/mg/ - Math general

Depends on the gauge group [math]G[/math] as well as the manifold [math]M[/math] (obviously). In general the local tangent space [math]T_\alpha \mathcal{A}[/math] can always be constructed, but the difficulty comes when you try to patch together the locally-defined measures, and this requires compactification of the moduli space (unless you don't care about convergence).
The main point is that if your homotopy quotient [math]M \times_G G[/math] is nice enough (i.e. vanishing odd-degree cohomology or a stabilizing Leray spectral sequence), then you the path integration becomes just the equivariant fibre integration, and the stationary phase approximation works via Deligne-Verne. In some cases if the Thom class is trivial you can even identify the path integral with the Euler characterisitc.

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My bad, the homotopy quotient is [math]M \times_G EG[/math] where [math]EG\rightarrow BG[/math] is the classifying [math]G[/math]-bundle.

ncatlab.org/nlab/show/(infinity,1)-topos

What are the prerequisites necessary to understand this article

Is it worth memorizing n choose k values?

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No, undrestand the difference between combinations and permutations though.

Basic Mathematics by Serge Lang, along with G Chrystal's Algebra texts. Glad to be part of this intellectual general. I just got to surds.

Sure, I've already got that down.

I just think it might comfy to be able to expand powers of binomials quickly and without having to draw one of those meme triangles.

Why?

where do y'all go when you want to look at futanari porn?

Doing group theory, on quotient groups atm. It's ok. When does the really cool stuff start though?

Are there any professors in /mg/. My linear algebra professor from last semester seems like the kind of guy that would post here (he had a Ph.D. in math and an anime girl desktop background on his laptop).

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>cringe
cringe

Not a prof but I feel like most people in math are on Veeky Forums, we're a fucked up breed.

hello there

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Starting this book. I've heard good things about it (from other physics shitters)

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It's good if you want good visual examples from differential geometry, but unecesarry for full blow diff geo in manifolds. Do it if you have the time, but don't lose a lot of time if your goal is riemannian geometry

Nice find on Chrystal.

>Serge Lang
>intellectual

...

This is a common question but if I want to learn math from the beginning (not counting or arithmetic) do I read the books assigned in the sticky or can I do Khan Academy instead? And to what level, most people recommend Khan Academy up until precalc then read a proofs book and understand mathematical logic, then jump into single variable calculus, linear algebra and then multivariate calculus.

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>This is a common question but if I want to learn math from the beginning (not counting or arithmetic) do I read the books assigned in the sticky or can I do Khan Academy instead?
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.

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).

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).

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.

If you're serious, how long would this take?

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).

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).

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.

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)

Stop posting that garbage analcyst curriculum

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I haven't read a math book for almost 1 month /mg/. Please suggest a mathematical way to kill myself. Thanks.

Read a physics book

That would indeed be a cruel way to die

How can I work out pic related? I've got the normal vector field, but what else do I need for GC?

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This.
This.
This.

you know that it's international give mercy to a brainlet day, right?
p-please.

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quotient objects are the coolest things breh

This is amazing senpai, but where is combinatorial set theory?

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Finishing calc 2 so improper integrals, trig substitution and simple fraccion separation.

abelprisen.no/seksjon/vis.html?tid=73018

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>Formally, given any five points in the plane in general linear position, meaning no three collinear, there is a unique conic passing through them, which will be non-degenerate;
>which will be non-degenerate
why?

What are degenerate conics? For example a single point is one, but that can't happen because you have five distinct points (due to non-colinearity). Similarly, you could have a line which is also degenerate as a conic, but then you would have five (and thus any three) points be colinear. Think of the other degenerate cases through.

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this shit get's posted everytime someone asks for academic advice and it's unreasonable and frankly unnecessary. Going through this rigorous of an outline for years straight at such an early age will kill someone. Assuming the freshman - senior correspond to undergraduate standings, no one is learning such a clusterfuck of topics when they're 18.

