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\title{External Aggregation of Mathematics}

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\maketitle

\vspace{.25in}

2020 session program
The aggregation test program is not written as a lesson plan. It describes a
set of knowledge that the candidate must master and know how to demonstrate. It includes redundancies when concepts come naturally from different points of view. The program evokes
examples sometimes, which are given for information only and can be replaced by
others that would also be relevant.

In the sections 1 to 5 which follow, all the fields (denoted K in general) are assumed to be commutative.

\section{Linear Algebra}

\subsection {Vector spaces}
\begin{itemize}
\item  Vector spaces, linear functions, products of vector spaces, subspaces, images and kernels of linear function, quotient spaces, sums of subspaces, direct sums. Free families, generative families, basis, linear vector space endomorphism algebras, linear group GL(E)
\item Stable subspaces of an endomorphism. Proper value, proper vectors, proper subspaces
\item Linear representation of a group. Irreducibility. In finite dimensions: examples of decomposition of a linear representation in direct sum of sub-representations, Shur’s lemma.
\end{itemize}
\subsection{Finite dimensional vector spaces}
\begin{itemize}
\item Finite dimensional vector spaces. Existence of basis, isomorphisms with Kn. Existence of supplement subspace. Rank of a linear function, rank of a system of vectors. Dual space, rank of a system of linear equations, transpose of a linear function. Dual basis. Biduality. Orthogonality. 
\item Multilinear functions. Determinant of a system of linear functions, of an endomorphism. Special linear group SL(E). Orientation of an R-vector space.
\item Matrix with coefficients from a commutative ring. Elementary operations on rows and columns of a matrix, determinant, invertibility. Matrices with coefficients from a field. Matrix representations of a linear function. Change of basis,Gaussian elimination, row-echelon form. Solving systems of linear equations, calculating determinants, inverting square matrices, calculating the rank of a matrix, finding equations that define a vector subspace. 
\item Stable subspaces under endomorphism, kernel lemma. Characteristic polynomial, endomorphic polynomials. Annihilating polynomials. Minimal polynomials. Cayley-Hamilton theorem. Diagonalization, trigonalisation, characteristic subspaces, Dunford decomposition, exponentiation of real and complex matrices.
\end{itemize}


\section{Groupes}
The different notions of group theory introduced in the following paragraphs are to be applied in geometric situations. 
\begin{itemize}
\item Groups, group morphisms. Direct group products. Subgroups, generative subgroups, order of an element, distinguished and normal subgroups, group quotients, group action on a set, stabilizers, orbites, quotient spaces, class formulas, conjugation class. Function to determine the isometric groups of a regular polytope in 2 and 3 dimensions
\item Cyclic groups. Finite dimensional abelian groups. Groups with complex roots and nth of unity, primitive roots.
\item Permutation groups (for finite sets). Decomposition of a permutation in product of transpositions, into products of cycles to support disjoints. Signature. Alternating groups. Determinants.
\item Classic automorphic groups on a finite dimensional vector space. General linear group, special linear group, orthogonal group, special orthogonal group, unitary group, special unitary group. 
\item Representations of a finite group on a C-vector space. Abelian case. Orthogonality of irreducible characteristics. Dual group. Fourier transforms. Convolution. Maschke theorem. Characteristics of a finite dimensional representation. Central functions on the group, orthonormal basis of irreducible elements. Examples of low cardinality groups. 
\end{itemize}

