tao says it's the mathematical subject that he finds most challenging
That's a common thing among non-mathematicians.

Is this the math general thread?

So I'm learning about tangent vectors for smooth manifolds and I found the definition in terms of derivations to be pretty cool. I was surprised to see that from such a barebones axiomatic definition they end up behaving just like partial derivatives. In other words, by merely requiring that they be linear and obey the Leibniz product rule they end up being isomorphic to directional derivatives in R^n

Does this imply that derivatives can be defined in a similarly simple axiomatic way?

>>9698203
>Is probability worth learning? Endgame is topology
Probability isn't a mathematical topic so it has no bearing on topology whatsoever.

>>9698696
>Probability isn't a mathematical topic so it has no bearing on topology whatsoever.
This is a non-sequitur.

>>9698705
>This is a non-sequitur.
This is a non-sequitur.

>>9698696
define "mathematical topic"

>>9698710
Definitions of non-mathematical terminology such as "mathematical topic" are meant to be discussed at >>>/lit/.

>>9698715
If "mathematical topic" is undefined then "Probability isn't a mathematical topic" is a meaningless notion.

>>9698734
>"mathematical topic" is undefined
This is not true unless you assume it is. Refer to >>>/lit/ for a more detailed discussion of these matters.

>>9698739
>This is not true unless you assume it is.
It's true as long there is no definition.

>>9698744
>definition
Refer to >>>/lit/ for the metaphysics general.

>>9698751
>Refer to >>>/lit/ for the metaphysics general.
What does metaphysics have to do with it?

>>9698005

>>9698904
>Basic Mathematics
>tough
is it really, considering the intended level of course? Most of the exercises seem piss easy (I'm halfway through) and I though I was struggling with proving several things so far only because I'm a brainlet.

>>9698959
sure, most of the exercises are pretty straightforward, but there are still some tough exercises for someone only doing precalc

>>9698203
It is worth knowing a bit of probability. It is pretty interesting.

>>9698904
>look I posted the memelist again!

About Eilenberg-Mac Lane spaces, how does one construct [math]K(G, 1)[/math] for an arbitrary group [math]G[/math]? For abelian groups, one can use a free resolution which will correspond to two bouquets of spheres of the desired dimension and a mapping cone, but this doesn't quite work for arbitrary groups.

Can we take a moment and laugh at the fact that one of the math degrees offered by my university (the BA) doesn't require anything higher than proof-based Linear Algebra, e.g. Axler.

>>9699989
Which developing country is this?

>>9698020
>Stewart

Might as well eat our own shit.

Redpill me on vector bundles /mg/.

>>9700182
Texas.

>>9700357

>>9700421
To be fair, there are very few mutts in the math department. They stick to COLA/Business.

>>9700357
Which university?

>>9699989
>proof-based
As opposed to what?

>>9700718
>As opposed to what?
Not being proof-based.

>>9700720
And why would you even mention that in this thread? There are other boards for non-mathematical stuff.

>>9700718
Matrix arithmetic

>>9700721
Are you mentally deficient in some way?

>>9700721
>And why would you even mention that in this thread?
Why wouldn't I?

Why do most elementary geometry books exclude trig as I was taught it (functions, graphs, identities, side-angle-side, law of cosine, etc)?

>>9700730
What constitutes "elementary geometry?"

>>9700732
Scheme theory

>>9700722
>Matrix arithmetic
And how would that be related to linear algebra?

>>9700745
Most (bad) universities call the class teaching engineering students how to find eigenvalues "linear algebra."

>>9700745
>And how would that be related to linear algebra?
https://en.wikipedia.org/wiki/Linear_algebra#Matrix_theory

>>9700748
I couldn't care less about what bad universities consider to be linear algebra. "Matrix arithmetic" and other such trash belongs elsewhere.
>>9700751
>wikipedia
Stopped reading right there.

>>9700753
And I really don't understand what you're point is. There's a common abuse of language that occurs, and I was explaining it because someone had apparently never heard it used before. You can hold whatever opinion you like on the importance of matrix arithmetic.

>>9700755
>And I really don't understand what you're point is.
It's pretty easy to understand, actually. Off-topic trash does not belong in these threads.
>There's a common abuse of language that occurs
It might be common in engineering departments and other places where engineers gather as I've never heard it used anywhere. Maybe you could try pushing that terminology elsewhere.

