User Rating: 5 / 5

Course Code    : PMAT 42833

Title                   : Measure Theory

Pre-requisites   : PMAT 42793

 Learning Outcomes:

At the end of the course the student should be able to demonstrate knowledge of the concepts and theorems of abstract Measure Theory and to apply them in Lebesgue integrals.

 Course Contents:

Measure Theory: Algebra, -algebra, additivity properties of a set function, Measure, Borel sets, Lebesgue measure, outer Measure, measurable subsets, measurable functions, Integral, Properties that hold almost everywhere, integrable functions, Additivity Theorem, Monotone convergence theorem, Dominated convergence theorem, Fatou's lemma, Relation of Riemann and  Lebesgue integrals, Modes of convergence.

 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.

User Rating: 5 / 5

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.

User Rating: 5 / 5

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

User Rating: 4 / 5

Course Code    : PMAT 41403

Title                   : Topology

Pre-requisites   : PMAT 21263

 Learning Outcomes:

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

1. analyze and interpret basic topological concepts introduced in this course

2. discuss and work with abstract topological spaces and develop tools to characterize them at a depth suitable for someone aspiring to study higher-level mathematics

3. formulate tools to identify when two topological spaces are equivalent (homeomorphic)

4. differentiate between functions that define a metric on a set and those that do not

5. describe the hereditary of topological properties under continuous maps

6. summarize the basis of point-set topology.

 Course Contents:

Topological spaces: Definition, Open sets, closed sets, Basis (any topology/given topology), Finite-Closed topology, Euclidean topology

Limit Points: Definition, Boundary, Closure, Dense sets, Neighborhoods, Connectedness

Homeomorphisms: Definition, Subspaces, Non-Homeomorphic spaces, Hausdorff space

Continuous Mappings: Definition, Intermediate Value Theorem

Metric Spaces: Definition, Convergence (of a sequence), Completeness, Contraction Mappings, Baire Spaces

Compactness: Definition, Heine-Borel Theorem

Product Topology

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

 Assessment:  Based on problem sets, presentations and end of course examination  

Recommended Reading:

1. Munkres, J.R. (2015). Topology, a first course, Prentice-Hall, India.
2. Janich, K. (1984). Topology, Springer.
3. Armstrong, M.A. (1983). Basic Topology, Springer.
4. Morris, S. A. Topology Without Tears. https://www.topologywithouttears.net/topbook.pdf