# Undergraduate mathematics in mathlib

This gives pointers to undergraduate maths topics that are currently covered in mathlib. The list is gathered from the French curriculum. There is also a page listing undergraduate maths topics that are not yet in mathlib.

To update this list, please submit a PR modifying docs/undergrad.yaml in the mathlib repository.

#### Linear algebra

Finite-dimensional vector spaces finite-dimensionality, isomorphism with $K^n$, rank of a linear map, rank of a set of vectors, isomorphism with bidual.

Endomorphism polynomials annihilating polynomials, minimal polynomial, characteristic polynomial, Cayley-Hamilton theorem.

Structure theory of endomorphisms eigenvalue, eigenvector, generalized eigenspaces.

Exponential matrix exponential.

#### Group Theory

Classical automorphism groups general linear group, special linear group, orthogonal group, unitary group.

#### Ring Theory

Fundamentals ring, subrings, ring morphisms, ring structure $\Z$, product of rings.

Ideals and Quotients ideal of a commutative ring, quotient rings, prime ideals, maximal ideals, Chinese remainder theorem.

#### Bilinear and Quadratic Forms Over a Vector Space

Low dimensions cross product, triple product.

#### Affine and Euclidean Geometry

General definitions affine space, affine function, affine subspace, barycenter, affine span, affine groups.

#### Single Variable Real Analysis

Real numbers definition of $\R$, field structure, order.

Sequences of real numbers convergence, limit point, recurrent sequences, limit infimum and supremum, Cauchy sequences.

Numerical series Geometric series, convergence of $p$-series for $p>1$, alternating series.

Real-valued functions defined on a subset of $\R$ continuity, limits, intermediate value theorem, image of a segment, continuity of monotone functions, continuity of inverse functions.

Taylor-like theorems Taylor's theorem with Lagrange form for remainder.

Elementary functions (trigonometric, rational, $\exp$, $\log$, etc) polynomial functions, rational functions, logarithms, exponential, power functions, trigonometric functions, hyperbolic trigonometric functions, inverse trigonometric functions, inverse hyperbolic trigonometric functions.

#### Single Variable Complex Analysis

Functions on one complex variable holomorphic functions, analyticity of a holomorphic function, maximum principle.

#### Multivariable calculus

Differential equations Cauchy-Lipschitz Theorem, Grönwall lemma.

#### Measures and integral calculus

Fourier analysis Parseval theorem.

#### Probability Theory

Definitions of a probability space probability measure, events, independent events, sigma-algebra, independent sigma-algebra, Borel-Cantelli lemma (easy direction), conditional probability.

Convergence of a sequence of random variables convergence in probability, $\mathrm{L}^p$ convergence, almost surely convergence, Markov inequality, Chebychev inequality, strong law of large numbers.

#### Distribution calculus

Spaces $\mathcal{S}(\R^d)$ Schwartz space of rapidly decreasing functions.

#### Numerical Analysis

Approximation of numerical functions Lagrange interpolation, Lagrange polynomial of a function at (n + 1) points.