Eötvös University Budapest
BSc in Mathematics
BSc in Mathematics:
one year (partial) program in English for visiting students
Each term 5-6 of the following courses will be offered in English, based on students’
interest. The courses are (almost) independent, thus they can be taken in (almost) any
order during a one year program.
Please, note that the content of the courses may differ slightly from the descriptions
given below: first of all, the students’ background will be taken into account, secondly in
some descriptions a wider course material is listed and only a selection of topics will be
offered.
1. Applied linear algebra (2+2 hours, 5 credits)
Algebraic applications. Generalized inverses, application of generalized inverses to solutions
of systems of linear equations (exact solution and best approximation). Jordan normal form,
taking the power of matrices, solving linear recurrences, Markov chains. Positive matrices,
stochastic matrices, the Perron-Frobenius theorem. Fast matrix multiplication.
Applications in graph theory. Graphs and associated matrices. The Cauchy-Binet formula. The
number of spanning trees, Cayley’s theorem about the number of labelled trees. Catalan
numbers. Shannon capacity.
Geometric and combinatorial applications. Volume and determinants. Applications of
Vandermonde's determinant. The solution of Hilbert’s third problem: Dehn’s theorem about
equidecomposability. Points sets with a small number of distances. Extremal set theory, block
systems.
2. Groups and rings (2+2 hours, 5 credits)
Groups. Examples: additive and multiplicative structure of known objects (vector spaces,
residue classes, matrices etc.), groups of symmetries of geometric objects. The symmetric and
the alternating groups. Small examples: Klein group, quaternion group, dihedral groups.
Subgroups, generated subgroups, cyclic groups, cosets, Lagrange’s theorem. The order of
group elements, subgroup structure of cyclic groups.
Permutation groups, orbit, stabilizer, transitivity. Homomorphisms and isomorphisms of
groups. Normal subgroups, factor groups. The isomorphism theorems. Conjugation as an
inner isomorphism. Simple groups.
Direct product of subgroups, the fundamental theorem of finite abelian groups. Free groups,
defining relations. Normal and composition series of groups, solvable groups, the JordanHölder theorem. p-groups, Sylow subgroups, Sylow’s theorems.
Rings. Basic examples: polynomials, n × n matrices, residue classes. Subrings, left, right and
two-sided ideals, homomorphisms, factor rings. Unit element, left and right inverse, zerodivisor. Chain conditions. Simple rings; the ring of endomorphisms over a finite dimensional
vector space (i.e. the ring of n × n matrices over a division ring) is simple. The Wedderburn–
Artin theorem. Euclidean rings, unique factorization domains. Field of quotients. The additive
order of elements in rings without zero divisors. Characteristic, the prime field. Quaternions.
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Eötvös University Budapest
BSc in Mathematics
3. Galois theory (2+2 hours, 5 credits)
Field extensions. Revision of basic facts about rings, PID-s and factor rings. The notion of
field extensions, the minimal polynomial of an element, algebraic and transcendental
extensions. The structure of simple extensions. The degree of field extensions; the degree of
two consecutive extensions. The algebraic elements in a field extension form a subfield; the
field of algebraic numbers, it is algebraically closed. The existence of algebraically closed
field extensions.
Galois theory. Splitting fields, normal extensions. Separanle extensions, perfect fields. Finite
extensions of perfect fields are simple. Relative automorphisms of field extensions, the Galois
group. Connection between subgroups of the Gaois group and intemediate fields.
Finite fields. Construction, existence, uniqueness. Finite fields are perfect. The multiplicative
subgroup of a finite field is cyclic. Wedderburn’s theorem about the commutativity of finite
division rings.
Construction by ruler and compass. Algebraization of the geometric problem. Constructible
numbers, their degree is a power of 2. Classical unsolvable problems: doubling the cube,
trisection of angles, squaring the circle. The cyclotomic field. Construction of regular n-gons.
