Education - Csaba Vincze

Courses

Geometry 1. Lecture

Registration for the exam is allowed only if passing the seminar.

Incidence structures

  • Incidence plane/space; the four point model, proofs based on incidence axioms.

Absolute geometry

  • Ruler axiom (the coordinatization of lines and half-lines, segment construction theorem).

  • Plane separation axiom and Pasch theorem.

  • Protractor axiom, congruence axiom: Pons Asinorum, congruence theorems.

  • Existence and unicity theorem of the perpendicular line.

  • Sufficient conditions of parallelism, existence theorem of the parallel line.

  • Euclidean parallel axiom and its equivalent forms.

Follow-up questions: absolute geometry

Euclidean geometry

  • Theorems of parallels, similarity of triangles.

  • Height and leg theorems, Pythagorean theorem.

  • Euclidean plane isometries, fundamental theorem and classification.

  • Euclidean space isometries.

  • Fixed point theorem of non-isometric similarities.

  • A general concept of congruence and similarity.

  • Area function and area axioms, Jordan measure and area of a circle.

  • Volume function and volume axioms, volume of a sphere.

Follow-up questions: Euclidean geometry

Compulsory/Recommended readings

Geometry 1. Seminar

Registration for the exam is allowed only if passing the seminar. Attendance is obligatory. The

exam can be retaken up to only one extra time.

Basics of triangles: Points, lines and circles associated with a triangle. Euler line, Feuerbach circle.

Fermat-Torricelli point. Orthic triangle (Fagnano’s problem).

Trigonometry and its applications: sine- and cosine rules, computation of unreachable distances.

Basics of circles: tangent lines, choords, inscribed and circumscribed quadrilaterals. Power of a point to a circle.

Coordinate geometry and its applications: Points, lines and circles associated with a triangle.

Intersection problems.

Construction with ruler and compass: operations with line segments (addition, multiplication, square

roots). Golden ratio. Regular 5- and 10-gons.

Inversion in a circle: Apollonian problems.

Basics of conics: an elementary treatise. Coordinate geometry of conics.

Geometry in 3D: surface area and volume. Conic sections.

The sphere: basics of spherical geometry

Compulsory/Recommended readings

Convex Geometry Lecture and Seminar

  • Cs. Vincze: Convex Geometry, University of Debrecen, 2013, TÁMOP-4.1.2.A/1-11/1-2011-0025.

College Geometry Seminar

General computational skills. Numbers, polynomials, functions. Equations in one variable, quadratic equations, inequalities.

Elementary geometry. Right triangles, Pythagorean and related theorems. Trigonometry in right triangles. The extension of trigonometric expressions. General triangles, lines and circles in a triangle, sine and cosine rules. Quadrilaterals, sum of the interior angles, area. Special quadrilaterals. Polygons, decomposition into triangles, sum of the interior angles, area. Regular polygons. Circles, tangent and bitangent segments. The area of a circle, tangential and cyclic quadrilaterals. Geometrical transformations, isometries and similarity transformations.

Coordinate geometry. The analytic model of Euclidean geometry. Distance between points in the coordinate plane. Equation of lines (slope-intersect form) and circles. Parallelism and perpendicularity. Distance of a point from a line. Intersections (line-line, line-circle, circle-circle). Tangent lines to a circle from an external point. Conics (ellipse, hyperbole, parabole). Coordinate geometry on the sphere: longitudes and latitudes.

  • Samples

  • L. Kozma and Cs. Vincze, College Geometry, University of Debrecen, 2014, TÁMOP-4.1.2.A/1-11/1-2011-0098.

  • Zs. Juhász, Teach Yourself Mathematics, Studium '96, Debrecen, 1998.

Mathematics 3. Lecture

  1. Complex numbers: addition, multiplication, conjugation, division. Algebraic, trigonometric and exponential forms. 

  2. Elementary topology in the plane: interior point, boundary point, accumulation point, isolated point. Open, closed and connected sets, paths.

  3. Convergent and Cauchy sequences. Convergence and absolute convergence of series. Uniformly convergent sequences and series of functions.

  4. Power series, geometric series and applications: D'Alambert and Cauchy criteria of  conver-gence. Exponential, sine and cosine functions. 

  5. Differentiation. Holomorphic functions and Cauchy-Riemann equations.

  6. Integration. Cauchy’s integral theorem and Cauchy’s integral formula. Taylor expansion. 

  7. Laurent series. The residue of a function at an isolated singularity. Residue theorem.

  8. Vector space, inner products (real and complex case). Norm, distance and angle: Cauchy-Bunyakovsky-Schwarz inequality.

  9. Examples: the space of sequences, the space of continuous functions. Completion.  

  10. Orthogonality. The Gramm-Schmidt orthogonalization. The problem of the best approximation in inner product spaces: Fourier coefficients. 

  11. Orthogonal polynomials.

  12. Fourier series.  Sufficient conditions for the convergence.

  13. Fourier and Laplace transforms. Linearity, shifting, scaling, differentiation in time domain and convolution. 

  14. Integral transform of differential equations.

Mathematics 3. Seminar

  1. Operations wirth complex numbers.  

  2. The topology of the complex plane. 

  3. Sequences and series.  

  4. Geometric series. D'Alambert and Cauchy criteria of  convergence.

  5. Sequences and series of functions. The complex exponential function. 

  6. Differentiability and  Cauchy-Riemann equations.

  7. Integration along curves in the complex plane.  

  8. The applications of Cauchy’s integral theorem.

  9. Taylor and Laurent series. Residue.

  10. Inner products. Norm, distance and angle. The Gram-Schmidt orthogonalization.  

  11. Orthogonal polynomials. 

  12. The Fourier system. Fourier series. 

  13. Integral transforms: Fourier and Laplace transforms.

  14. Integral transform of differential equations.

Last update: 2023. 07. 04. 12:14