Fundamental Algebra and Analysis II

Prerequisites: 

Fundamental Algebra and Analysis I

Course description:

This course is the continuation of Fundamental Algebra and Analysis I. It is designed for first year students who intend to pursue studies in fundamental or applied mathematics and is also open to all students who are interested in modern mathematics. Throughout the semester, students will engage with key analytical principles that are essential for understanding complex mathematical theories and applications. To fully grasp the main results discussed in this course, students will need to familiarize themselves with several preliminary concepts from linear algebra and topology. Topics focus on multilinear algebra, differential geometry in Euclidean spaces, integral analysis, and Fourier theory. This course will employ a variety of teaching methods, including lectures, class discussions, problem solving sessions, and independent reading.

Learning Objectives:

Upon completion of this course, students will be able to:

construct and manipulate multilinear algebraic structures, including tensor products, exterior algebras, and determinants, while applying key theorems such as the Cayley-Hamilton theorem and Jordan decomposition;

analyze linear and affine geometric structures, including bilinear forms, orthogonality, and convexity, and classify quadratic hypersurfaces using spectral methods;

compute and apply advanced integration techniques in multiple dimensions, including Fubini-Tonelli, change of variables, and differential forms, while proving fundamental theorems (Green, Gauss, Stokes);

decompose functions using Fourier series, demonstrating convergence in Lˆ2, pointwise, and normal senses, and applying Parseval's theorem and Dirichlet's results;

synthesize algebraic, geometric, and analytic methods to solve problems in higher-dimensional spaces, connecting spectral theory, measure theory, and harmonic analysis.

Detailed topics covered:

Part I. Multi-linear algebra

Multilinear mappings, multilinear forms, action of the symmetric group.

Tensor product of modules, extension of scalars, tensor algebra

Alternating forms, exterior power, exterior algebra.

Determinant of an endomorphism, multiplicativity, determinant of a matrix.

Eigenvalue, eigenvectors, invariant subspaces, diagonalizable endomorphisms.

Reduction of endomorphisms, characteristic polynomial, Cayley-Hamilton theorem, minimal polynomial, diagonalisation, decomposition of Dunford.

Jordan canonical form.

Part II. Geometry

Bilinear forms, sequilinear forms, inner product, orthogonality.

Orthogonal complement, orthogonal basis, Gram-Schmidt algorithm.

Adjoint of a linear mapping, self-adjoint operator, signature.

Affine subspace in a vector space, associated vector subspace, lines, planes, hyperplanes.

Affine mapping, projection, affine group, symmetries, translation and dilation.

Barycentre, convex subset, image and preimage by an affine mapping, convex envelop.

Geometry of Euclidean plane, conics.

Quadratic hypersurfaces, classfication.

Part III. Multi-dimensional integral analysis

Change of variables for Lebesgue integrals.

Higher order differential forms and their integrals along subvarieties.

Formulae of Green, Gauss and Stokes.

Part IV. Fourier series

Square integrable functions on a multi-torus, trigonometric functions.

Expansion of a square integrable function into a Fourier series.

Convergence in Lˆ2, Parseval's theorem.

Pointwise convergence of Fourier series, Dirichlet's theorem.

Normal convergence of Fourier series.

Applications.