Online Course: https://canvas.harvard.edu/courses/90105/assignments/syllabus
I ended up back here coincidentally, and I think my intuition about this was on the right track. I just need to review it a bit. This is entirely a note to self.
Required Reading: https://math.stackexchange.com/questions/3005409/koch-curve-from-cantor-sets-paradox
I don’t know where the coefficients when finding volume of spheres, cones and pyramids come from in 3-dimensional geometry.
I just discovered the WordPress app.
So I guess we start with a circle, which I don’t know how to find the area of either.
I guess we start with a square.
Area is a measure of how much space something is taking up. Physicists call this Mass. We will use these interchangeably for now. Area and Mass are now mathematically interchangeable.
Required Reading: Chapter 1 of Hartshorne, Algebraic Geometry
Grothendieck, A. (1969), “Standard Conjectures on Algebraic Cycles”, Algebraic Geometry (Internat. Colloq., Tata Inst. Fund. Res., Bombay, 1968) (PDF), Oxford University Press, pp. 193–199, MR 0268189.
Grothendieck, A. (1958), “Sur une note de Mattuck-Tate”, J. Reine Angew. Math., 1958 (200): 208–215, doi:10.1515/crll.1958.200.208, MR 0136607, S2CID 115548848
In mathematics, an algebraic cycle on an algebraic variety V is a formal linear combination of subvarieties of V. These are the part of the algebraic topology of V that is directly accessible by algebraic methods. Understanding the algebraic cycles on a variety can give profound insights into the structure of the variety.
The most trivial case is codimension zero cycles, which are linear combinations of the irreducible components of the variety. The first non-trivial case is of codimension one subvarieties, called divisors. The earliest work on algebraic cycles focused on the case of divisors, particularly divisors on algebraic curves. Divisors on algebraic curves are formal linear combinations of points on the curve. Classical work on algebraic curves related these to intrinsic data, such as the regular differentials on a compact Riemann surface, and to extrinsic properties, such as embeddings of the curve into projective space.
In mathematics, the gamma function (represented by Γ, the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except the non-positive integers. For any positive integer n,
Derived by Daniel Bernoulli, for complex numbers with a positive real part, the gamma function is defined via a convergent improper integral:
The gamma function then is defined as the analytic continuation of this integral function to a meromorphic function that is holomorphic in the whole complex plane except zero and the negative integers, where the function has simple poles.
The gamma function has no zeroes, so the reciprocal gamma function 1/Γ(z) is an entire function. In fact, the gamma function corresponds to the Mellin transform of the negative exponential function:
Other extensions of the factorial function do exist, but the gamma function is the most popular and useful. It is a component in various probability-distribution functions, and as such it is applicable in the fields of probability and statistics, as well as combinatorics.
The gamma function can be seen as a solution to the following interpolation problem: “Find a smooth curve that connects the points (x, y) given by y = (x − 1)! at the positive integer values for x.”
Prove/Disprove that every simply connected, closed 3-manifold is homeomorphic to the 3-sphere. (MATH 544/FREN 544)
Introduction to Topological Manifolds, John M. Lee
http://www.numdam.org/numdam-bin/feuilleter?id=BSMF_1902__30_ (Page inexistante)
Translating these papers satisfies the foreign language requirement, solves the statement and will grant you a phd. Good luck!
Proof: For Stevie Wonders Eyes Only