\section{ Rings, fields and polynomials}
\begin{itemize}
\item Unitary rings, ring morphisms, subrings. The ring Zn on the integers. Ideal of a commutative ring, quotient rings, first ideals, maximal ideals. Chinese remainder theorem. Commutative ring algebra.
\item Polynomial algebra on one or several indeterminates on a commutative ring. Roots of a polynomial, multiplicity. Relationship between the coefficients and the roots of a split polynomial. Newtonian sums. Derived polynomials. Decomposition into sums of homogeneous polynomials. Symmetric polynomials. 
\item Fields, subfields, Characteristics, Frobenius morphisms, field extensions, integral ring fractional fields. Field Q of rational numbers. Field R of real numbers. Field C of complex numbers. d’Alembert-Gauss theorem. Algebraic elements  transcendentals. Algebraic extensions. Closed algebraic fields. Rupture and decomposition fields. Finite fields. 
\item Divisibility in commutative integral rings. Irreducible elements, invertible elements, first elements among them. Factorial rings. Greatest common divisor, least common multiple. Factorability of A[X] when A is a factorial ring. Principle ring. Bezout’s theorem. Euclidean rings. Euclid’s algorithm. Case of ring Z and K[X] algebra of polynomials on the field K. Irreducible polynomial. Cyclotomic polynomials in Q[X], Einstein’s criteria. 
\item Congruence in Z. Prime numbers. Z/nZ and its invertible elements. Eurler’s indicator function.
\item Rational fraction fields with one indeterminate on the field. Decomposition into simple elements. Real and complex cases.
\end{itemize}

\section{Bilieanar Forms and Quadratics on a Vector Space}
\begin{itemize}
\item Bilinear forms. Alternating bilinear forms. Symmetric bilinear forms, quadratic forms, polar quadratic form (with characteristic difference of 2). Orthogonal elements, geometric interpretation. Non-degenerative forms. Adjoint to an endomorphism. Matrix representation. Change of basis. Rank of a bilinear form.
\item Orthogonality. Subspace isotropes. Decomposition of a quadratic form into sums of squares. Sylvester’s inertia theorem. Classification in the case of R or C. Orthogonalization process. 
\item Euclidian vector spaces, hermitian vector spaces. Isomorphisms of a Euclidian vector space with its dual. Orthogonal supplement. Cauchy-Schwarz inequality. Norm. Orthonormal basis. 
\item Orthogonal group, special orthogonal group. Examples of orthogonal group generators: decomposition by an orthogonal automorphism via reflexion products. Symmetric endomorphisms, normal endomorphisms. Diagonalisation of a symmetric endomorphism. Simultaneous reduction of two quadratic real forms, one being positive definite. Polar decomposition in GL(n,R). Euclidean vector spaces in 3 dimensions. Classification of elements in O(3,R). Mixed product, vector product.
\item Unitary group, special unitary group. Diagonalization of normal endomorphisms. Polar decomposition in GL(n,C)
\end{itemize}

\section{5 Géométries affine et euclidienne}
\begin{itemize}
\item All spaces considered in this section are finite dimensional
\item Affine spaces and associated vector spaces. Affine functions and linear associated functions. Affine subspaces, barycenters. Affine spans , equations of affine subspaces. Affine groups, notions of affine properties. Homothetic transformation groups, affinity. Convex parts, convex envelopes of part of an affine real space. Extrema. 
\item Isometries of a Euclidian affine subspace. Euclidian affine space isometry group. Euclidean affine space isometries. Isometries that do and do not preserve orientation. Direct and indirect similarities of the plane. Isometric classification in two and three dimensions.
\item Angles in two dimensions : angles of vectors, right angles, inscribed angle theorem, cocyclicity. 
\item Group of isometries that leave stable part of the plane or space. Regular polygons. Metric relations in the triangle. Using complex numbers in plane geometry. 
\item Application of quadratic forms to study proper conic sections of the affine euclidean plane (foyer, eccentricity) and of quadratics to study 3 dimensional euclidean affine spaces
\end{itemize}

\section{ Single Variable Real Analysis}
\subsection{ Real Numbers}
The field R of real numbers. Topology of R. Additive subgroups of R. Series of real numbers: convergence, adherence value, recurrent series. Limit infimum and supremum. Completeness of R. Bolzano-Weierstrass theorem. Compact parts of R. Caucy series. Connected parts of R.

\subsection{ Numerical Series}
Convergence of real value series. Geometric series, Riemann series. Positive valued series. Summation of comparison relations. Comparison of a series and an integral. Error estimation. Absolute convergence. Products of series. Alternating series. 