>>9700755
>importance of matrix arithmetic
Nobody was discussing the importance (or lack thereof) of a non-mathematical subject in a mathematical thread.

>>9700760
Someone made a point of mathematics education at their university (clearly related to the topic of the thread) Someone else asked a question about that post. I answered. I don't see that as off topic.

Again, I don't really care what your opinion on the terminology. I'm not going to get into an argument about the nature of language on this board. To give an example of the incorrect conflation of these two things: https://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/

>>9700730
>Why do most elementary geometry books exclude trig as I was taught it
Because trig isn't mathematical in nature while elementary geometry (EGA1, EGA2) is.

>>9696126
Are you stupid?

>>9700765
>Someone made a point of mathematics education at their university
Clearly it isn't mathematics education if they consider "matrix arithmetic" to be linear algebra. So that makes it pretty blatantly off-topic.
>To give an example of the incorrect conflation of these two things
I couldn't care less about engineers and other groups affiliated with engineering using incorrect terminology.

>>9700775
I'm going to stop responding to you because it's clear now that you're pretending to be retarded to rile me up.

>>9700778
>you're pretending to be retarded
How so? I'm just explaining that off-topic trash isn't welcome here. You don't need to get riled up over easily understandable facts.

>>9700775
define "mathematics education"

>>9700780
>define
Try making a thread on >>>/lit/. They should be able to help you with defining non-mathematical terms.

>>9700784
Undefined terms have no place in a mathematics thread. If you want to discuss ambiguous notions you should try making a thread on >>>/lit/.

>>9700789
The process of defining non-mathematical terms has no place in a mathematics thread. The proper place for such discussions is >>>/lit/.

>>9700789
>mathematics
What do yo mean?

>>9700799
Clearly, the final arbiter of what is and is not "mathematical" should be left to random fuck on the internet, rather than someone with expertise in the area or the general process by which language changes.

>>9700799
>What do yo mean?
The study of sets.

>>9700803
>The study of sets.
Refer to >>>/lit/.

>>9700802
>expertise in the area or the general process by which language changes
Sounds like something >>>/lit/ would specialize in. Try discussing this deep topic with them. I'm sure they'll be able to help you.

>>9700802
>Clearly, the final arbiter of what is and is not "mathematical" should be left to random fuck on the internet, rather than someone with expertise in the area or the general process by which language changes.
Irregardless of whether language changes, what is the current definition of "mathematical"?

>>9700806
>what is the current definition of "mathematical"
"mathematical" is a non-mathematical term so it's definition has no place in these threads. Try discussing it at boards such as >>>/lit/ and websites about logic and philosophy in general.

Actually, yes. I agree with you about the deeper ontology of this. But I have a question. Is this truly justified philosophically speaking?

>>9700806
>definition of "mathematical"
This is a meaningless notion.

>>9700809
>"mathematical" is a non-mathematical term
What do you mean?

>>9700809
>"mathematical" is a non-mathematical term so it's definition has no place in these threads.
You want "its" here.

>>9700820
>You want "its" here.
How so?

>>9700818
Can you keep up?

>>9700824
>Can you keep up?
What do you mean?

>>9700821
What do you mean?

>>9700821
>How so?
Contractions are best discussed on >>>/lit/.

>>9700825
Are you following this conversation?

>>9700831
>Are you following this conversation?
Yes.

>>9700817
>This is a meaningless notion.
What do you mean?

>>9700832
The past three thousand years of world history prove you wrong.

>>9700833
I mean what I mean.

>>9700836
>I mean what I mean.
Which is?

>>9700834
>The past three thousand years of world history prove you wrong.
What do you mean?

>>9700839
You have much to learn.

>>9700837
The ontological meaning of my sentence.

>>9700841
>You have much to learn.
About what?

>>9700844
>The ontological meaning of my sentence.
Which is?

>>9700849
Oh yes.

>>9700852
>Oh yes.
Oh yes what?

>>9700853
Brace for impact!

>>9700865
>Brace for impact!
What is coming?

What's the official /mg/ 'end of proof' indicator? The tombstone, Q.E.D., B.Y.U., something else?

>>9700866
This is my world.

>>9700877
>This is my world.
In what sense?

>>9700887
In the sense that it is my own invention. I am law here.

