**Bachelor Degree in Mathematics**

**General Principles**

Accordingly to the national requirements the academic degree **Bachelor
**is acquired in four academic years with a total minimum of
3000 academic hours ( 25 hours per week at an average). There
are compulsory and optional courses. Some fundamental courses
are taught in two variants. Variant B guarantees the acquisition
of the fundamental knowledge in the corresponding discipline whereas
variant A offers additional knowledge and is intended for students
having specific interest in the corresponding scientific area.

**Compulsory Courses**

(All courses are one term courses. Below is shown the number of
hours per week for each course.)

1. Linear Algebra 3+2

2. Analytic Geometry 2+2

3. Differential and Integral Calculus -1 4+4

4. Differential and Integral Calculus -2 4+4

5. Algebra -1 2+2

6. Programming 2+2

7. Mathematical Analysis 1 (A/B) 4+2

8. Data Structures and Programming 2+2

9. Algebra -2 2+2

10. Discrete Mathematics 2+2

11. Mathematical Analysis -2 (A/B) 4+2

12. Differential Equations 3+3

13. Complex Analysis 4+2

14. Differential Geometry 4+2

15. Mechanics 4+2

16. Numerical Methods 4+2

17. Physics/ Mechanics 4+2

18. Mathematical Logic 2+2

19. Probability and Statistics 4+2

20. Partial Differential Equations 4+2

Compulsory Courses

First Semester

No. | Course | Code | hours per term
(lectures+seminars) |
hours per week
(lectures+seminars) | |

1 | Analytic Geometry | AG | 30+30 | 2+2 | |

2 | Linear Algebra | LA | 45+45 | 3+3 | |

3 | Differential and Integral Calculus-1 | DIC-1 | 60+60 | 4+4 | |

4 | Sports | Sts | 0+30 | 0+2 | |

Total:9+11 (20) |

**Second Semester**

No. | Course | Code | hours per term
(lectures+seminars) |
hours per week
(lectures+seminars) |

6 | Differential and Integral Calculus-2 | DIC-2 | 60+60 | 4+4 |

7 | Introduction to Programming | IP | 30+30 | 2+2 |

* | Sports | Sts | 0+30 | 0+2 |

* | Practice 1 | P1 | 0+30 | 0+2 |

Total: 8+12 (20) |

**Third Semester
**

No. | Course | Code | hours per term
(lectures+seminars) |
hours per week
(lectures+seminars) |

8 | Mathematical Analysis | MA1 | 4+2 | |

9 | Data Structures and Programming | DSP | 2+2 | |

10 | Algebra-2 | A2 | 30+30 | |

* | Sports | Sts | 0+30 | |

* | Practice-2 | P2 | 0+30 | |

* | Optional Course 1 | EC1 | 45+30 | |

Total: 11+12 (23) |

**Fourth Semester
**

No. | Course | Code | hours per term
(lectures+seminars) |
hours per week
(lectures+seminars) |

11 | Discrete Mathematics | DM | 30+30 | |

12 | Mathematical Analysis -2 | MA2 (A/B) | 60+30 | |

13 | Differential Equations | DE | 45+45 | |

* | Sports | Sts | 0+30 | |

* | Practice 3 | P3 | 0+30 | |

* | Optional Course 2 | EC2 | 45+0 | |

23) |

**Fifth Semester
**

No. | Course | Code | hours per term
(lectures+seminars) |
hours per week
(lectures+seminars) |

14 | Complex Analysis | CA | 60+30 | |

15 | Differential Geometry | DG | 60+30 | |

16 | Numerical Methods | NM | 60+30 | |

* | Optional Course 3 | EC3 | 45+30 | |

* | Study Seminar 1 | SS1 | 0+45 | |

26) |

**Sixth Semester
**

No. | Course | Code | hours per term
(lectures+seminars) |
hours per week
(lectures+seminars) |

17 | Partial Differential Equations | PDE | 60+30 | |

18 | Mechanics | M | 60+30 | |

19 | Mathematical Logic | ML | 30+30 | |

* | Optional Course 4 | EC4 | 45+30 | |

* | Study Seminar 2 | SS2 | 0+45 | |

24) |

**Seventh Semester
**

No. | Course | Code | hours per term
(lectures+seminars) |
hours per week
(lectures+seminars) |