\subsection{Real-valued functions defined on part of R}
\begin{itemize}

\item Continuity
Limites, continuity. Intermediate value theorem. Image of a segment. Study of continuity of monotonic functions. Continuity of reciprocal functions.
\item Differentiability 
Derivative at a point, differentiable functions. Derivative of a composite function. Derivative of a reciprocal function. Rolle’s theorem and intermediate value theorem. Derivatives of higher order functions. Applications to Ck, to piecewise Ck. Leibniz formula. Taylor’s theorem in integral and differential form. Limit and asymptote calculations. 
\end{itemize}

\subsection{“Usual” functions (trig, rational, exp, log, etc).}
Polynomial functions, rational functions. Logarithms. Exponential. Power functions. Circular and hyperbolic functions. Reciprocal circular and hyperbolic functions.

\subsection{ Integration}
\begin{itemize}
\item Integral over a segment of piecewise continuous functions
Calculation of primitives. Riemann sums. Primitives of a continuous function. Usual methods of
calculating integrals. Change of variable. Integration by parts.
\item Generalized integrals
Absolutely convergent integrals. Integration of comparison relationships. Semi-converging integrals.
\end{itemize}

\subsection{Sequences and series of functions}
Simple convergence, uniform convergence. Continuity and differentiability of the limit. Case of series of
functions; normal convergence.
Polynomial Weierstrass and trigonometric Weierstrass approximation theorems.

\subsection{Convexity}
Convex functions of a real variable. Continuity and differentiability of the convex functions. Characterizations of convexity. Convexity inequalities.



\section{ Single Variable Complex Analysis}
\subsection{ Complex Valued series }
\begin{itemize}
\item Radius of convergence. Properties of sums of complex valued series on their disks of convergence. Continuity, differentiability with respect to the complex variable, primitives
\item Complex exponentials. Extension of circular functions to the complex plane. Development of trig, exponential, log, and rational functions among others.
\end{itemize}

\subsection{ Functions on one complex variable }
\begin{itemize}
\item Holomorphic functions. Cauchy-Riemann conditions. Line integrals of continuous functions in C. Primitives of a holomorphic function. Representations of the log function on C. Theorem of holomorphic functions under integral domains
\item Winding number of a closed curve in C with respect to a point
\item Cauchy formulas. Analyticity of a holomorphic function. Principle of isolated zeros. Cauchy formulas. Analyticity of a holomorphic function. Principle of analytic continuation. Maximum principle.
\item Isolated singularities. Laurent series. Meromorphic functions. Residue theorem. 
\item Sequences and series of holomorphic functions. Holomorphic stability by uniform convergence. 
\end{itemize}

\section { Topology}
\subsection{Topology and Metric Spaces}
\begin{itemize}
\item Topology of a metric space. Induced topology. Finite product of metric spaces.
\item Series. Limit/cluster/accumulation points. Limits. Continuous functions. Homeomorphisms.
\item Compactness. Equivalence of definitions in terms of cluster points (Bolzano-Weierstrass) or open covers (Borel-Lebesgue). Connectedness. Connected components. Arc connectedness.
\item Lipschitz functions, uniformly continuous functions. Heine-Cantor theorem
\item Complete metric spaces. Fixed point theorem for continuous functions.
\end{itemize}

\subsection{ Normed vector spaces on R and C }
\begin{itemize}
\item Topology on a normed vector space. Equivalent norms. Finite dimensional spaces.
Normes ||· ||p on Rn and Cn. Banach spaces. Absolutely convergent series on Banach spaces
\item Continuous linear functions, norme of a continuous linear function
\item Norme of uniform convergence. Continuous bounded Banach space valued linear functions on a metric space.
\item Compactness of parts of a normed vector space. Riesz’ Theorem, d’Ascoli’s Theorem. 
\end{itemize}