Solving equations by roots. Solvability of the equation and solvability of the corresponding
Galois group. There is no solution formula for equations of degree higher than 4.
4. Chapters in number theory (2+2 hours, 5 credits)
Text: Niven–Zuckerman: An Introduction to the Theory of Numbers
Introductory concepts: divisibility, euclidean algorithm, greatest common divisor,
fundamental theorem of arithmetic. Congruences; Fermat’s theorem, Wilson’s theorem.
Linear congruences, diophantine equations. Congruences of higher degree, index, primitive
root. Number theoretic functions, additive and multiplicative functions. Prime numbers,
bounds for π(x). Riemann’s ζ function. Dirichlet’s theorem on prime numbers in arithmetic
progressions. Selected topics in algebraic number theory.
5. Multivariable analysis (2+2 hours, 5 credits)
Text: Marsden-Tromba: Vector calculus
Geometry of Euclidean space, limits- and continuity- in several variables. Partial and
directional derivative, gradient, mean value theorem, iterated partial derivatives, Taylors
theorem, local extrema of functions, Hesse matrix, constrained extrema and Lagrange
multipliers, implicit function theorem.
Vector fields, divergence and curl, gradient field, potential function, length of curves, work of
a vector field, line integral, conservative vector fields.
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Eötvös University Budapest
BSc in Mathematics
Double integral over a rectangle, changing the order of integration, applications of the
integral, change of variables theorem, Green's theorem, triple integral.
6. Measure theory (2+2 hours, 5 credits)
Text: Bruckner-Bruckner-Thomson: Real Analysis
One-dimensional Lebesgue Measure. Measure Spaces. Outer Measures. Measurable
Functions. Integration of Nonnegative Functions. Integration of Complex Functions.
Convergence Theorems.
7. Complex analysis (2+2 hours, 5 credits)
Complex differentiability, Cauchy-Riemann equations. Complex power series. The complex
exponential function and the complex logarithm. Complex trigonometric functions. Complex
integration. Cauchy's fundamental theorem, Cauchy-s integral formula. Maximum principle.
Schwarz's lemma. Liouville's theorem.
Laurent series. Isolated singularities. Residue theorem and its applications. Rouché's theorem.
Conformal mappings. Theorems of Weierstrass, Hurwitz and Montel. Riemann mapping
theorem. Reflection principle. Schwarz-Christoffel formula. Harmonic functions. The Poisson
kernel.
8. Topology (2+2 hours, 5 credits)
Continuity and metric spaces. Open and closed sets. Continuity and topological spaces.
Examples of topological spaces. Intermediate value theorem and connectedness. Separation
axioms. Limit of sequences. Countability axioms. Compactness. The notion of homotopy and
the fundamental group. Compact surfaces. Manifolds. Homotopy equivalences. Simply
connected spaces. The fundamental group of the circle. Covering spaces and maps. Group
actions. Seifert-van Kampen Theorem. The Dirac experiment and the no-wind theorem.
9. Elementary Lie theory (2+2 hours, 5 credits)
Text: J. Stillwell:Naive Lie theory, undergrad texts in math, Springer, 2008
Geometry of complex numbers and quaternions, isometries of euclidean spaces and
reflections, the groups SU(2) and SO(3), rotations of R4 and pairs of quaternions.
Generalized rotation groups:orthogonal, special orthogonal groups, unitary and symplectic
group, maximal tori.
The exponential map of a square matrix, tangent space of matrix groups (O(n), SO(n), U(n),
SU(n), Sp(n), classical non-simple Lie algebras, simplicity of sl(n,C) and su(n) sp(n) and
so(n) for n>4.
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Eötvös University Budapest
BSc in Mathematics
The matrix logarithm, the Campbell-Baker-Hausdorff theorem
10. Functional analysis(2+2 hours, 5 credits)
Normed spaces. Continuity of linear maps. Hahn-Banach extension and separation theorems.