20 | Probability and Statistics | P&S | 60+60 | |

21 | Physics/ Mechanics | P/M | 60+30 | |

* | Optional Course 5 | EC5 | 45+30 | |

* | Optional Course 6 | EC6 | 45+30 | |

24) |

**Eighth Semester
**

No. | Course | Code | hours per term
(lectures+seminars) |
hours per week
(lectures+seminars) |

* | Optional Course 7 | EC7 | 45+30 | |

* | Optional Course 8 | EC8 | 45+30 | |

* | Optional Course 9 | EC9 | 45+30 | |

* | Optional Course 10 | EC10 | 45+30 | |

Total: 12+8 (20) |

*Notes:*

1. The compulsory courses are those that have numbers. Students must attend those courses and pass the exams at the end of the semester.

2. Optional courses may be done in any semester and several optional courses may be done in one semester. The same is also valid for seminars and for computer practical courses.

3. Each student is obliged to pass at least two or three exams
per semester (six exams per year) in order to complete the year
successfully.

Apart from the compulsory courses the students must attend:

1. Two humanity courses selected by the students from the list provided below.

2. Optional courses in Mathematics, Applied Mathematics and Computer Science - The number of selected optional courses depends on the number of hours for these courses. However, at least two courses from each group have to be selected together with at least two courses from Mathematics - Level II.

3. Three computer practical courses selected from the list provided below.

4. At least two, but not more than three, study seminars.

**Optional Courses in Mathematics - Level I
**

1. Galois Theory

2. Introduction to Number Theory

3. Coding Theory

4. General Topology (set-theoretic topology)

5. Graph Theory

6. Introduction to Topology of Two Dimensional Surfaces

7. Lebesgue Integral

8. Analysis on Manifolds

9. Dynamical Systems

10. Calculus of Variations

11. Introduction to Functional Analysis

12. Hilbert spaces

13. Set Theory

14. Fundamentals of Arithmetic

15. Mathematical Optimisation

16. Approximation Theory

17. Numerical Methods for DE

18. Stability and Control of Mechanical Systems

19. Insurance Mathematics

**Optional Courses in Mathematics - Level II**

(Also valid for M.Sc. and PhD)

1. Commutative Algebra

2. Lie's Algebras (classification)

3. Finite Groups and Representations

4. Infinite Dimensional Lie's Algebras and Representations

5. Algebraic Number Theory

6. Riemannian Geometry

7. Algebraic Geometry I (algebraic curves)

8. Algebraic Geometry II

9. Algebraic Topology I

10. Algebraic Topology II

11. Differential Geometry and Mechanics

12. Analysis on Manifolds II

13. Riemann Surfaces and Theta Functions

14. Distributions Theory and Boundary-Value Problems

15. Spectral Theory of Operators

16. Functional Analysis

17. Nonlinear Integrable Equations

18. Algorithms, Recursive Functions, Theory of Programming (variants)

19. Complexity of Algorithms

20. Gödel's Incompleteness Theorem

**Optional Courses in Computer Science - ** from the Programme
for Computer Science degree

**Computer Practical Courses**

1. Introduction to Programming

2. Data Structures and Programming

3. Office Systems: word processing, spreadsheets

4. Numerical Methods for Differential Equations

5. The "Mathematica" system

**Optional Courses in Applied Mathematics - **from the Programme
for Applied Mathematics degree

**Humanity Courses**

1. Micro-economics

2. Macro-economics

3. Patent Law

4. Financial Law

5. Financial Marketing

6. Genetics

7. Geology

8. Civil Law

9. Commercial Law

**Seminars**

1. Analysis

2. Algebra

3. Topology

4. Differential Equations

5. Geometry

6. Mathematical Logic

7. Mechanics

8. Probability Theory

9. Numerical Methods

10. Optimisation

Each of the seminars is offered once per two semesters (or once
per two years). The students prepare talks and present them on
the seminar. The topics are submitted by the chair of the seminar.

**Physics **(variants A,B)

(**Electrodynamics and Elements of the Relativity
Theory)**

** type: **compulsory

** weekly hours: **4 lectures + 2 seminars

** prerequisites: **

** degree: **Mathematics

** offers:
**

Annotation

The aim of the course is twofold. On one hand, it acquaints the
students with the most important field theory in Classical Physics
- Electrodynamics which development has brought about the modern
notions of space and time together with the creation of Special
and General relativity. On the other hand, since many initial
value and boundary value problems for partial differential equations
arise naturally in Electrodynamics the second aim of the course
is to give students more thorough knowledge of Mathematical Physics
and its applications. This is done through following the connection
and mutual influence between the mathematical and physical theories.