>>9700888
>I am law here.
Mathematicians use "we", not "I".

>>9700890
Why am I so sinful?

>>9700890
We am the law here

>>9700893
>Why am I so sinful?
Self-loathing is best discussed on >>>/r9k/.

>>9700922
Could it be that you have already witnessed the end?

>>9700933
>Could it be that you have already witnessed the end?
The end of?

>>9700938
Herein lies your undoing.

>>9701016
>Herein lies your undoing.
Undoing what?

>>9701020
Please answer the question.

>>9701026
>Please answer the question.
What question?

How do I map one set of barycentric coordinates onto another set of barycentric coordinates? I have a pair of nested triangles that share an edge, each with coordinates mapped onto them, and I need to map the outer triangle's coordinates onto the inner triangles coordinates in a succinct way.

What are some fields of mathematics that aren't for babies?

>>9701051
Is it sufficient to use an affine map sending vertices to vertices? I think it should be, but I haven't played with those for a long time.

>>9700806
>Irregardless

>>9700227
just my masters thesis on them, AMA

>>9700872
that's that

>>9700848
About everything

>>9700431
Im a nigger what does COLA stand for my dude?

>>9701384
>Im a nigger
Then you are not welcome in these threads.

>>9699081
The classifying space BG is a K(G,1).

>>9700227
what kind of vector bundles

>>9701384
College of Liberal Arts

>>9700464

>>9701508
Not true

http://catalog.utexas.edu/undergraduate/natural-sciences/degrees-and-programs/bachelor-of-arts-plan-i/mathematics/

You mean the teaching option?

>>9701526
In retrospect, I was probably remembering the BA as the actuary option (http://catalog.utexas.edu/undergraduate/natural-sciences/degrees-and-programs/bs-mathematics/).). To be fair, though, the BA and the BSA (http://catalog.utexas.edu/undergraduate/natural-sciences/degrees-and-programs/bachelor-of-science-and-arts/mathematics/)) only require the easy version of analysis I (361K), which isn't much better.

>>9701434
And how is BG constructed?

>>9701567
By induction over the class of topological spaces.

>>9701606
Oh thank you for this answer. It really explained a lot of things to me.

>>9701567
Geometric realization of the simplicial set [n] |---> G^n

G^n the n-fold cartesian product

Can we talk about mathematics education in this thread?

>>9701852
I'll allow it

>>9701852
>Can we talk about mathematics education in this thread?
define "mathematics education"

>>9701875
Definitions aren't math, so its off topic.

>>9701878
>Definitions aren't math
What do you mean?

>>9701882
Words aren't math, so your post is off-topic. Could you encode that in the real numbers and post again?

>>9701885
>Words aren't math
What do you mean?

>>9701889
Yeah. Math is clearly the study of Formal Systems; so, English being a formal system, ought to be considered mathematics.

>>9701905
>Formal Systems
define "Formal Systems"

>>9701907
As defined by Emil Post in the 1920s

>>9701920
>As defined
As what?

>>9701925
Please encode your post in the real numbers and try again

>>9701930
>Please encode your post in the real numbers and try again
In what sense?

>>9701931
I cannot provide a definition for you, because definitions are not math and thus cannot be spoken of.

>>9701937
>definitions are not math
What do you mean?

>>9701938

>>9701882

>>9701942
Yes.

>>9701946
The cycle is complete.

>>9701508
More proof that UT is worse than A&M

>>9700769
How is trig not mathematical? That’s an absurd claim so I’m discrediting this as an answer.

Bump

>>9701051
>>9701085
This is what I wound up with:

[math]
\begin{bmatrix}
x_1 & x_2 & x_3 \\ y_1 & y_2 & y_3
\end{bmatrix}
\cdot
\begin{bmatrix}
b_{11} \\ b_{12} \\ b_{13}
\end{bmatrix}

=

\begin{bmatrix}
\begin{pmatrix}
\begin{bmatrix}
x_1 & x_2 & x_3 \\ y_1 & y_2 & y_3
\end{bmatrix}
\cdot
\begin{bmatrix}
b_{21} \\ b_{22} \\ b_{23}
\end{bmatrix}

\end{pmatrix}
\begin{matrix}
x_2 & x_3\\ y_2 & y_3
\end{matrix}
\end{bmatrix}
\cdot
\begin{bmatrix}
b_{31} \\ b_{32} \\ b_{33}
\end{bmatrix}

[/math]

D-does anyone know how to solve for [math]b_3[/math]?

