Algebraic Geometry(MAT475)
| Course Code | Course Name | Semester | Theory | Practice | Lab | Credit | ECTS |
|---|---|---|---|---|---|---|---|
| MAT475 | Algebraic Geometry | 8 | 4 | 0 | 0 | 4 | 5 |
| Prerequisites | |
| Admission Requirements |
| Language of Instruction | |
| Course Type | Elective |
| Course Level | Bachelor Degree |
| Course Instructor(s) | Meral TOSUN mtosun@gsu.edu.tr (Email) |
| Assistant | |
| Objective | The aim of this course is to understand affine and porjective geometry and to learn the relation between algebraic notions and geometric structures |
| Content |
Ring theory and fields (summary), Polynomials and afine space, Affine algebraic sets, İdeals of algebraic sets, Hilbert’s Nullstellensatz theorem, Radical ideals and Nullstellensatz theorem; Zariski topology and irreducible algebraic sets, Decomposition of an algebraic set, Polynomial mappings and polynomial functions, Coordinate ring of an algebraic set, Affine change of coordinates, Rational functions and local rings; Projective space, Projective algebra-geometry dictionary, Homogeneous coordinate ring and function field, Projective change of coordinates, Dehomogenization and homogenization of polynomials, Affine-projective transfer of algebraic sets, Multiprojective space and Segre product; Algebraic set of a monomial ideal, Hilbert function and dimension, Dimension of a projective algebraic set, Elementary properties of dimension; Tangent spaces and singularities, blow-up, Smooth algebraic sets, Blow-up of curves and surfaces, Examples. |
| Course Learning Outcomes |
To know the definitions and the properties of algebraic sets in affine and projective space To be able to examine the properties of geometric structures using the properties of ideals |
| Teaching and Learning Methods | Lecture, discussion, problem solving |
| References |
A Primer of Algebraic Geometry, Huishi Li Ideals, Varieties and Algorithms, D. Cox, J. Little, D. O’Shea |
Theory Topics
| Week | Weekly Contents |
|---|---|
| 1 | Ring theory and fields (summary), Polynomials and afine space |
| 2 | Affine algebraic sets, Ideals of algebraic sets |
| 3 | Hilbert’s Nullstellensatz theorem, Radical ideals and Nullstellensatz theorem |
| 4 | Zariski topology and irreducible algebraic sets, Decomposition of an algebraic set |
| 5 | Polynomial mappings and polynomial functions, Coordinate ring of an algebraic set |
| 6 | Affine change of coordinates, Rational functions and local rings |
| 7 | Projective space, Projective algebra-geometry dictionary |
| 8 | Homogeneous coordinate ring and function field, Projective change of coordinates |
| 9 | Dehomogenization and homogenization of polynomials, Affine-projective transfer of algebraic sets |
| 10 | Multiprojective space and Segre product |
| 11 | Algebraic set of a monomial ideal, Hilbert function and dimension |
| 12 | Dimension of a projective algebraic set, Elementary properties of dimension |
| 13 | Tangent spaces and singularities, Blow-up, Smooth algebraic sets |
| 14 | Blow-up of curves and surfaces, Examples |
Practice Topics
| Week | Weekly Contents |
|---|
Contribution to Overall Grade
| Number | Contribution | |
|---|---|---|
| Contribution of in-term studies to overall grade | 5 | 50 |
| Contribution of final exam to overall grade | 1 | 50 |
| Toplam | 6 | 100 |
In-Term Studies
| Number | Contribution | |
|---|---|---|
| Assignments | 4 | 10 |
| Presentation | 0 | 0 |
| Midterm Examinations (including preparation) | 1 | 40 |
| Project | 0 | 0 |
| Laboratory | 0 | 0 |
| Other Applications | 0 | 0 |
| Quiz | 0 | 0 |
| Term Paper/ Project | 0 | 0 |
| Portfolio Study | 0 | 0 |
| Reports | 0 | 0 |
| Learning Diary | 0 | 0 |
| Thesis/ Project | 0 | 0 |
| Seminar | 0 | 0 |
| Other | 0 | 0 |
| Toplam | 5 | 50 |
| No | Program Learning Outcomes | Contribution | ||||
|---|---|---|---|---|---|---|
| 1 | 2 | 3 | 4 | 5 | ||
| 1 | understands principles of deductive reasoning; has experience to verify well-foundedness and exactness of mathematical statements in systematic ways; | X | ||||
| 2 | can properly state and use concepts and results of major mathematical interest; | X | ||||
| 3 | masters current computational techniques and algorithms; has a good ability in their use; can identify relevant tools, among those one has learned, suitable to solve a problem and is able to judge whether or not one is in possession of these tools; | X | ||||
| 4 | is able to express one’s mathematical ideas in an organised way both in written and oral forms; | X | ||||
| 5 | understands relations connecting substantial concepts and results; can switch from one viewpoint to another on mathematical objects (pictures, formulae, precise statements, heuristic trials, list of examples,...); | X | ||||
| 6 | has followed individually a guided learning strategy; has pursued steps toward the resolution of unfamiliar problems; | X | ||||
| 7 | has a theoretical and practical knowledge in computer science well adapted for learning a programming language; | |||||
| 8 | has investigated the relevance of modeling and using mathematical tools in natural sciences and in the professional life; is conscious about historical development of mathematical notions; | |||||
| 9 | has followed introduction to some mathematical or non-mathematical disciplines after one’s proper choice; had experience to learn selected subjects according to one’s proper arrangement; | X | ||||
| 10 | masters French language as well as other foreign languages, to a level sufficient to study or work abroad. | |||||
| Activities | Number | Period | Total Workload |
|---|---|---|---|
| Total Workload | 0 | ||
| Total Workload / 25 | 0.00 | ||
| Credits ECTS | 0 | ||