\subsection {Hilbert Spaces}
\begin{itemize}
\item Hilbert projection theorem. Orthogonal projection onto closed vector subspaces. 
\item Dual space, Riesz representation theorem. l2 and L2 cases. Hilbert bases (in the separable case). Examples of trigonometric polynomial bases and orthogonal polynomials. Lax-Milgram theorem. 
\item H10(|0,1|) and its application to the Dirichlet problem in one dimension
\end{itemize}


\section{ Differential Calculus }
\subsection{ Differential Calculus }
\begin{itemize}
\item Differentiable functions on an open subset of Rn. Differentials (linear tangent functions).
Derivatives with respect to a vector
\item  Partial derivatives. Jacobian matrix, gradient vector, Hessian matrix. Composition of differentiable functions. Mean value theorem. Differentiable functions
\item  Functions that can be differentiated k times. Kth partial derivative. Inversion of differentiation order. Taylor’s theorem in differential and integral form
\item  Local study of real valued functions. Series representations. Local extrema, concavity of functions on an open convex subset of Rn.
\item  Diffeomorphisms. Inverse function theorem. Implicit function theorem. 
\end{itemize}

\subsection{Differential equations}
\begin{itemize}
    \item  differential equations of the form X’ = f(t, X) on I × Ω where I is an open interval in R and Ω
Is an open subset of Rn. Cauchy-Lipschitz Theorem. Maximal solutions. Grönwall lemma.
Exit theorem of a compact subspace
\item  Autonomous differential equations. Phase portraits, qualitative behavior. Stability of equilibrium points (linearisation theorem).
\item  Linear differential systems. Method of constant variation (Duhamel’s formula). Constant coefficient case. Solving systems of differential equations of order > 1
\end{itemize}

\subsection{9.3 Differential Geometry}
\begin{itemize}
\item  Generalizations to Rn. Equivalent definitions: local graphs, local parameterization, local equation, tangent space, gradient. R3 case, position with respect to the plane of the tangent.
\item  construction of curves/planes represented by a parametric equation. Metric study of curves, line integrals, curve length. 
\item Extrema, Lagrange multipliers.
\end{itemize}

\section{Integral Calculus}
\subsection{Notions of measure theory}
Definition of measurable spaces, sigma-algebras and their special cases, positive measure, 
special cases of counting measure, Lebesgue measure (construction
allowed) and probability measures. Definition of a product measure (construction allowed). Definition of measurable functions, approximation by step functions.

\subsection{ Integration }
\begin{itemize}
\item Integrals of positive measurable functions, monotonic conversion, Fanou’s lemma, integrable functions, dominated convergence theorem. 
\item  finite dimensional vector-valued integrable functions, continuity, differentiability of parametric integrals
\item  $L^p$ spaces where $1 \leq p \leq \infty$ , completeness, Holder’s, inequality, 
\item  Fubini’s theorem, change of variables for multiple integrals, polar coordinate case, spherical coordinate case. 
\item Convolution. Regularization and approximation by regularization and approximation by convolution.
\end{itemize}

\subsection{Fourier Analysis}
\begin{itemize}
\item Fourier series  of locally integrable periodic real-valued functions, Riemann-Lebesgue lemma, convolution product of periodic functions, Dirichlet, Fejer, and Parseval theorems
\item Fourier transforms on $L^1(R^d)$ and  $L^2(R^d)$, Plancherel’s theorem
\end{itemize}

\section{ Probabilities }
\subsection{ Definitions of a probabilistic space}
Events, measure of probability, independent events, sigma-algebra, 0-1 law, Borel-Cantelli lemma, conditional probability, total probability formulae

\subsection{ Random variables and their laws}
\begin{itemize}
\item Discrete law, law of absolute continuity, probability density function, law of joint probability, independence of random variables, mean and variance of a real-valued random variable, transfer theorem, moments, examples of Bernoulli, binomial, geometrique, Poisson, uniform, exponential, and Gaussian distributions
\item characteristic equation, generator functions, functions on sums of independent random variables.
\end{itemize}

