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)
Start to learn math as a hobby
Thank you good sir, you are a wealth of knowledge
>Thank you good sir, you are a wealth of knowledge
I'm not a "sir", but you are welcome.
Yeah, only someone autistic enough to type all that out could also be autistic enough to have gender dysphoria
Speed Mathematics Simplified (Dover Books) by Edward Stoddard
Elements of Algebra by Euler
A Transition to Advanced Mathematics by Smith, Eggen, and St. Andre
it's a pasta newredditfriend :^)
Oh look that faggoty pasta.
What is that pasta supposed to achieve?
Anyways, I think most students (american at lest) will go through exactly what you're talking about in calc II. I did and then I went on to tutor calc II and every single person I helped had the exact same problems.
It's because, in all honesty, math teachers throughout k-12 are generally stupid fucks that don't actually understand mathematics and do a shit sloppy job trying to teach their curriculum. It's a catch 22 because anybody who CAN understand and study mathematics and the more advanced applications isn't going to take one of the worst jobs in america (k-12 teacher).
So you stumble through arithmetic, learn a couple algorithms to solve very specific problems, you'll stumble through geometry, algebra, trig, maybe high school offers some calculus.
But you won't know or understand the purpose of complex numbers, what sin and cos functions are aside from how to solve a triangle or see a little wave form and you'll probably be able to factor or expand terms if you're explicitly told to, you might know some exponential and log properties but again. Nothing other than explicit academic exercises.
Calc I is the highest level of math most non stem fields will require therefore it needs to be easy enough for some liberal arts fag to get a "c".
Calc II Is one of your first real stem courses and will expect you to not only have mastered the previous courses but to have somewhat of an understanding so that you can solve problems with fewer directions. You will then go on to actually learn what you where supposed to learn over the last 12+ years in the course of one semester.
>realize I don't actually understand arithmetic
you don't have to understand it, you just have to be able to do it well enough to learn higher concepts.
then you can go back armed with the skills to actually understand arithmetic.
this is basically the way math is taught; in "passes" where you understand a little bit more each time.
which is the best way for kids because their brains need time to develop abstraction anyways
Posting the meme list
There probably aren't more than a dozen little things you have to know from arithmetic. Some are "understand" things and some are just "that's what the notation means" things. But if you skip learning *any* of them, you're completely fucked, even in algebra 1. None of it is hard. Just learn them and then you'll see why you didn't understand crap, and how simple it really is. And stop glossing over shit. You can learn all of it in like a day of study.