Applied Algebra (ΑΛ3)

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General

School	School of Science
Academic Unit	Department of Mathematics
Level of Studies	Graduate
Course Code	ΑΛ3
Semester	2
Course Title	Applied Algebra
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 student to the basic concepts, results, tools and methods of the theory Polynomials of Many Variables over fields and its applications to Elimination Theory, Projective Algebraic Geometry, Robotics, Invariant Theory of Finite Groups an the Automated Proof of Geometric Theorems. In addition, we will give an introduction to the symbolic algebra package Macaulay2. 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 in various fields of Mathematics and related sciences, to be able to apply them to derive new elementary consequences in various thematic fields, and finally to be able to perform some (not so obvious) calculations related to the construction and analysis of ideals of Polynomial Rings.

General Competences

The course aims at enabling the graduate to acquire the ability to analyse and synthesizebasic knowledge of the theory of Polynomials of Many Variables over fields, which is an important topic of modern Algebras, with further aim the applications to Algebraic Geometry, Invariant Theory of finite fields, Elimination Theory and elsewhere. In particular, the course studies the basic computational technique of Groebner Bases, and the elementary theory of Projective Algebraic Geometry. When the graduate student comes in for the first time in connection with the basic theoretical and computational notions of the theory of Polynomials of Many Variables over fields, she/he strengthens her/his creative, analytical and inductive thinking, and her/his ability to apply abstract knowledge in different areas of central interest with numerous applications.

Syllabus

• Polynomial Rings over fields
• Introduction to the computer algebra program Macaulay2
• Groebner Bases
• Elimination Theory
• Selected Topics in Algebraic Geometry
• Automatic Geometric Theorem Proving
• Applications to Invariant Theory of Finite Groups
• Applications to Projective Algebraic Geometry
• Applications to Robotics

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

• Χαραλάμπους, Χ., & Παπαδάκης, Σ. 2023. Εισαγωγή στην Αντιμεταθετική Άλγεβρα. Κάλλιπος, Ανοικτές Ακαδημαϊκές Εκδόσεις. https://repository.kallipos.gr/handle/11419/9536?&locale=el .
• Cox, David A.; Little, John; O’Shea, Donal. 2015. Ideals, varieties, and algorithms. An introduction to computational algebraic geometry and commutative algebra. 4th revised ed. Springer.
• Schenck, Hal. 2003. Computational algebraic geometry. Cambridge University Press.