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Course Code     : PMAT 42433

Title                   : Group Theory

Pre-requisites    : PMAT 21263

 Learning Outcomes:

After the completion of this course unit, the student will be able to:

1. analyze and use basic definitions in Abstract algebra including binary operations, groups, subgroups, homomorphism, group actions and Sylow theorems

2. examine the fundamental principles including the laws and theorems arising from the concepts covered in this course

3. develop and apply the fundamental properties of abstract algebraic structures, the substructures, their quotient structures and their mappings

4. build experience and confidence in proving theorems about the structure size and nature of groups, subgroups, group Isomorphism, group actions, Sylow theorems and the associated mappings

5. apply course materials along with techniques and procedures covered in this course to solve problems

6. develop specific skills, competencies and thought processes sufficient to support further studies or work in this or related fields

7. demonstrate knowledge of the structure of Groups and to apply the knowledge in solving problems in different areas in Algebra.

 Course Contents:

Groups: Definition and Examples, Basic properties, Subgroups, symmetric Groups, Dihedral Groups, Cyclic Groups, Abelian Groups, Finite & Infinite Groups, order of a group, order of an element, Structure theory of finite Abelian Groups

Normal subgroups: Definition and examples, Quotient groups, Cosets, Lagrange theorem, Internal direct product

Group isomorphism: Definition of Group Homomorphism, Kernel of a homomorphism, Image of a homomorphism, Group Isomorphism, Isomorphism Theorems (First, Second and Third Isomorphism Theorems)

Group action: Orbits and Stabilizers, Orbit Stabilizer theorem, Burnside's theorem, Fixed point convergence theorems, Cauchy’s theorem, Fermat’s Little theorem

Sylow Theorems: Sylow’s p- subgroups, First, Second and Third Sylow’s Theorems, Applications of the Sylow’s Theorem, Groups of small order

 Method of Teaching and Learning: A combination of lectures, tutorial discussions and presentations.

 Assessment: Based on tutorials, tests, presentations and end of course examination.

 Recommended Reading: 

1. Khanna, V.K. & Bhambri, S.K. (2016). A Course in Abstract Algebra, Vikas Publishing House.

2. Frakeigh, J.B. (2003). A first course in Abstract Algebra, Pearson Education India.

3. Baumslag, B. & Chandler, B. (1968). Group theory, McGraw-Hill, New York.

4. Narayan, S. & Pal, S. (1992). A Textbook of Modern Abstract Algebra, S. Chands, India.

5. Rotman, J.J. (4th Ed., 2014). An Introduction to the Theory of Groups, Springer-Verlag.

6. Gilbert, L. (8th Ed., 2014). Elements of Modern Algebra, Cengage Learning.

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Course Code    : PMAT 43976

Title                   : Research Project

 Learning Outcomes:

At the end of this course, the student should be able to

1. plan a research project in the field of Mathematics

2. investigate a research problem

3. develop a theoretical/practical/conceptual framework towards achieving the research objectives

4. document a research project proposal, and progress in detail

5. solve a research problem using theoretical principles and practical knowledge

6. analyze research results critically

7. communicate research findings, information, and solutions to a specialized audience in a form of dissertation and oral presentations

8. practice research ethics, technical skills.

 Course Contents:

An undergraduate research project is an inquiry, investigation, or creation produced by a final year honours degree undergraduate that contributes to the discipline and reaches beyond the traditional curriculum. An undergraduate research project is designed to provide students with the opportunity to develop and practice advanced discipline-specific projects in collaboration with senior academics in the department.

 Method of Teaching and Learning: self-studies, discussions and student presentations, seminars and colloquiums

 Assessment:  A combination of self-study, seminars, presentations, reports and dissertation

 Recommended Reading:

1. Robson, C. (2nd Ed., 2016). How to do a research project - A guide for undergraduate students, Wiley

2. Reading list and material relevant for each selected topic to be provided at the beginning of the academic year by the supervisor

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Course Code    : PMAT 42423

Title                   : Measure Theory

Pre-requisites   : PMAT 42363

 Learning Outcomes:

At the end of the course the student should be able to

1. Develop familiarity with measures, Integration theory and convergence as presented in the course

2. Understand the construction of the integral and know its key properties

3. Describe and apply the notion of measurable functions and sets and use Lebesgue monotone and dominated convergence theorems and Fatou’s Lemma

