Mathematical horizons *

2022/2023
Programme:
Mathematics Education
Year:
4 ali 5 year
Semester:
first
Kind:
mandatory
ECTS:
5
Language:
slovenian
Hours per week – 1. semester:
Lectures
3
Seminar
0
Tutorial
2
Lab
0
Content (Syllabus outline)

Lecturer chooses topics that complement mathematical knowledge of a high school mathematics teacher. The topics include:
analysis (Morse theory on surfaces, functional equations, theory of function series, discrete dynamical systems etc.),
discrete mathematics (mathematical games, graphs, geometric configurations etc.),
geometry (geometry of hyperbolic plane, tesselations, geometry of curves and surfaces, classification of surfaces),
algebra (topics in linear algebra, ordered structures, structural algebra),
number theory (elementary, analytic).

Readings

J. Bračič: Uvod v analitično teorijo števil, Podiplomski seminar iz matematike 26, DMFAS, 2003
M. Hladnik: Povabilo v harmonično analizo, Izbrana poglavja iz matematike in računalništva 26, DMFAS, Ljubljana 1992
B. Lavrič: Delno urejene grupe in delno urejeni kolobarji, Podiplomski seminar iz matematike 21, DMFAS, 1993
A. Ramsay, R. D. Richtmyer: Introduction to hyperbolic geometry, Springer, 1995
B. Zalar: Strukturna algebra za podiplomce in nespecialiste, Podiplomski seminar iz matematike 25, DMFAS, 2002

Objectives and competences

The course is aimed at the students of Mathematics education. It covers topics that build on their previous mathematical knowledge and are connected with the topics covered in high school curriculum.

Intended learning outcomes

Deeper knowledge of select fundamental mathematical topics which are connected to high school mathematics.
Better foundations and improved intuition of a high school teacher for topics taught to high school students. This is essential for motivating and educating all high school students and especially those above average.

Learning and teaching methods

lectures, recitations, homeworks, consultations

Assessment

Course grade consists of a single grade.

Two midterm exercise-based exams or final exercise-based exam.
Theoretical knowledge exam.

Grades: 6-10 (pass), 5 (fail) (according to the Statute of UL).

Lecturer's references

Primož Potočnik:
POTOČNIK, Primož, SPIGA, Pablo, VERRET, Gabriel. On the nullspace of arc-transitive graphs over finite fields. Journal of algebraic combinatorics, ISSN 0925-9899, 2012, vol. 36, no. 3, str. 389-401 [COBISS-SI-ID 16162137]
KNOR, Martin, POTOČNIK, Primož, ŠKREKOVSKI, Riste. The Wiener index in iterated line graphs. Discrete applied mathematics, ISSN 0166-218X. [Print ed.], 2012, vol. 160, iss. 15, str. 2234-2345 [COBISS-SI-ID 16409945]
POTOČNIK, Primož, SPIGA, Pablo, VERRET, Gabriel. Cubic vertex-transitive graphs on up to 1280 vertices. Journal of symbolic computation, ISSN 0747-7171, 2013, vol. 50, str. 465-477 [COBISS-SI-ID 16520537]

Sašo Strle:
STRLE, Sašo. Bounds on genus and geometric intersections from cylindrical end moduli spaces. Journal of differential geometry, ISSN 0022-040X, 2003, vol. 65, no. 3, str. 469-511. [COBISS-SI-ID 13135193]
OWENS, Brendan, STRLE, Sašo. A characterization of the Z [sup] n [oplus] Z([delta]) lattice and definite nonunimodular intersection forms. American journal of mathematics, ISSN 0002-9327, 2012, vol. 134, no. 4, str. 891-913. [COBISS-SI-ID 16408153]
GRIGSBY, J. Elisenda, RUBERMAN, Daniel, STRLE, Sašo. Knot concordance and Heegaard Floer homology invariants in branched covers. Geometry & topology, ISSN 1364-0380, 2008, vol. 12, iss. 4, str. 2249-2275 [COBISS-SI-ID 14892121]

Aleš Vavpetič:
VAVPETIČ, Aleš, ŽAGAR, Emil. A general framework for the optimal approximation of circular arcs by parametric polynomial curves. Journal of Computational and Applied Mathematics. 2019, let. 345, str. 146-158. [COBISS-SI-ID 18388057]
KANDIĆ, Marko, VAVPETIČ, Aleš. Topological aspects of order in C(X). Positivity. 2019, let. 23, št. 3, str. 617-635. [COBISS-SI-ID 18551897]
VAVPETIČ, Aleš. Commutators of cycles in permutation groups.
Ars mathematica contemporanea. 2016, let. 10, št. 1, str. 67-77. [COBISS-SI-ID 17731929]