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
- 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
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.)
- Abstract Algebra
- Group theory
- Finite group theory: See my BBK's lecture note!
- Lie theory
- Combinatorial and geometric group
- Ring theory
- Commutative Rings
- Algebraic geometry
- Non-commutative rings
- Representation theory
- Commutative Rings
- Field theory: http://alpha.math.uga.edu/~pete/FieldTheory.pdf
- Galois Theory: https://www.youtube.com/watch?v=ccc4EYeytYo
- Module theory: https://www.youtube.com/watch?v=HHyBTlUJyF0&list=PLY2U8CcmALO4HqPsKdghs0JMy0HUL7TLs
- Represenation theory: https://www.youtube.com/results?search_query=Represenation+theory+lecture
- Group theory
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.
- Outline of algebra
- List of abstract algebra topics
- List of algebraic structures
- Covered in MIT 18-703 Modern Algebra
- List of Boolean algebra topics
- List of category theory topics
- List of commutative algebra topics
- List of homological algebra topics
- List of group theory topics
- Included in MIT 18-703 Modern Algebra, but dig in more!
- List of representation theory topics
- List of linear algebra topics
- List of reciprocity laws
- TODO
- List of cohomology theories
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.
- Single Variable Calculus
- List of complex analysis topics
- List of functional analysis(vector spaces) topics
- List of integration and measure theory topics
- List of harmonic analysis(Fourier analysis) topics
- Partially included in Complex Variables with Applications, but dig in more!
- List of multivariable calculus topics
- List of q-analogs
- TODO
- List of real analysis topics
- List of variational topics
- TODO
- List of mathematical series
- TODO
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.
- List of geometry topics
- List of geometric shapes
- List of curves topics
- List of triangle topics
- List of circle topics
- List of general topology topics
- List of differential geometry topics
- List of algebraic geometry(algebraic surfaces) topics
- List of algebraic topology(cohomology theories) topics
- List of geometric topology topics
- List of knot theory topics
- TODO
- List of topology topics
- List of Lie group topics
- List of mathematical properties of points
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).
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.
- List of Boolean algebra topics
- List of first-order theories
- TODO
- List of large cardinal properties
- TODO
- List of mathematical logic topics
- TODO
- List of set theory topics
- TODO
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.