List of possible topics that may occur at the entrance exam

Algebra

Linear Algebra:

  • Vector spaces over R, C, and other fields: subspaces, linear independence, basis and dimension.
  • Linear transformations and matrices: constructing matrices of abstract linear transformations, similarity, change of basis, trace, determinants, kernel, image, dimension theorems, rank; application to systems of linear equations.
  • Eigenvalues and eigenvectors: computation, diagonalization, characteristic and minimal polynomials, invariance of trace and determinant.
  • Inner product spaces: real and Hermitian inner products, orthonormal bases, Gram-Schmidt orthogonalization, orthogonal and unitary transformations, symmetric and Hermitian matrices, quadratic forms.

Abstract Algebra:

  • Groups: finite groups, matrix groups, symmetry groups, examples of groups (symmetric, alternating, dihedral), normal subgroups and quotient groups, homomorphisms, Sylow theorems.
  • Rings: ring of integers, induction and well ordering, polynomial rings, roots and irreducibility, unique factorization of integers and polynomials, homomorphisms, ideals, principal ideals, Euclidean domains, prime and maximal ideals, quotients, fraction fields, finite fields.

Analysis

  • Real numbers as a complete ordered field. Extended real number system. Topological concepts: neighborhood, interior point, accumulation point, etc.
  • Sequences of real numbers. Convergent sequences. Subsequences. Fundamental results.
  • Numerical series. Standard tests for convergence and divergence.
  • Real functions of one real variable. Limits, continuity, uniform continuity, differentiation, Riemann integration, fundamental theorem of calculus, mean value theorem, L'Hopital's rule, Taylor's theorem, etc.
  • Sequences and series of functions. Pointwise and uniform convergence. Fundamental results. Power series and radii of convergence.
  • The topology of Rk. Connected and convex subsets of Rk.
  • Functions of several real variables. Limits, continuity, uniform continuity. Continuous functions on compact or connected sets. Partial derivatives. Differentiable functions. Taylor's theorem. Maxima and minima. Implicit and inverse function theorems.
  • Multiple integrals. Integrals in various coordinate systems. Vector fields in Euclidean space (divergence, curl, conservative fields), line and surface integrals, vector calculus (Green's theorem in the plane, the divergence theorem in 3-space).
  • Ordinary differential equations. Elementary techniques for solving special differential equations (separable, homogeneous, first order linear, Bernoulli's, exact, etc.). Existence and uniqueness of solutions to initial value problems (Picard's theorem). Linear differential equations and systems. Fundamental results.