almost three-hundred concepts to know for a pre-mathematician


Elementary maths (nowadays dubbed pre calculus) are those that you are learning since the awakening of your conscience and anyone should master no matter what profession. After you learn those elementary maths that each human mind should learn to perform diligently in a  competitive and cruel society, you would be prepared to begin to learn a set of concepts that serve to you to be considered a professional mathematician, a list of concepts and results is offered to do a contrast against you think you know or should know.

Independently of the flavor or kind of mathematician you gonna be, the following list is aimed for a universal person which is an aspirant mathematician who claim to be useful in the whatever environment where he or she evolves. Remember that as an amateur mathematician you are fated to be abstract minded and perhaps to apply these techniques to the reality, to  model matter and energy as well as all its tangible instances. Abstract reasoning is our essential spirit.

For a beginning professional mathematician an incomplete list of modern concepts and results is:

  1. sets and maps
  2. set operations
  3. symmetric difference
  4. basic types of maps: constant, injective, surjective, bijective
  5. image, pre-image, rank and fiber of a mapping
  6. equivalence relation
  7. equivalence class
  8. partitions
  9. abstraction lemma
  10. combinations
  11. binomial theorem
  12. division algorithm
  13. Euclides algorithm
  14. prime numbers
  15. arithmetic fundamental theorem
  16. co-prime integers
  17. modular arithmetics
  18. mathematical logic
  19. propositional and predicative calculus
  20. axiomatics
  21. deduction theorem
  22. Zorn lemma
  23. Zermelo-Fraenkel set theory axiomatics
  24. binary operation in a set
  25. algebraic structure
  26. semigroup
  27. monoid
  28. group
  29. ring
  30. field
  31. skewfield
  32. module
  33. vector space
  34. algebra
  35. groupoid
  36. subgroup
  37. normal subgroup
  38. coset
  39. quotient set
  40. quotient group or factor group
  41. group morphisms
  42. kernel of a morphism
  43. fundamental theorem on group morphisms
  44. group action
  45. equation class
  46. Lagrange theorem
  47. symmetric group
  48. permutations
  49. Cayley theorem
  50. Cauchy theorem on finite groups
  51. Sylow theorems
  52. structure theorem on abelian groups
  53. composition series
  54. subrings and ideals of rings
  55. algebra of ideals
  56. factor ring
  57. ring morphisms
  58. chinese remainder theorem
  59. integral domains
  60. principal ideal domains
  61. Euclidean rings
  62. prime and maximal ideals
  63. local rings
  64. unique factorization domains
  65. radical
  66. nilradical
  67. nilpotents
  68. polynomial rings
  69. subfields
  70. field extension
  71. Galois fields
  72. Galois correspondence
  73. Abel theorem
  74. quaternions
  75. submodules
  76. module morphisms
  77. noetherian modules
  78. linear combinations
  79. linear dependence
  80. linear transformations
  81. space of linear transformations inter vector spaces
  82. linear functional or covectors
  83. matrices
  84. linear equations
  85. eigen systems
  86. change of basis
  87. determinants
  88. Gram Schimdt
  89. orthonormal and unitary matrices
  90. inner product spaces
  91. Hamilton Calyley theorem
  92. Jordan canonical forms
  93. minimal polynomial
  94. rational canonical form
  95. bilinear maps
  96. quadratic forms
  97. euclidean geometry
  98. Pitagoras theorem
  99. Desargues theorem
  100. Hilbert axioms
  101. parallel axioms
  102. real numbers
  103. euclidean norm
  104. \varepsilon-neighbourhood
  105. bounded sets
  106. supremum and infimum
  107. bounded functions
  108. limits of sequences and limits of functions
  109. algebra of limits
  110. convergence \varepsilon-N and \epsilon-\delta
  111. Cauchy criterion for convergence
  112. real number completeness
  113. numerability or countability
  114. topology of euclidean spaces
  115. acummulation points
  116. partial sums and series
  117. Hadamard formula
  118. tests of convergence
  119. continuity
  120. algebra of continuity
  121. mean value theorem
  122. types of discontinuity
  123. uniform continuity
  124. derivative
  125. algebra of derivatives
  126. differentiability
  127. Rolle theorem
  128. L’Hopital rule
  129. Taylor and Mclaurin series
  130. curvilinear coordinates
  131. partial derivatives
  132. gradient
  133. directional derivative
  134. divergence
  135. rotational
  136. hessian
  137. extrema: critical points
  138. critical sets
  139. Lagrange multipliers
  140. jacobians
  141. chain’s rule
  142. inverse function theorem
  143. implicit function theorem
  144. Morse functions
  145. Riemann sums and integral
  146. fundamental theorem of integral calculus
  147. change of variable integral theorem
  148. Fubini theorem
  149. Lebesgue integral
  150. bounded variation
  151. complex number
  152. complex functions
  153. Cauchy-Goursat theorem
  154. residue theorem
  155. Cauchy integral formula
  156. Cauchy Riemann equation
  157. Harmonic functions
  158. winding number
  159. Laurent series
  160. Rouche theorem
  161. maximum modulus theorem
  162. conformal maps
  163. Riemann mapping theorem
  164. fractional linear transformations or Möbius transformations
  165. hyperbolic geometry
  166. Poincaré model
  167. ordinary differential equations
  168. 1st order ordinary differential equations
  169. initial valued problems
  170. boundary valued problems
  171. existence and unicity of solutions
  172. Laplace transform
  173. convolution
  174. 2nd order and special functions
  175. Bessel functions
  176. Laguerre, Legendre, Airy special functions
  177. hypergeometric functions
  178. partial differential equations
  179. 2nd order classification
  180. Cauchy problem
  181. Sturm Louville method
  182. Laplace equation
  183. diffusion equation
  184. wave equation
  185. maximum principle
  186. Green function
  187. variables separation
  188. line and surface integrals
  189. contour integration
  190. Green, Gauss, Divergence integral theorems
  191. Stokes theorem
  192. duality of vector spaces
  193. dual space
  194. Riesz representation theorem
  195. tensor product  of  linear transformations
  196. tensor product of  vector spaces
  197. multilinear algebra of inner product spaces
  198. parameterizations of curves and surfaces
  199. curvature
  200. torsion
  201. Serret Frenet moving frame
  202. Gauss map or shape operator
  203. Weingarten map
  204. covariant standard derivative or Levi-Civita connection
  205. Gauss equation
  206. Egregium theorem
  207. geodesics
  208. exponential map
  209. geodesic curvature
  210. Gauss-Bonnet theorem
  211. differential forms
  212. wedge product
  213. exterior derivative
  214. Poincaré lemma
  215. Grassmann (or exterior) algebra of a vector space
  216. topological structure
  217. topological space
  218. open and closed set
  219. closure
  220. relative topology
  221. homeomorphism
  222. completeness
  223. hausdorffness
  224. connectedness
  225. compactness
  226. Urysohn lemma
  227. compact open  topology
  228. topological classification of surfaces
  229. homotopy
  230. fundamental group
  231. fiber bundle
  232. tangent space
  233. tangent bundle
  234. differential
  235. Fourier series
  236. metric spaces
  237. spaces of bounded sequences
  238. Cauchy-Schwarz
  239. Jordan-Holder
  240. normed and Banach spaces
  241. Hilbert spaces
  242. Hahn Banach theorem
  243. numerical approximation
  244. interpolation
  245. splines
  246. divided differences
  247. Newton Rapson
  248. Gauss Seidel
  249. Runge Kutta
  250. midpoint rule
  251. trapezoidal rule
  252. Simpson rule
  253. finite element methods p.d.e.s
  254. generating functions
  255. sample spaces
  256. conditional probability
  257. random variables
  258. discrete and continuous distributions
  259. expectation
  260. correlation
  261. moment generating functions
  262. law of large numbers
  263. limit theorem
  264. limiting distributions
  265. linear model
  266. point estimation
  267. interval estimation
  268. confidence interval
  269. hypothesis testing
  270. Lie groups
  271. special and general linear groups
  272. orthonormal groups
  273. unitary groups
  274. Lie algebras
  275. topological manifolds
  276. differentiable manifolds
  277. cw-complexes
  278. Euler characteristic
  279. fundamentals of computer organization
  280. computer graphics
  281. PC architecture
  282. operating systems
  283. pseudo code
  284. c-language
  285. parallel programing
  286. numerical computation
  287. Thruston Nielsen classification of autohomeomorphims
  288. mapping class group
  289. free groups
  290. Schreier free subgroup theorem
  291. free products of groups
  292. Kurosch subgroup theorem
  293. amalgamated products
  294. Bass Serre theory
  295. linear programming
  296. simplex algorithm
  297. minimax
  298. variational calculus
  299. Euler-Lagrange equations
  300. combinatorial phenomena
  301. Catalan numbers
  302. vector fields
  303. tensor fields
  304. Riemannian metrics
  305. Minkowski metrics
  306. spacetimes
  307. gravitational fields
  308. connections

