Skip to content

Latest commit

 

History

History
194 lines (157 loc) · 15.3 KB

README.md

File metadata and controls

194 lines (157 loc) · 15.3 KB
Raw

Road-to-Maths-Degree

I hope this helps someone who has the same dream as me to make it come true!
** I'm broadly exploring various Maths topics. Also, I am essentially following the roadmap for Maths students@MIT

Existing organisation of Arxiv

  • AG - Algebraic Geometry
    • Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology
  • AT - Algebraic Topology
    • Homotopy theory, homological algebra, algebraic treatments of manifolds
  • CT - Category Theory
    • Enriched categories, topoi, abelian categories, monoidal categories, homological algebra
  • AC - Commutative Algebra
    • Commutative rings, modules, ideals, homological algebra, computational aspects, invariant theory, connections to algebraic geometry and combinatorics
  • GN - General Topology
    • Continuum theory, point-set topology, spaces with algebraic structure, foundations, dimension theory, local and global properties
  • GT - Geometric Topology
    • Manifolds, orbifolds, polyhedra, cell complexes, foliations, geometric structures
  • GR - Group Theory
    • Finite groups, topological groups, representation theory, cohomology, classification and structure
  • KT - K-Theory and Homology
    • Algebraic and topological K-theory, relations with topology, commutative algebra, and operator algebras
  • NT - Number Theory
    • Prime numbers, diophantine equations, analytic number theory, algebraic number theory, arithmetic geometry, Galois theory
  • OA - Operator Algebras
    • Algebras of operators on Hilbert space, C^*-algebras, von Neumann algebras, non-commutative geometry
  • RT - Representation Theory
    • Linear representations of algebras and groups, Lie theory, associative algebras, multilinear algebra
  • RA - Rings and Algebras
    • Non-commutative rings and algebras, non-associative algebras, universal algebra and lattice theory, linear algebra, semigroups

Format of the list below

I have organised all my notes by following the format below;

  • List of <TOPIC_NAME>: A link of Wiki page describing the various research topics in the domain
    • <NOTE_NAME>: A link of the course I followed(**My handwritten notes are stored in the dropbox below.)

New organisation of topics

Notes: My handwritten notes in Dropbox

Algebra

Algebra includes the study of algebraic structures, which are sets and operations defined on these sets satisfying certain axioms. The field of algebra is further divided according to which structure is studied; for instance, group theory concerns an algebraic structure called group.

Calculus and analysis

Fourier series approximation of square wave in five steps.

Calculus studies the computation of limits, derivatives, and integrals of functions of real numbers, and in particular studies instantaneous rates of change. Analysis evolved from calculus.

Geometry and topology

Ford circles—A circle rests upon each fraction in lowest terms. Each touches its neighbours without crossing.

Geometry is initially the study of spatial figures like circles and cubes, though it has been generalised considerably. Topology developed from geometry; it looks at those properties that do not change even when the figures are deformed by stretching and bending, like dimension.

Combinatorics

Combinatorics concerns the study of discrete (and usually finite) objects. Aspects include "counting" the objects satisfying certain criteria (enumerative combinatorics), deciding when the criteria can be met, and constructing and analyzing objects meeting the criteria (as in combinatorial designs and matroid theory), finding "largest", "smallest", or "optimal" objects (extremal combinatorics and combinatorial optimization), and finding algebraic structures these objects may have (algebraic combinatorics).

Logic

Venn diagrams are illustrations of set theoretical, mathematical or logical relationships.

Logic is the foundation which underlies mathematical logic and the rest of mathematics. It tries to formalise valid reasoning. In particular, it attempts to define what constitutes a proof.

Number theory

The branch of mathematics that deals with the properties and relationships of numbers, especially the positive integers. Number theory is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss said, "Mathematics is the queen of the sciences—and number theory is the queen of mathematics." Number theory also studies the natural, or whole, numbers. One of the central concepts in number theory is that of the prime number, and there are many questions about primes that appear simple but whose resolution continues to elude mathematicians.