Algebra II (ΑΛ2)

Graduate Courses Outlines - Department of Mathematics

General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Graduate
Course Code	ΑΛ2
Semester	2
Course Title	Algebra II
Independent Teaching Activities	Lectures (Weekly Teaching Hours: 3, Credits: 7.5)
Course Type	General 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 main purpose of the course is to introduce the main concepts, tools and methods of Commutative Algebra, and to indicate some of its direct applications to various areas of Mathematics, most notably in Algebraic Geometry, and related sciences. 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 of algebraic and geometric origin, to be able to apply them to derive new elementary consequences in various related fields, and finally to be able to perform some (not so obvious) calculations related to the thematic core of the course.

General Competences

The course aims at enabling the graduate to acquire the ability to analyse and synthesize basic knowledge of Commutative Algebra, which is an important part of modern Mathematics, in particular of contemporary Algebra and Geometry. When the graduate comes in for the first time in connection with the basic notions of Commutative Algebra and its applications to Algebraic Geometry, (s)he strengthens her/his creative, analytical and inductive thinking, and her/his ability to apply abstract knowledge in related fields.

Syllabus

• Introduction to Algebraic Geometry.
• Dimension.  
• Krull’s Principal Ideal Theorem.
• Regular sequences.
• Depth.
• Cohen-Macaulay rings.
• Gorenstein rings.
• Free resolutions.
• Hilbert functions.

Teaching and Learning Methods - Evaluation

Delivery	Face to face
Use of Information and Communications Technology	-
Teaching Methods	Activity Semester Workload Lectures 39 Student's study Hours 78 Exercises: Problem Solving 70.5 Course total 187.5
Student Performance Evaluation	The evaluation is based on the combined performance of the graduate student in: 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.

Attached Bibliography

• J.P. Serre: “Linear Representations of Finite Groups”, Springer-Verlag, (1977).
• B. Steinberg: “Representation Theory of Finite Groups: An Introductory Approach”, Springer, (2012).
• C.W. Curtis and V. Reiner: “Methods of Representation Theory: With Applications to Finite Groups and Orders”, Wiley, (1981).
• P. Etingof et al: “Introduction to Representation Theory”, Student Mathematical Library 59, AMS, (2011).
• J.L. Alperin and R.B. Bell: “Groups and Representations”, Springer (1995).
• M. Burrow: “Representation Theory of Finite Groups”, Academic Press, (1965).
• M. Liebeck and G. James: “Representations and Characters of Groups”, CUP, (2001).