When you take a second course.

What is a quotient group to you?

>Formally,

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>Going through this rigorous of an outline for years straight at such an early age will kill someone.
A literal retard spotted.

Sometimes you explain a concept inuitevly, or give an idea and then you give a more mathematical definition, you fucking autist.

Never seen such a thing. Any good definition is already "intuitive" and self-explanatory. Assuming you aren't brain-dead, of course.

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Wow you are so smart and cool. Is anime your secret?

>you are so smart
If you think that someone who is merely not retarded qualifies as being smart, then I have bad news for you. I don't think it's impressive. It would be sad if it was.
>and cool
Thank you.
>Is anime your secret?
It's mainly not visiting websites such as "reddit.com", "facebook.com" and the likes.

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Oh, so it's the autism.

Who ENS here?

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What the fuck is your problem? Your kind is not wanted here, you fucking simpleton.

Oh sorry I forgot my HILARIOUD animr pic and my cliche snarky remark
>mfw physicits don't even know delta-epsilon

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>2hu
>animr
You reek of and . We don't like your kind around here.

>>mfw physicits don't even know delta-epsilon
Who are you quoting?

Is this a new meme?

Or maybe you should fuck off to /jp/ for falling in such retarded bait.

Pretending to be retarded is just being retarded. You are not welcome anywhere outside of .

Refer to

Sounds like an IQ problem, sorry.

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I'm studying algebra one, course I have adhd and never learned math when I was in school.

The componentd if the first fundamental form. Find a parametric representation of your tangent plane at every point if the surface and plug and chug.

I memorized the definition of a manifold, but I still have no clue what it is. I can't envision the difference between a manifold and a non-manifold

Just signed up for a minor in Computational Mathematics. I've already taken diff eq, just gotta take:
>Linear Algebra
>Applied Numerical Methods
And I get to choose two of:
>Intro to Finite Element Method
>Numerical Linear Algebra
>Advanced Applied Numerical Methods
>Stochastic Simulation

What electives do you nerds recommend I take? (I am in mechanical engineering)

Why do you think that your question is appropriate for this thread? Ask in the engineering threads.

Well I'm 3/4 through the chapter, will fully flesh out the stuff about defining the coset operation by choosing a normal subgroup today. My understanding is a quotient group is just a group we get by defining an operation between the fibres of some homomorphism between groups.

Gee, I dunno, I'm asking about mathematics courses. This seems like a pretty good place for that.

None of the things on your list is even remotely related to mathematics.

Switch major to Math then take Real Analysis, Topology, and Probability in the same semester

>I'm asking about mathematics courses
Can you point me to the so-called "mathematics courses" in your post? I don't see any.

>Real Analysis
>Probability
Those are engineer courses, why would he need to switch majors to take them?
>Topology
I'm assuming you mean point-set garbage. That's also an engineering course.

welcome to /mg/. it's not math if it's not arrows.

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>everything is engineering

Everything which is engineering is engineering. Correct.

Everything is arrows, even engineering.

Thumbing through pic related atm.

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So it was a buttmad physishit, who would've thought.

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My economist sister aced her probability exam. It seems my superior intellect radiates itself onto people around me. What I touch turns to gold, where I walk flowers start growing through the snow layer. I am divine.

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To be fair, you don't have to be very intelligent to ace exams in subjects such as "probability" where you can simply make shit up as you go along.

What should I read to start understanding this?

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>abelprisen.no/seksjon/vis.html?tid=73018
Based.

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>mathematics courses
No such thing.

What do you know already?

But she's not intelligent.

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Wew.

Just realised, the area of a circle makes complete intuitive sense when it's described as:

[math]\frac{1}{2}cr[/math]

[math]\pi r^2[/math] makes no intuitive sense at all.

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Thinking it makes any difference what you call certain constants is what makes you an actual brainlet.

Smart people see beauty in tau and ugliness in pi.

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does it take long to learn latex? is it difficult?