4. Apply theory of integral and understand the definition and basic properties of integral with respect to a measure

5. Analyze and understand how the major theorems presented in the course depend on preliminary results

6. Demonstrate knowledge of the concepts and theorems of abstract Measure Theory and to apply them in Lebesgue integral.

Content:

Sets and classes: Set inclusion, Number sets, Operations on sets, Unions, Intersections, Complement, Differences, Symmetric differences, Monotone sequence of sets, Limits, Inferior and Superior limits, Set of extended real numbers, Countable sets

Abstract Measure Theory: Algebra, 𝜎-Algebra, Measure, Additivity properties of a set function, Borel sets, Lebesgue measure, Outer measure, Extension of measures ( Measure on a larger class of subsets), Measurable subsets, Measurable functions, Integration with respect to a measure, Properties of integral, Properties that hold almost everywhere, Integrable functions, Additivity theorem, Monotone convergence theorem, Fatou’s lemma, Lebesgue Dominated convergence theorem, Relation between Lebesgue and Riemann integrals, Modes of convergence

Further results in Measure and Integration: Jordan-Hahn Decomposition theorem, Absolute continuity, Radon Nikodym theorem

 Method of Teaching and Learning: A combination of lectures, tutorial discussions and presentations.

Assessment: Based on tutorials, tests, presentations and end of course examination.

 Recommended Reading:

1. Cohn, D.L., (2^nd edition, 2015) Measure Theory, Springer New York.
2. Barra, G., (2^nd edition, 2003) Measure Theory and Integration, Elsevier.

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Course Code    : PMAT 44962

Title                   : Research Methodology

 Learning Outcomes:

At the end of this course, the student should be able to 

1. recognize the importance of research and discuss the methods of research designs and process

2. demonstrate knowledge and skills acquired in research methods necessary for undertaking and completion of a research related to the field of study

3. assess the key characteristics of both quantitative and qualitative research methods in the field of study

4. demonstrate the awareness of the development of the area of study

5. critically analyze, synthesize, and utilize information and data related to the field of study

6. apply research methodology and /or scholarly inquiry techniques specific to the field of study

7. critically analyze, verify, and interpret the results and provide valid conclusions

8. exercise research ethics and respect for other cultural perspectives in scientific research

9. proficiently communicate and disseminate information in a manner relevant to the field and intended audience.

Course Contents:

1. Fundamentals of research: Meaning and objective of research, understanding about Research, Applications of Research, Criteria and Characteristics of Research, Research strategies, Types of Research, Research design, Research process and steps involved in research, Research proposal.

2. Literature survey and documentation: Methods of literature survey, Use library and web resources (books, journals, e-journals, thesis), importance of documentation, documentation techniques

3. Data collection, Sampling techniques, Descriptive and Inferential methods: Classification of data, methods of data collection, Questionnaire, Sampling procedure and methods, Data processing and graphical representation of data, Estimation and Hypothesis testing, Using statistical software/packages in data analysis.

4. Research ethics, plagiarism, and impact of research: Research ethics, responsibility and accountability of researchers, Plagiarism and use of plagiarism detection software.

5. Technical writing and reporting of research: Types of research report, Structure, and organization of research reports, use of reference managing software, Impact factor, rating, indexing and citation of journals.

6. Publishing research and Research grants: Conferences, Journals, applying for research grants.

7. Developing presentation skills: structuring the presentation, how to improve presentation skills, available software.

 Method of Teaching and Learning: A series of seminars by senior academic members in the department.                                            

 Assessment: Submission of a research/study proposal.

 Recommended Reading:

1. Zina, O. (2 nd Ed., 2021). The essential guide to doing your research project, Sage.

2. Cohen, L., Manion, L. & Morrison, K. (2017). Research methods in education, Routledge. Curriculum Revision- Department of Mathematics- 2022 57

3. Mishra, S.B. & Alok, S. (2017). Handbook of research methodology.

4. Singh, Y.K. (2006). Fundamental of research methodology and statistics. New Age International.

5. Flick, U. (2 nd Ed., 2015). Introducing research methodology: A beginner's guide to doing a research project, Sage. 6. Kumar, R. (2018). Research methodology: A step-by-step guide for beginners. Sage.