>>9701963
UT is top 14 in math, A&M's department is not strong. Haven't looked at minimum requirements.

Wait, is this the real /mg/?

>>9703401
Specifically they are not in the Amer. Math Society list of top 25 public schools:

http://www.ams.org/profession/data/annual-survey/group_i

This thread wasn't very productive. Try harder next time guys!

>>9695028
> plus the not translated Kollar-Mori and Kawamata-Matsuki-Masuda
Absolute highlight of this list. When you're feeling underchallenged with reading an English text.

>>9703401
They may have smart professors, but that doesn't translate to being a good school. I've never personally taken a class with Caffarelli, but I know enough people who have to never want to.

bump

are there two distinct terms, one which refers to 'smaller' in the sense of x < y, and the other in the sense of |x| < |y|?

>>9696016

does Misha shitpost on /sci/? come on!

what about G***in?

>>9706287
No

>tfw no high school p-adic numbers class

man the guy >>>lit trolling ITT and saying
>define x
and
>[actual math subject] is not a mathematical topic and that trash doesn't belong here
is pretty annoying
but I guess it's fine to talk about easily understandable facts

>>9698287
Kek. You won.

>>9710332
>man the guy >>>lit trolling ITT and saying
>>define x
>and
>>[actual math subject] is not a mathematical topic and that trash doesn't belong here
>is pretty annoying
>but I guess it's fine to talk about easily understandable facts
I'm not a "guy".

Rip!

>>9710854
Wut, he is still alive right?

>>9706287
Well for |x| < |y|, you may say that x is closer to 0 than y

>>9710858
If you checked wiki, it's been updated now.

>>9711093
I'm sad.

I'm struggling to do part b) here. I really don't have any idea on how to start. Complex analysis has turned out to be much harder than I thought...

>>9711852
What can you say about the localization of the zeroes ? Assuming that there were infinitely many, what could you deduce about f ?

>>9695018
>Sets containing only one element are sometimes called *one-element sets*

>>9712117
>>Sets containing only one element are sometimes called *one-element sets*
He/she's not wrong.

>>9712117
Sounds retarded. Is this from mathematical didactics or some equally retarded place?

>>9698366
Wrong.
Algebra is math for autists with a fucked up visual cortex.

>>9698691
Tangent vectors are just the derivatives of a curve in a differentiable manifold.

>>9700730
They shouldn't. It's part of the high school geometry standards, unless you're in some flyover state that let the jesus freaks take over the schools.

>>9700753
>I'm le language game man

>>9712215
Calculus-Volume-I-One-Variable-Calculus-with-an-Introduction-to-Linear-Algebra Apostol

>>9698691
The differential of a function, df, can be defined completely algebraically.

A= ring of (germs of) smooth functions about p on R^n

Then A is a local with maximal ideal m. The cotangent space at p is m/m^2 .

A/m^2 splits canonically as R⊕m/m^2.

The differential map d is then the quotient map A --> A/m^2 followed by the projection onto the second factor m/m^2 under the canonical splitting.

How do you prove that dx/dy = (dy/dx)^-1?

>>9712386
you don't

>>9712388
why?

>>9712400
because

How do I solve improper integrals containing terms like this
[math]\int_{0}^{\infty} e^{-x} x^n f(x)dx[/math]
for some random n and f(x)
chugging in inf leads to 0*inf and I don't know what to do with it and I don't think I can use L’Hopitals rule here

I was re-reading a chapter on the types of convergences in [math]\mathcal{L}(X, Y)[/math] (i.e. uniform, strong, weak), and I had this thought. All our notions of convergence and completeness in, let's use [math]\mathcal{L}(X, Y)[/math] as the example, depend on our notions of convergence in the normed vector spaces [math]X[/math] and [math]Y[/math]. But convergence in those spaces, in turn, rely on how we talk about convergence in their underlying fields, for instance by what topologies we choose for them. In most cases we deal with real or complex vector spaces, and so how we construct the topological space [math](\mathbb{R}, \mathcal{O}_\mathbb{R})[/math] trickles its way up into how we fundamentally talk about vector spaces over [math]\mathbb{R}[/math], operator spaces on those vector spaces, and further generalizations (superoperators etc.).