**Mathematical Logic**

** type: **optional

** weekly hours: **lectures + seminars.

** prerequisites: **

** degree: **Mathematics

** offers:
**

Annotation

The course provides fundamental knowledge about the most important
logical systems. The emphasize is on the First Order Predicate
Calculus.

**Main Topics**

1. First Order Predicate Calculus - the language and its semantics. Identical truth and equivalence of formulae. Satisfaction of a set of formulae.

2. Representation of formulae in prefix form. Skolemisation.

3. Compactness theorem and theorem of Lowenheim-Skolem.

4. Predicate calculus with equality. Rules for substitution. Compactness theorem and theorem of Lowenheim-Skolem for predicate calculus with equality.

5. Formal deduction systems in predicate calculus. Completeness
theorem.

**Discrete Mathematics**

** type: **optional

** weekly hours: **lectures + seminars

** prerequisites: **

** degree: **Mathematics

** offers:
**

Annotation

The course acquaints the students with mathematical topics related
to finite and countable sets as well as to functions defined on
these sets. In addition to general topics like sets, functions,
graphs etc. , the course goes into topics of theoretical models
of computing devices as well as discusses some general issues
from the theory of formal languages and grammars. Among the computing
devices' models the course considers are Turing machines and finite
automata. Binary functions considered in the course correspond
to devices acting as logical transformers. The course acquaints
the students with major classes of formal languages and builds
a connection between these classes and classes of computing devices.

**Numerical Methods for Differential Equations
(variants
A, B)**

** type: **optional

** weekly hours: **3** **lectures + 3 seminars

** prerequisites : **

** degree: **Mathematics

** offers: ** Associate professor Peter Binev

Annotation

The course introduces the basic numerical methods for solving of ordinary and partial differential equations, namely differential methods and method of the finite elements. The courses presents also the basic methods for solving of the resulting algebraic equations systems.

Essential topics:

The **A variant of the course **includes:

One-step methods (Euler's method and Runge-Kutta's methods) for
the Cauchy's problem for first order ODE and for systems of such
equations. Multi-step methods (Adams' methods) for the Cauchy's
problem for first order ODE. Linear difference equations with
constant coefficients, stability. Finite difference methods for
the boundary value problem for second order ODE, solving of the
system of difference equations. Approximation, stability and convergence
of the finite differential methods. Finite difference methods
for the Poison equation, canonical representation of the FDM
- scheme, Maximum principle, convergence of the method, methods
for solving of the resulting system of difference equations.
FDM for the heat equation, stability and convergence. FDM for
the string equation. Variation methods for equation solving. Method
of Ritz for boundary value problem for second order ODE and for
the equation of Poison. Finite elements method, linear finite
elements.

The** B variant of the course** includes the same topics but
without theoretical study of the problems for stability and convergence.
It emphasizes on the algorithmic realisation of the methods. Stability
and convergence are studied numerically in the seminar in Numerical
Methods, which is compulsory for this version of the course.

One-step methods (Euler's method and Runge-Kutta's methods) for
the Cauchy's problem for first order ODE and for systems of such
equations. Multi-step methods (Adams' methods) for the Cauchy's
problem for first order ODE. Finite difference methods for the
boundary value problem for second order ODE, solving of the system
of difference equations. Finite differential methods for the Poison
equation, canonical representation of the FDM - scheme, Maximum
principle, methods for solving of the resulting system of difference
equations. FDM for the heat equation. -FDM for the string equation.
Variation methods for equation solving. Method of Ritz for boundary
value problem for second order ODE and for the equation of Poison.
Finite elements method, linear finite elements.

**Infinite Dimensional Lie's Algebras and
Representations**

** type: **optional

** weekly hours: **3** **lectures + 0 seminars

** prerequisites : **

** degree: **Mathematics

** offers: ** Prof. Emil Horozov

Annotation

The course includes the following topics: Virasoro's algebra, Heisenberg's algebra and their oscillator representations. Verma's modules. Lie's algebras of infinite matrices. Fock's spaces. Boson-Fermion correspondence. Vertex operators. Kondomtsev-Petviashvili's hierarchy. Rational and soliton solutions. Kac-Moody's algebras and their representations in Fock's spaces. Determinant formula of Kac.

