# 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
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
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
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
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

### 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.

• juanmarqz

una forma es estudiar las ligas en linea para ligarlas a la lista

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!

• juanmarqz

bien, muy bien,,, voy a editar tu colaboración para hacerla sinóptica:

• juanmarqz

DONE

3. Omar Rojas

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

• juanmarqz

habría que separar por B.Sc. , M.Sc. y Ph.D. ,,,

4. c-qit

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