I'm just kinda rambling but it always strikes me as strange that, despite how much we abstract these notions, they're still ultimately rooted within our "comfortable" world of the 1D reals (or fields in general).

>>9712485
nigger if anyone could tell you the answer to this without you specifying what f you're speaking about they would literally solve every integral in the world

>>9712755
he said random not arbitrary

What's the intuition behind infinite dimensional vector spaces? What is their physical and geometric interpretation?

too bad this list doesn't have recommended books.

>>9712761
>random
No such thing.

>>9713097
>too bad this list doesn't have recommended books.
But it does, look in the parentheses.

>>9712545
Keep in mind that we are finite beings that only know how to do very few things: solve linear systems, count, do arithmetic, do symbolic and numeric computation, etc.
Consequently, even though we build very complicated structures, these are only attempts to describe complicated objects in terms of the few things we can somewhat understand: Real numbers are very complicated, but the fact that the rationals are dense ensures that we are never very far from computing with rationals. Finite dimensional vector spaces are complicated, but the maps between them are described by a finite set of numbers. Separable vector spaces are pretty complicated but the fact that they have a countable dense set ensures that you can hope to make "algorithmic" constructions, (pre-)weak topologies are nice because they have lots of compact sets (which are not very far from finite) and these allow for easy existence theorems, etc.
Basically, people define these things in order for them to stay "rooted within our comfortable world of 1D reals" (and we could actually reduce to finite sets) as you say (otherwise, they would be much too complicated to handle.
It's actually interesting how many difficult theorems boil down to linear algebra, combinatorics, number theory or constructing algorithms.

>>9712773
Same as finite dimensional (basically things that you can take linear combinations of), but bigger.
The typical examples are spaces of "functions" of some sort (continuous functions on a compact space, L^p spaces, holomorphic functions on some region etc.)

You have to have a special kind of autism to think this list is anywhere close to realistic.

>>9713983
>You have to have a special kind of autism to think this list is anywhere close to realistic.
What's not realistic about it? Is something missing?

>>9713966
>Finite dimensional vector spaces are complicated

>>9714009
well real dimensional vector spaces are obviously at least as complicated as the reals

>>9713983
Not autism, just a propensity for IQ signaling and unabashed faggotry.

>>9713983
This. It's merely a draft and pretty much obviously so.

>>9713966
>stay rooted within our comfortable world of 1D reals
>our
Speak for yourself. Not everyone here is an engineer.

>>9698287
THE CHAMP

>>9712237
How so? Algebraic geometry and diagrams make algebra pretty visual.

anyone here read Lang's Basic Mathematics? Are some of his exercises in the geometry section difficult or am I just a brainlet for having problems with proving some of them? I had some college math years ago (most advanced being multivariate calculus and continuous probability and statistics, I think), but nothing involving proofs

>>9714664
>I had some college math years ago
You didn't. None of the things you listed are math.

>>9714724
applied math is still math

>>9714724
>None of the things you listed are math.
define "math"

>>9714776
Math applied in math is still math. None of what he listed is used in math.

>>9714776
>applied math
Refer to >>>/x/.
>>9714782
Refer to >>>/lit/.

is this book a meme for someone with no background in (abstract) algebra

>>9714817
>is this book a meme
Yes.

>>9714817
The book is trash regardless of your background.

>>9714838
>>9714842
Why

>>9714810
Mathematics - The abstract science of number, quantity, and space, either as abstract concepts (pure mathematics), or as applied to other disciplines such as physics and engineering (applied mathematics) [Oxford dictionary]

>>9714852
It takes hundreds of pages to introduce something which should be introduced in the beginning. It doesn't even study group actions.

>>9714859
>applied to other disciplines such as physics and engineering
Engineering and physics are distinct from mathematics. If what you're studying isn't applied in mathematics, it's not mathematics. Refer to the engineering and physics generals if you wish to discuss these topics.

>>9714878
>If what you're studying isn't applied in mathematics, it's not mathematics.
But it is, see >>9714859

>>9714883
>[Oxford dictionary]
Discuss such artistic topics at >>>/lit/ and >>>/his/.

>>9714886
what is wrong with you, man?