Prior knowledge of the theory of Lie's algebras is not necessary,
but it is of help if the basic definitions are known.

**Databases**

** type: **optional

** weekly hours: **3** **lectures +2 seminars

** prerequisites : **Programming, Data Structures

**degree: **Mathematics

** offers: ** Senior Lecturer Monika Philipova

Annotation

The course aims at introducing to the fundamental concepts in database theory and practice. The most popular approaches to databases are considered. The main attention is paid to the classical data models - basically on the relational model. However, the hierarchical and CODASYL's network models are also concerned. The course goes also into some other data models commonly used in the design at the conceptual level like, for example, the ER model. Some of the data management problems solvable by the Database management systems are also discussed, e.g. :

- security and integrity of data;

- management of the parallel access to data;

- query optimisation.

The material is illustrated by examples from some of the widespread DBMS like INFORMIX, ORACLE, and INGRES.

The aim of the seminars is to help the students to acquire practical
skills for design and qualified usage of databases for solving
specific applied problems. The students are expected to acquire
some knowledge about modern database query languages - SQL, QUEL,
and 4GL.

**Genetics**

** type: **optional

** weekly hours: **lectures + seminars

** prerequisites : **

**degree: **Mathematics

** offers: ** Faculty of Biology

Annotation

The aim of the course is to provide basic knowledge in the area
of classical (formal) Genetics. It also prepares the students
methodically and practically in this particular area. The course
acquaints with contemporary achievements in Genetics as well.
There is belief that as far as the morphological, physiological
and biochemical characteristics of every organism are held under
close genetic control, knowledge in this area is of great general
educational value. This belief affected the course content.

**Introduction to Functional Analysis**

** type: **optional

** weekly hours: **3 lectures + 2 seminars

** prerequisites : **Mathematical Analysis and Linear Algebra

**degree: **Mathematics

** offers: **

Annotation

This course requires good knowledge in the standard courses in
Mathematical Analysis and Linear Algebra. The following topics
are included: metric spaces, norm spaces, spaces of linear operators,
the Baire category theorem and its corollaries, the Hahn-Banach
theorem, Hilbert spaces, compact operators, the spectral theorem
for compact self-adjoint operators.

**Introduction to Lebesgue Integral**

** type: **optional

** weekly hours: **3 lectures + 2 seminars

** prerequisites : **Mathematical Analysis I and II

**degree: **Mathematics

** offers: **

Annotation

This course requires good knowledge in the standard courses in
Mathematical Analysis I and II. The course is intended for third
and fourth year students. The theory of the Lebesgue integral
is developed for finite dimensional spaces. The course includes
the following topics: Lebesgue null sets, necessary and sufficient
condition for Riemann integrability, Lebesgue integrable functions
and measurable sets, theorems of limits in integrals of Beppo-Levi,
Fatou and Lebesgue, Fubini Theorem, differentiating of functions
of bounded variation, Lebesgue-Stieltjes integral.

**Analysis in Norm Spaces and Variation Calculus**

type: optional

hours: 3 lectures + 2 seminars

prerequisites: Mathematical Analysis 1 and 2

degree: M

offers:

Annotation

This course requires good knowledge in standard courses in Mathematical
Analysis I and II. It is intended for second and third year students
and provides the fundamentals of differential calculus in norm
spaces (derivatives of Frechet and Gateau, implicit function theorem
in norm spaces), Newton's method for solving equations, variation
problems, equations of Hamilton-Jacobi.

**Theory of Operators**

type: optional

hours: 3 lectures + 2 seminars

prerequisites: Mathematical Analysis 1 and 2 and Introduction to Functional Analysis

degree: M

offers:

Annotation

This course requires good knowledge in standard courses in Mathematical
Analysis I and II and Introduction to Functional Analysis. It
is intended for third and fourth year students and provides the
fundamentals of spectral theory of operators, spectral theory
of compact operators with Fredholm's alternatives, Gelfand and
Gelfand-Naimark's theorems for algebras of operators, spectral
theorem for self-conjugate operators.

**Fourier Transformation and Signal Processing**

type: optional

hours: 3 lectures + 0 seminars

prerequisites: Mathematical Analysis 1 and 2

degree: M

offers:

Annotation

The course requires good knowledge in standard courses in Mathematical
Analysis I and II. It is intended for third and fourth year students
and provides the fundamentals of the many variations of Fourier
transformation (Fourier series, Fourier transformation, discrete
Fourier transformation) as well as the connections between them.
Applications to signal analysis are considered.

