Here is a list of my teaching experience during my PhD at Eindhoven University of Technology.
You can also find some notes on Finite semifields and related structures in finite geometry, which I wrote in 2006 for a few lectures I gave as part of the course Finite Geometry at Ghent University.
Galois Theory (University of Primorska)Undergraduate course
Content: - Classical formulas - Polynomials and field theory - Fundamental theorem on symmetric polynomials and discriminants - Roots of unity and cyclotomic polynomials - Solvability by radicals - Basic elements of Galois theory. Automorphisms. Galois extensions - Fundamental theorem of Galois theory - Classical theorems by Abel, Galois, Gauss, Kronecker, Lagrange, and Ruffini - Applications and examples
Objectives: The student gets to know the historical context and the modern approach to the problem of solving polynomial equations with radicals.
General competences: The student learns mathematical thinking and recognizes the rigorous mathematical language. In this way he achieves a sovereign and a critical attitude towards different ways of addressing specific mathematical content.
Course-specific competences: The student learns the foundations and modern applications of Galois Theory, widely regarded as one of the most elegant areas of mathematics.
Prerequisites: Analysis I - Foundations of Analysis, Algebra I - Matrix Calculus, Algebra II - Linear Algebra, Algebra III - Abstract Algebra, Discrete Mathematics I - Set Theory.
Algebraic Curves - Selected Topics in Algebra (1) (University of Primorska)Graduate course
Content: Algebraic sets, ideals, Hilbert basis Theorem, irreducibility, nullstellensatz, coordinate rings, polynomial maps, rational functions, local rings, multiplicity, tangent lines, intersection numbers, affine and projective space, plane curves, Bezout's theorem, varieties, dimension of varieties, resolution of singularities, Riemann-Roch Theorem.
Objectives: the students become acquainted with more advanced concepts of algebra.
Algebra IV - Rings and Fields (University of Primorska)Undergraduate course
Content: - Rings - Ideals - Homomorphism of rings - Factor rings - Integral domain - Euclidean rings - Principal ideal rings - Gaussian integers - Chinese remainder theorem - Unique factorisation domains - Euclidean domains - Polynomial rings - Irreducibility criteria - Fields - Subfields - Extensions - Finite extension - Finite fields - The degree theorem - Algebraic extensions - Splitting field - Constructions with ruler and compass - Squaring the circle - Angle trisection - Doubling Cube - Constructions of regular polygons - Primitive element theorem - Basics of Galois theory.
Coding Theory (University of Primorska)Undergraduate course
Content: - mathematical background (groups, rings, ideals, vector spaces, finite fields); - basic concepts in coding theory; - algebraic methods for the construction of error correcting codes; - Hamming codes; - Linear codes; - Binary Golay codes; - Cyclic codes; - BCH codes; - Reed-Solomon codes; - bounds (Hamming, Singleton, Johnson's bound , ...)
Algebra I - Groups and Rings (Sabancı University)Graduate course for math students
Contents:
Groups: definition and examples, homomorphisms, subgroups, normal subgroups, quotient groups, cyclic groups, symmetric group, group actions, Sylow subgroups, direct sums, free abelian groups, finitely generated abelian groups, free groups, solvable groups, finite simple groups.
Rings: definitions and examples, homomorphisms, ideals, commutative rings, factorization, polynomial rings, formal power series, groups rings, localization, Groebner bases.
All theory will be build from scratch, and a basic (linear) algebra course and some prior experience with abstract mathematics should allow a student to follow the course. Almost all material which will be treated in the course can be found in any graduate level algebra book, some example of which are: [1] Abstract algebra, by Dummit and Foote; [2] Algebra, by Hungerford; [3] Algebra, by Lang.
This is the first part of the two-semester basic algebra course for graduate students. The aim is to strengthen students' familiarity with basic algebraic structures which are commonly used in all parts of mathematics. These structures include groups, rings, vector spaces, modules and fields.
Algebra II - Modules, Fields, and Galois Theory (Sabancı University)Graduate course
Contents: modules over principal ideal domains, algebraic field extensions, splitting fields, algebraic closures, separable and inseparable extensions, cyclotomic extensions, automorphisms, Galois theory, finite fields.
This is the second part of the two-semester basic algebra course for beginning graduate students. The aim is to strengthen students' familiarity with basic algebraic structures which are commonly used in all parts of mathematics. These structures are groups, rings, modules and fields.
In Algebra I, the focus lies on groups and rings. Algebra II deals with modules, fields and Galois theory.
Introduction to Algebraic Geometry (Sabancı University)Graduate course
Contents: Projective and affine algebraic varieties, Zariski topology, projective equivalence, normal rational curves, Veronese varieties, Segre varieties, coordinate rings, product varieties, graph of a regular map, cones and projections, classification of quadrics, rational function fields, rational maps, birational equivalence, Noether's normalisation theorem, Hilbert's nullstellensatz, irreducible decomposition, Grassmann varieties, Plücker coordinates, the Klein correspondence, tangent spaces, singular locus of a variety, dimension and degree.
This course aims to prepare the students for more advanced courses on algebraic geometry. The students are introduced to the basic notions of algebraic varieties, and their associated coordinate rings and functions fields. Classical examples of algebraic varieties are studied. The course includes a detailed analysis of the correspondences between algebra and geometry, highlighting the fundamental ideas behind algebraic geometry.
The course is intended to allow the students to develop a good algebraic-geometric intuition before engaging in more abstract algebraic geometry.
Introduction to Coding Theory (Sabancı University)Graduate course
Contents: The course covers basic theory of error-correcting codes.