>>9714894
I'm not a "man"

>>9714886
>definitions are only valid in my special snowflake sense and what's generally assumed doesn't matter

>>9714886
What definition are you using under which applied maths is not maths?

>>9714896
ah, your that mentally ill tranny who spams 'Lang is a meme' everywhere. it explains a lot

>>9714903
>mentally ill tranny
Can you refrain from petty insults?

>>9714903
>your
I don't understand you. Do you mean "you're"?

>>9714910
>I don't understand you.
Mathematicians use "we", not "I".

>>9714906
but mental illness is not an insult and being a tranny is mental illness

>>9714914
Could you please define "mathematician" for we?

>>9714894
Why do you think there is something wrong with me? I'm just telling him to discuss dictionaries at the proper board.
>>9714899
>what's generally assumed doesn't matter
What's generally assumed by non-mathematicians doesn't matter when it comes to mathematics, you are correct. Feel free to fuck off to a physics thread to discuss physics.

>>9714873
I’m taking upper division honors algebra in the fall, it should be a sufficient intro for self study over the summer shouldn’t it? or is it legitimately that bad? If so, what would you recommend?

>>9714918
see >>9714859

>>9714900
>>9714918
Refer to >>>/lit/ for philosophical discussions on the definitions of non-mathematical terms.

>>9714927
>non-mathematical terms
I don't understand you. What do you mean?

>>9714930
>I don't understand you
Refer to
>>9714914

>>9714930
I mean that subhuman trannies should kill themselves at >>>/lit/.

>>9714920
it's generally assumed by mathematicians too

>>9714933
>I mean that subhuman trannies should kill themselves at >>>/lit/.
Do you need to swear?

>>9714946
insufficient evidence

>>9714946
Engineering and physics being a part of mathematics isn't something which is generally assumed by mathematicians. You can fuck off to >>>/lit/ and physics thread if you wish to discuss the deeper philosophy behind this.

>>9714952
>>9714956
post the evidence that it's generally assumed by mathematicians as it's only your claim and it opposes the general definition

>>9714956
>Engineering and physics being a part of mathematics isn't something which is generally assumed by mathematicians.
This is false.

>>9714961
>This is false.
This is false.

>>9714950
refer to >>>/lgbt/ and >>>/hm/

>>9714971
But we're neither lgbt nor interested in handsome men.

>>9714923
It depends on your goals. If you want to learn algebra in any meaningful way then it's pretty bad. Courses like "upper division honors algebra" usually don't have a whole lot to do with actual algebra so it might be fine for that. If you're interested in learning algebra, try "Abstract Algebra" by Pierre Antoine Grillet. The book by Dummit and Foote isn't too bad either, but it's 900 pages long and still doesn't cover a lot.

>>9714923

>>9714979
it’s at ucla (math 110ah) if that narrows it down
i’ve been reading through pinter and it doesn’t seem horrible. what makes it so bad?

>>9714982
>ucla (math 110ah)
The book the course follows actually seems to be a lot better - https://www.math.ucla.edu/~rse/algebra_book.pdf
>it doesn’t seem horrible
Perhaps to someone who doesn't know the subject yet.
>what makes it so bad?
It's structured in a retarded manner for no legitimate reason and it literally doesn't even study modules and group actions, these things are ubiquitous in modern algebra and math in general.

>>9714817
>This book addresses itself especially to the average student, to enable him or her to learn and understand as much algebra as possible.

>>9714998
>>9714994
>>9714982
"This book is a basic algebra text for first-year graduate students, with some
additions for those who survive into a second year. It assumes that readers know
some linear algebra, and can do simple proofs with sets, elements, mappings,
and equivalence relations. Otherwise, the material is self-contained. A previous
semester of abstract algebra is, however, highly recommended."-Grillet

>>9715003
>first-year graduate students
You can give up right now if this somehow stops you from reading it as an undergrad.
>A previous semester of abstract algebra is, however, highly recommended.
The usual disclaimer for retards.

>>9715008
>Recommending a book by a retarded author .

>>9715016
Can you not read English? How is placing a disclaimer for retards who might buy your book make an author retarded?

>>9715017
lol kid if I wasn't sure you are a roleplayer I would ask why you are wasting your time reading grillet before you even took your first algebra class

>>9715027
Your post doesn't make much sense. It's fine for a first class in algebra assuming you aren't mentally impaired.