**Representation Theory**

type: optional

hours: 3 lectures + 0 seminars

prerequisites: Linear Algebra and Mathematical Analysis

degree: M

offers: Associate Professor Vasil Tsanov

Annotation

The theory of Lie semi-simple groups and algebras is built and
Cartan's classification theorem is proved. From the theory of
system of roots and weights the full classification of non-reductive
finite representation of complex Lie semi-simple groups and algebras
is deduced. Basic knowledge of classification of homogeneous spaces
and theorems of Borel-Weil-Bott type is given. The course starts
from an elementary level - only basic knowledge of Linear Algebra
and Mathematical Analysis is required.

**Mathematical Morphology and Applications**

type: optional

hours: 3 lectures + 0 seminars

prerequisites: Mathematical Analysis 1 and 2, Algebra 1 and 2 and Mathematical Programming and Optimisation (recommended)

degree: M

offers: Senior Assistant Professor Antonii Popov

Annotation

Mathematical Morphology is a discipline, initiated in the early 80s by the fundamental works - George Materon's "Random Sets and Integral Geometry" and Jean Serre's "Image Analysis and Mathematical Morphology". The course covers theoretical results, connected with the theory of lattices, convex sets and functions, fuzzy sets, fractal surfaces and Hausdorff's metric, as well as direct applications to image recognition and 3D geometric modelling. Connections between objects of differential geometry like Voronoy diagrams and morphological construction "set skeleton" are considered.

Transformations known as "Euclid granulometrics" and their applications to the analysis of computer images can be considered by the will of the students. Also, as a choice, different problems from so called "Mutation equations" with applications to visual interactive systems can be considered.

This course is intended for students from any degree in Faculty
of Mathematics and Computer Science: for Mathematics students
who specialise in Mathematical Analysis, Geometry, Topology and
Numerical Methods with interests in signal and video information
processing, as well as, basically, for students in Computer Science
and Applied Mathematics, willing to learn a modern approach in
signal processing, requiring richer mathematical culture than
the general methods traditionally used by engineers. Prerequisites
- standard Mathematical Analysis 1 and 2, Algebra 1 and 2, and
Mathematical Programming and Optimisation (recommended) courses.

**Gˆdel Theorems for Incompleteness**

type: optional

hours: 3 lectures + 0 seminars

prerequisites:

degree: M

offers: Assoc. Prof. Petyo Petkov

Annotation

Basic facts from semiotics. Inductive, axiomatic and descriptive definitions. Languages of formal semiotics: E0 (equality, inequality, concatenation, conjunction, disjunction, bounded quantifiers); E1 (addition of non-bounded existential quantifier); E2 (addition of negation); E3 (addition of implication and universal quantifier). Provability, defined through the theorem of non-expressiveness of truth, as well as first and second Gˆdel incompleteness theorems. Transferring results to related languages of arithmetic of natural numbers.

Accessible to students without previous preparation on Mathematical
Logic.

**Fundamentals of Arithmetic**

type: optional

hours: 3 lectures + 1 seminar

prerequisites:

degree: M

offers: Associate Professor Petyo Petkov

Annotation

Course objectives:

a) to demonstrate the possibility for creating of unified entity of mathematical science;

b) to demonstrate the most fundamental principles of mathematical logic, methodology and philosophy;

c) to demonstrate different methods for building one and the same mathematical theory paying attention to the questions of their equivalence.

Consists of: elements of semiotics; theories of integers, natural,
rational and real numbers (classical, constructive, non-standard
approach).

**Differential Forms and Geometry**

type: optional

hours: 4 lectures + 0 seminars

prerequisites:

degree: M

offers: Geometry, Senior Lecturer S. Ivanov and Senior Lecturer
G. Grancharov

Annotation

E. Cartan's method for non-stationary basis will be developed
on the basis of the elementary theory of differential forms. Through
this method, the geometry of immersions of compact surfaces into
R^{n} as well as their internal geometry will
be considered.

**Matrix Groups and Geometry**

type: optional

hours: 3 lectures + 0 seminars

prerequisites:

degree: M

offers: Geometry, Senior Lecturer S. Ivanov and Senior Lecturer
G. Grancharov

Annotation

Basic examples of transformation groups will be described in detail.
Special attention will be paid to the Skinner group in a connection
with a further study of geometry of Dirac operators on surfaces
in R^{n}.