Banach spaces. Dual spaces and transposes. Uniform Boundedness Principle and its
applications. Closed Graph Theorem, Open Mapping Theorem and their applications.
Spectrum of a bounded operator. Examples of compact operators on normed spaces. Inner
product spaces, Hilbert spaces. Orthonormal basis. Projection theorem and Riesz
Representation Theorem. Adjoints of bounded operators on a Hilbert space, Normal, selfadjoint and unitary operators, their spectra and numerical ranges. Compact operators on
Hilbert spaces. Spectral theorem for compact self-adjoint operators.
11. Differenctial equations (2+2 hours, 5 credits)
Differential equations in physics, chemistry, biology and economics. Existence and
uniqueness of solutions. Methods for solving differential equations. Linear systems, the
exponential of matrices. Higher order linear equations. Autonomous differential equations,
dynamical systems. Stability, Liapunov's method. Poincaré-Bendixson theory. Stability of
periodic orbits, the Poincaré map. Discrete time dynamical systems. Boundary value
problems. Elements of the calculus of variation.
12. Numerical analysis (2+2 hours, 5 credits)
Mathematical modelling, Error analysis, Some elements of the functional analysis, numerical
interpolation and its applications, numerical differentiation and integration, Numerical
solution of nonlinear equations, and systems. Direct and iterative solution of system of linear
algebraic equations, numerical solution of Cauchy problems for the first order ordinary
differential equations, Runge-Kutta methods, multistep methods, consoistency, stability and
convergence, introduction to the numerical solution of partial differential equations,
application of Matlab to the numerical methods, numerical modelling of some applied reallife problems (chemical process, air pollution modelling)
13. Chapters in geometry (2+2 hours, 5 credits)
Text: Elmer G. Rees, Notes on Geometry, Springer Universitext, 1983
Prerequisite: linear algebra, elements of group theory
Euclidean geometry: n-dimensional Euclidean space and its isometries. Reflections.
Classification of isometries in low dimensions. Finite groups of Euclidean isometries and
Platonic solids. Spherical geometry and quaternions. Projective geometry: projective spaces
and transformations, incidence theorems, cross-ratio. Conics and quadratic forms, polarity.
Hyperbolic geometry: projective, conformal, and quadratic form models. Isometries of
hyperbolic space. Area and trigonometry in hyperbolic plane.
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Eötvös University Budapest
BSc in Mathematics
14. Differential geometry (2+2 hours, 5 credits)
Text: Do Carmo, Differential Geometry of Curves and Surfaces
Balázs Csikós: Differential Geometry
Prerequisite: analytical geometry, multivariable calculus, linear algebra
Parameterized curves in the Euclidean space, tangent lines, osculating planes, Frenet frame,
curvature functions, fundamental theorem of curve theory. Osculating circles, evolute,
involute. The rotation number theorem, convex curves, the four vertex theorem.
Parameterized hypersurfaces in a Euclidean space, fundamental forms and the Weingarten
map, principal curvatures. Surfaces of revolution. Ruled and developable surfaces.
Fundamental equations and the fundamental theorem of hypersurface theory, Theorema
Egregium. Geodesic curves on a hypersurface.
15. Chapters in graph theory (2+2 hours, 5 credits)
Graphs, trees. Eulerian and Hamiltonian graphs. Planar graphs, Euler’s formula, Kuratowski's
theorem. Graph coloring, chromatic number. Matchings, perfect metchings. The theorems of
Frobenius, Hall, Konig and Tutte. Algorithmic questions: augmenting paths. Connected and
k-(edge)-connected graphs, Menger's theorems. Flows in networks, Ford-Fulkerson theorem.
Ramsey theory. Extremal graph theory: Turan's theorem, graphs without 4-cycles.