a) cf

b) prepa or prelicenciatura maths

c) more on elementary

8 responses to “almost three-hundred concepts to know for a pre-mathematician

  1. Daniel

    Muy buena propuesta la del maestro Omar. Me gustaria cooperar.

  2. Alonso Castillo Ramírez

    Muy bien la lista. Agregaría, sobre todo en el área de álgebra:
    cyclic group,
    fundamental theorem of cyclic groups,
    alternating groups,
    simple group,
    characteristic of a ring,
    Gauss’ Lemma,
    Eisenstein’s Criterion,
    finite field extension,
    algebraic and transcendental field extensions,
    normal field extension,
    splitting field,
    algebraically closed field,
    finite field,
    Hilbert’s Basis Theorem,
    Hilbert Nullstellensatz,
    Artinian modules,
    Krull dimension.

    Saludos!

  3. Propongo lo siguiente. Tomar cada uno de los conceptos e irlos definiendo ya sea en el blog juanmarqz o en uno nuevo. Esto puede ayudar a tener una referencia mínima de conceptos útiles a la mano. Y si, si creo que todos deben ser conocidos por cualquier matemático

  4. c-qit

    Follow to the W.Thurston’s: “ON PROOF AND PROGRESS IN MATHEMATICS” in

    http://arxiv.org/PS_cache/math/pdf/9404/9404236v1.pdf

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