>>9715036
It's really not.
>hurr durr if it dosnt categoriy it not math

>>9715041
It's fine to be mentally retarded if you don't try to drag other people down to your level. Maybe someday you will understand, although your type of people never learn.
>>hurr durr if it dosnt categoriy it not math
Who are you quoting? What do categories have to do with this?

>>9715043
It's not nice to troll people into reading somthing they won't be able to understand by saying "You're retarded if you can do this just stop being a brainlet skip about 3-6 months of work and just jump into it. Calculus who needs that shit just read rudin"

>>9715053
If you require 3-6 months of studying algebra to begin to understand that book, you are genuinely mathematically retarded. There is nothing necessarily wrong with that, but I think it's a fact everyone who plans to study mathematics should know about themselves. And why are you assuming he wouldn't be able to understand it? Don't project your experiences and the low intelligence level of "people" around you (yourself included) onto others.

>>9713966
That didn't answer my question at all.

>>9714982
Bruinfag here: there are two sections of 110AH offered next fall. Try to get the one with David Gieseker, not the post doc one. Also if you haven't done linear yet I highly recommend 115AH in the fall with Richard Elman.

>>9715066
>Why do you assume that not everyone has the knowledge equivalent to an undergraduate math degree with a semester of abstract algebra.
Are you implying that you read this book before your first algeba class? Why not just take a graduate level course at that point?

>>9715085
Incoming transfer; planning on doing 115ah and 110ah together

Why would I get vectors that are colinear with one another when using Gram-Schmidt to construct an orthonormal basis for a subspace of R^n? My matlab code looks right.

>>9715109
Are you from OCC? I'd recommend getting in 131AH either the fall or winter; the advantage of fall is you can take honors complex if you do 13ah in fall

>>9715122
how’d u guess? i plan to do real in the winter, then i’ll have linear done with plus 110ah only seems to be offered in the fall so if i waited id have to do the non honors one (could be wrong)

>>9715132
I spent 3 years at OCC, started at math 182H through 290H with Art Moore. I'm grading calculus on manifolds (290H) right now. Been a part of the community (met my wife there). Transferred out spring 2016

>>9715137
hey ryan. small world lmao. talked to you the other day about ucla courses

>>9715093
>the knowledge equivalent to an undergraduate math degree with a semester of abstract algebra.
Such knowledge isn't needed to start reading the book, unless you consider the ability to manipulate basic mathematical constructions and follow basic proofs being equivalent to an undergrad math degree. It's pretty fucking sad if that's the case where you are from.
>Are you implying that you read this book before your first algeba class?
I didn't know about this book at the time, but I was reading notes and books which present equivalent information. I don't see the relevance of this though. How do my actions change the fact that it's fine for a first course in algebra assuming you aren't mathematically retarded?

>>9715139
Nice Connor; glad to know we both browse this shit hole. I'll be around in person if you have questions

>>9715137
She's not your wife yet you absolute shitlord

>>9715137
Ryan, I feel like I have to tell you that she isn't your "wife" anymore. She's with me now.

>>9715153
I always wanted to get cucked.

>>9715140
Glad you were wasting your time reading material you couldn't understand while you were getting a c in your UNDERGRADUATE algebra class

>>9715171
Why are you this intent on projecting your retardation onto others?

(((>>9715183)))

>>9715185
I wish.

>>9715141
Sounds good. Right now I’m just trying to figure out the best way to go about studying in the summer so I’m well prepared for the upper divs

>>9715083
Okay I admit I kinda went on a tangent there. What was your question exactly ? How come every notion of convergence seems to boil down to the topology of R right ? (if not, my bad)
But what I meant to say is that it's not like these definitions were god given and we found that they turned out to have nice compatibility properties. They were *set* this way because they have nice compatibility properties.
There are many ways to topologize a given space, but people choose the topology in order to get nice properties.
And one of the things we want is to be able to express convergence in the new space ideally in terms of convergence of real sequences (that works for a normed space), or at least a bunch of real-valued functions (in the case of a topology defined by seminorms), because these are things we can check in practice.
Not sure if that helps at all but here.

>>9715116
your randomly generated vectors are not random enough. That is, your basis from which you are producing the new vectors do not span the space.