Specialized Topics in Algebra (ΑΛ6): Διαφορά μεταξύ των αναθεωρήσεων

Τελευταία αναθεώρηση της 16:28, 15 Ιουνίου 2023

Ελληνική Έκδοση
Graduate Courses Outlines
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Department of Mathematics
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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Graduate
Course Code	ΑΛ6
Semester	2
Course Title	Specialized Topics in Algebra
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 7.5)
Course Type	Special Background
Prerequisite Courses	-
Language of Instruction and Examinations	Greek
Is the Course Offered to Erasmus Students	Yes (in English)
Course Website (URL)	See eCourse, the Learning Management System maintained by the University of Ioannina.

Learning Outcomes

Learning outcomes	The aim of the course is for the postgraduate student to reach a good level of theoretical background on topics related to the theory of commutative rings.
General Competences	The aim of the course is to empower the postgraduate student to analyse and compose basic notions of Commutative Algebra.

Syllabus

Topics of Commutative and Combinatorial Algebra: Hilbert's Basis theorem, Primary Decomposition, Localization, Dimension, Hilbert Series, Groebner Bases, Simplicial complexes and homology, Stanley-Reisner ideals, Hilbert's Nullstellensatz theorem.

Teaching and Learning Methods - Evaluation

Delivery

Face to face

Use of Information and Communications Technology

-

Teaching Methods

Activity	Semester Workload
Lectures	39
Study of theory	78
Solving of Exercises	70.5
Course total	187.5

Student Performance Evaluation

Written exam at the end of semester (obligatory), problem solving or/and intermediate exams (optional)

Attached Bibliography

Μαλιάκας Μιχάλης, Εισαγωγή στην Μεταθετική Άλεβρα, Εκδόσεις Σοφία, 2008
Atiyah, M. F.; Macdonald, I. G., Introduction to commutative algebra. Addison-Wesley Publishing Co., 1969 ix+128 pp.

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-[[Graduate Courses Outlines]] - [https://math.uoi.gr  Department of Mathematics]
+* [[Ειδικά Θέματα Άλγεβρας (ΑΛ6)|Ελληνική Έκδοση]]
+{{Course-Graduate-Top-EN}}
+{{Menu-OnAllPages-EN}}
 === General ===
@@ Γραμμή 27: / Γραμμή 29: @@
 |-
 ! Course Type
-| General Background
+| Special Background
 |-
 ! Prerequisite Courses
@@ Γραμμή 49: / Γραμμή 51: @@
 ! Learning outcomes
 |
-The main purpose of the course is to introduce the student to the basic concepts, results, tools and methods of the Representation Theory of Finite Groups and its applications to other areas of Mathematics, mainly in Group Theory, and other related sciences, e.g. in Physics.
+The aim of the course is for the postgraduate student to reach a good level of theoretical background on topics related to the theory of commutative rings.
-At the end of the course, we expect the student to understand the basic concepts and the main theorems that are analysed in the course, to understand how these are applied to concrete examples arising from different thematic areas of Mathematics and related sciences, to be able to apply them to derive new elementary consequences in various fields, and finally to be able to perform some (not so obvious) calculations related to several problems arising in Group Theory.
 |-
 ! General Competences
 |
-The course aims at enabling the graduate student to acquire the ability to analyse and synthesize basic knowledge of the basic Representation Theory of Finite Groups, which is an important part of modern Mathematics with numerous applications to other sciences, for instance in Physics.  When the graduate student comes in for the first time in connection with the basic notions of representation theory and its applications to group theory, (s)he strengthens her/his creative, analytical and inductive thinking, and her/his ability to apply abstract knowledge in different areas of central interest in Mathematics and related sciences.
+The aim of the course is to empower the postgraduate student to analyse and compose basic notions of Commutative Algebra.
 |}
 === Syllabus ===
-* Representations and characters of groups.
+Topics of Commutative and Combinatorial Algebra:  Hilbert's Basis theorem, Primary Decomposition, Localization, Dimension, Hilbert Series, Groebner Bases, Simplicial complexes and homology, Stanley-Reisner ideals, Hilbert's Nullstellensatz theorem.
-* Groups and homomorphisms.
-* FG-modules και group-algebras.
-* Schur’s Lemma and Maschke’s Theorem.
-* Group-algebras and irreducible modules.
-* Conjugacy classes and characters.
-* Character tables and orthogonality relations.
-* Normal subgroups and lifted characters.
-* Elementary examples of characters tables.
-* Tensor products. Restricting representations to subgroups.
-* Applications.
 === Teaching and Learning Methods - Evaluation ===
@@ Γραμμή 92: / Γραμμή 82: @@
 | 39
 |-
-| Student's study Hours
+| Study of theory
 | 78
 |-
-| Exercises: Problem Solving
+| Solving of Exercises
 | 70.5
 |-
@@ Γραμμή 104: / Γραμμή 94: @@
 ! Student Performance Evaluation
 |
-The evaluation is based on the combined performance of the graduate student in:
+Written exam at the end of semester (obligatory), problem solving or/and intermediate exams (optional)
-* Weekly homeworks,
-* Presentations during the semester,
-* Major Homework at the end of the course,
-* Written examination at the end of the courses in Greek with questions and problems of development of theoretical topics and problem solving.
 |}
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