\subsection{ Convergence of series of random variables }
\begin{itemize}
\item Probabilistics convergence, in Lp, almost surely, markov inequality, Tchebychev inequality, Levy’s theorem
\item Strong and weak laws of large numbers, central limit theorem.
\end{itemize}

\section{ Probability distribution calculus}
For dimension $d>1$, we assume the “integration by parts” formula 
$\int_{\Omega} \delta_x f(x)dx = \int_{d\Omega}f(x)\nu_j(x)d\sigma (x)$
On the domain $\Omega\subset R^d$, with border  “sufficiently regular”,
 with $\delta\Omega$ being the measure of the Lebesgue on 
 $\partial \Omega$ and v(x) being the unitary exterior vector on $x \in \partial \Omega$.
  On the other hand, it is expected to have a certain familiarity with manipulating such functions, for example in the case where $\Omega$ is a ball in $R^d$. 





\subsection{ Distributions on Rd }
\begin{itemize}
\item vector spaces on C with compact support, stability by derivation, stability by multiplication by a function on Coo, partitions of unity, constructing approximations of probability density functions in spaces of common functions (trig, exp, rational, log, etc). 
\item Distributions. Examples of distributions, locally integrable functions, dirac measures, Cauchy principal values, multiplication by a function in Coo, probability distribution function from a dataset, convergent distribution series, definitions, examples, notions of support for a distribution, “punctual” support
\end{itemize}

\subsection {spaces $S(R^d)$ and $S’(R^d)$}
\begin{itemize}
\item Schwartz space S (Rd) of rapidly decreasing functions as well as all their derivatives, gaussian functions are their derivatives, stability by derivation, stability by multiplication by a function Coo of slow growth, Fourier transforms on S(Rd), convolution of two functions of S(Rd)
\item Tempered distribution spaces in Spaces $S(R^d)$, linear forms of T on $S(R^d)$ such that there exists $C > 0$ and $k \in N$ such that $|hT|\phi i| \leq C sup |x^α \partial^{\beta \phi(x)}$|, $x ∈ Rd$, $|α| ≤ k, |β| ≤ k$ forall φ ∈
S (Rd). examples of tempered distributions, functions L2 and Riesz representation, Lp functions, periodic case, Dirac comb, derivation of tempered distributions, multiplication by a function Cooof slow grows
\item Fourier transforms on $S^0(Rd)$, inverse formula, Fourier transform and derivation, Fourier transform on a product of convolution. 
\end{itemize}

\subsection{ Functions }
Find the derivative and the Fourier transform of a distribution. Poisson’s formula, using convolution and Fourier-Laplace transform to solve one dimensional linear differential equations. Notion of constant coefficient differential operator elementary solution (laplacien case). Notion of weak solution of partial derivative equation. Functions, for example,solving the laplace equations, heat equations, wave equations. 

\section{Numerical methods}
\subsection{ Solving systems of linear inequalities}
Notion of conditioning. 
Gershgorin-Hadamard theorem. Gauss’s pivot, LU decomposition, iterative methods 
(for example Jacobi, gauss-seidel), convergence analysis, convergence analysis,
 spectral ray, singular value decomposition, example of discretisation matrix by 
 finite differences of the laplacian in one dimension

\subsection{ iterative methods of solving systems of real and vector valued equations}
linear systems case, iterative methods, proper element search, brute force method, 
optimization of convex function in finite dimension, gradient descent, square root, 
nonlinear problems with real and vector values, bisection method, Picard method, 
Newton’s method, rate of convergence and estimation of error

\subsection{ Numerical integration}
Rectangle method, error estimation. Monte-Carlo method: rate of convergence,
application to the calculation of multiple integrals.

\subsection{ Approximation of numerical functions }
Lagrange interpolation: Lagrange polynomial of a function at (n + 1) points, estimation
of the error.

\subsection{ Ordinary differential equations }
Numerical aspects of Cauchy's problem: explicit Euler method, consistency, stability,
convergence, order.

\subsection{ Fourier transform }
Discrete Fourier transform on a finite abelian group. Fast Fourier transform.

\end{document}