16. Coding theory (2+2 hours, 5 credits)
Codes, minimum distance, information rate. Bound for codes: Hamming bound, Singleton
bound, Gilbert-Varshamov bound. Linear codes, generator matrix, parity check matrix. Perfect
codes, MDS codes. Important classes of codes: Hamming-codes, (generalized) Reed-Solomon
codes, BCH-codes, Reed-Muller codes. Decoding algorithms. Binary and ternary Golay codes
and their connection with designs.
17. Set theory (2+2 hours, 5 credits)
Text: A. Hajnal, P. Hamburger: Set Theory.
Naive and axiomatic set theory. Subset, union, power set. Pair, ordered pair. Cartesian
product, function. Cardinals, their comparison. Equivalence theorem. Operations on sets,
cardinals. Cantor's theorem, Russell's paradox. Axiom of choice, its use. Ordered, well
ordered sets. Ordinals, ordinal comparison. Axiom of replacement. Theorems of transfinite
induction and recursion. Well ordering theorem. Trichotomy of cardinal comparison. Zorn
lemma. Alephs, collapse of cardinal arithmetic. Cofinality, Hausdorff's theorem. König
inequality. Axiom of foundation, cumulative hierarchy. Stationary sets.
18. Operations research (2+2 hours, 5 credits)
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Eötvös University Budapest
BSc in Mathematics
Shortest paths, conservative weightings (algorithms of Dijkstra and Ford), Critical path
method. Assignment and transportation problems, Kuhn's Hungarian method. Algorithms for
maximum flows and feasible circulations. Linear inequality systems and the Fourier-Motzkin
elimination. Basic and strong basic solutions. Polyhedra and polytopes. Farkas lemma,
optimality criteria, duality theorem. The simplex method. Totally unimodular matrices and
their applications in network optimization.
19. Probability theory (2+2 hours, 5 credits)
The notion of probability, elementary properties. Combinatorial calculation of probabilities.
Conditional probability, properties, calculation. Bayes theorem. Independence. Random
variable and its distribution, joint distributions. Independent random variables. Random walk
and ruin probabilities. The most important discrete distributions. Mean and variance,
properties, calculation, inequalities. Median, moments. Covariance and the coefficient of
correlation. Conditional distributions, conditional expectation. Distribution- and density
functions. The distribution of sums of independent random variables (convolution). Absolute
continuous distributions and their properties, examples. Weak law of large numbers. Central
limit theorem. Normal and multivariate normal distribution.
20. Mathematical statistics (2+2 hours, 5 credits)
Elements of descriptive statistics. Sample, sample space. The most important statistics, their
properties. Glivenko-theorem. Estimation. Unbiasedness, consistency, efficiency. Sufficient
statistics. Rao-Blackwell theorem. Cramer-Rao inequality. Maximum likelihood method.
Confidence intervals. Hypothesis testing. The construction of the most powerful test. Famous
parametric and nonparametric tests. Sequential probability ratio test. Linear regression.
Analysis of variance. Monte-Carlo methods. Design and analysis of experiments.
21. Problem solving: chapters in elementary mathematics (2+2
hours, 5 credits)
The course aims at acquiring skills in problem solving and mathematical thinking, based on
secondary school curriculum. Furthermore, an overview of problems from international
mathematical competitions (e.g. International Mathematical Olympiad, Putnam Mathematical
Competition, Competitions of the former Soviet-Union etc.) will be given.
According to the Hungarian traditions of mathematical education, which is based on problem
solving, the basic topics of secondary school mathematics (such as logic, number sets,
functions, arithmetics and number theory, algebra, geometry, probability theory, statistics)
will be studied through problems. Several types of problem solving methods will be
discussed, analyzed, practiced and compared.
Recommended Reading:
[1] Pólya, György: How to solve it. A new aspect of mathematical method
[2] Pólya, György: Mathematical Discovery. On understanding, Learning, and Teaching
Problem Solving
[3] Mathematical and Physical Journal for Secondary Schools
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