Discrete Mathematics (Sabancı University)Undergraduate course for engineering and computer science students
Contents: Foundations and basic structures. Algorithms, growth of functions. Number theory and cryptography. Sequences, summation, induction, recursion. Counting, solving recurrence relations. Relations, graphs and trees. Applications.
This course aims to introduce basic ideas of discrete mathematics and formal mathematical reasoning techniques. The course gives students training to develop their mathematical skills, analytical and critical thinking abilities, their ability to apply these capabilities to practical problems, and to communicate their knowledge of these areas. Part of the theory is illustrated using the mathematics software system SageMath.
Projective Geometry (Sabancı University)Graduate course
Contents: projective spaces over fields, collineations, correlations, perspectivities, elations, homologies, cross ratio, algebraic varieties, Klein correspondence, cubic surfaces, quadric, Hermitian varieties, polar spaces, incidence geometry, projective planes.
Linear Algebra (Sabancı University)Undergraduate course for engineering and computer science students
Contents: matrices, vector spaces, bases, linear independence, linear transformations, eigenvalues, eigenvectors, eigenspaces, diagonalisation, orthogonal matrices, orthonormal bases.
This course aims to introduce basic concepts of linear algebra such as vector spaces, bases, linear transformations, eigenvalues and eigenspaces. The course gives students training to develop their mathematical skills, analytical and critical thinking abilities, their ability to apply these capabilities to practical problems, and to communicate their knowledge of these areas. Part of the theory is illustrated using the mathematics software system SageMath.
Introduction to Algebra (Sabancı University)Graduate course
Contents:
Finite Geometry (Sabancı University)Graduate course
Contents:
Fondamenti di Algebra Lineare e Geometria (University of Padua) (in Italian)Undergraduate course for engineering students
Contents: complex numbers, polynomials, matrices, vector spaces, bases, linear independence, linear transformations, eigenvalues, eigenvectors, eigenspaces, diagonalisation, orthogonal matrices, orthonormal bases.
Linear Algebra and Finite Fields (Ghent University)An elective course for the Master of Mathematical Informatics at Ghent University.
This course is a remediating course for the courses Coding Theory and Cryptography and Computational Group Theory. The aim is to:
1. make the students acquainted with the basic terminology of the theory of linear algebra, polynomials and equations over finite fields;
2. give the students a profound insight into the combinatorial aspects of projective and affine spaces over finite fields;
3. provide the students, by means of an assignment (project), with the necessary experience and skill to analyse problems characteristic for the area, and present the possible solutions in a well-founded and convincing way.
Further details can be found on the electronic learning environment Minerva (for which you need to login).
Projectieve Meetkunde (Ghent University)General course for 2nd Bachelor in Mathematics at Ghent University.
Position of the Course: The students should apply their pre-knowledge on geometry from the courses Linear Algebra and Analytic Geometry to the study of projective spaces. The way of setting up the course, will stimulate the student to be skilled in handling more abstract mathematical reasoning.
Contents: Starting with the general definition of a projective space over fields (including homogeneous coordinates, the theorems of Desargues and of Pappus, collineations, correlations and perspectivities) a detailed study is given of projective lines over fields (isomorphisms between projective lines, cross ratio, the theorem of the complete 4-gon, involutions). After that a general study of polarities will be treated, including classification and standard forms; emphasizing the polarities in projective spaces over the reals, the complex numbers and a finite field. In the special case of polarities in projective planes over finite fields, the study of conics and hermitian curves will be treated, as well as generalizations such as ovals, hyperovals and unitals. There is also a treatise on some polar spaces of low rank. The last chapter of the course contains an axiomatic study of projective and affine planes; the coordinatisation method; the relation with ternary rings; as well as some simple non-Desarguesian planes will be discussed.
Finite semifields and related structures in finite geometry (Ghent University)The first part is aimed at understanding the connection between translation planes and spreads; the so-called Andr{\'e}-Bruck-Bose construction. Next we define the notion of semifields (first studied by L. E. Dickson (1906)) and use a vector space representation to obtain semifield spreads, after which we can apply the Andr{\'e}-Bruck-Bose construction to obtain semifield planes. We define the notion of isotopy and mention the important result proved by A. A. Albert in 1960, which says that two semifields are isotopic if and only if the corresponding planes are isomorphic. All of this belongs to, let's say, the "classical" part of the notes. It gives us the necessary background to be able to deal with some more recent topics in Finite Geometry which have received quite a lot of attention during the last two decades. We start this "modern" part with a particular configuration of subspaces which turns out to be equivalent to semifields, i.e., each such configuration can be used to construct a semifield and vice versa, each semifield gives rise to such a configuration. We conclude these notes with an example of a well-studied structure in finite geometry related to semifields, namely translation ovoids (sometimes called semifield ovoids) of the parabolic quadric in 4-dimensional projective space. We show their correspondence with semifields by constructing them from a particular case of the configuration of subspaces mentioned above.
Coding Theory and Cryptography
(University of Canterbury, Christchurch, New Zealand)I gave part of this course during my visit to the University of Canterbury as an Erskine Fellow. The other part was given by Geertrui Van de Voorde. The course deals with the mathematical ideas underlying modern cryptography, including algebra, number theory and probability theory. The course covers basic coding theory, including Hamming codes, weight enumerators, sphere-packing bound, Singleton bound, MacWilliams identities, parity check matrix, dual codes, mutually orthogonal latin squares, decoding algorithms. The second part is an introduction to cryptography, starting with the discrete log problem.
Other teaching experience