Research in Mathematics

1. Research in Mathematics By DrKishorAdhav

2. Question? Question? Question? Question? Question?

3. When an infant born…. Ko-Ham? Ko-Ham?

4. According to Vedic philosophy when a child is born it cries, “KO-HAM, KO-HAM” that is who am I? • After Dhyan and Tapashcharya we come to know that “SO-HAM, SO-HAM” in Sanskrit it means “He is I” that is God says “You are same as I am.” • In short We only want to point out that search for ourselves which starts from the birth itself.

5. Ram, Laxman and two other brothers went to Vashishtha’sKuty (Hut) When SamratDashrathasent his sons Ram and Laxman to Guru Vashishtha . The door of the Kuti i.e. hut was closed. But, there was sound from inside “Who is there?” . Ram & Laxman both replied,” we are here in search of the same question WHO AM I?”

6. If some one of us was there as a common man like us, we may have replied that we are the sons of great SamratDashratha but that wasnot the answer. • It shows their modesty before Guru and hunger for the new search, new knowledge. • Of course, it is agreed that neither we are so modest nor our research level is so high but still just have a look on our research in pure mathematics.

7. What is Research? • Research is a organized and systematic way of finding answers to some questions. • ORGANIZEDin the sencethat there is a structure or method in going about doing research. It is a planned procedure, not a spontaneous one. It is focused and limited to a specific scope. • SYSTEMATIC because there is a definite set of procedures and steps which you have tofollow. • FINDING ANSWERSis the end of all research. Whether it is the answer to a hypothesis or even a simple question, research is successful when we find answers.

8. Research is like “Carving Statues out of stones”. • Better methods and tools lead to a Better statues. • Research is a game of innovative ideas that are more significant. • The significance of ideas could lie in any of the following: Beauty, Utility, Enhancement of knowledge.

9. Scientific research is the methodological study of certain problem. • It has some protocol which may differ from subject to subject. • This protocol contains Introduction, Review of literature, methodology, findings its analysis, Appendix and References.

10. What is Mathematics? • What is Mathematics? : Its a logical and abstract study of different patterns. • It’s the science of correct thinking. • Mathematicians resolve the truth or falsity of conjectures by mathematical proof. • A distinction is often made between pure mathematics and applied mathematics. However pure mathematics topics often turn out to have applications.

11. Pure mathematics: • Quantity: Number theory • Structure: group theory, graph theory, • Space: Geometry, Trigonometry, Differential Geometry, Topology, fractal theory • Change: Calculus, vector calculus etc Applied Mathematics: • A branch of mathematics that concerns itself with mathematical methods that are typically used in science, engineering, business, and industry.

12. Why people do research in Mathematics? • Need from own carrier point of view. • But this is not the only reason. Beauty of Mathematics and the joy of discovering new patterns within complex abstract patterns is one of the primary motivations behind many research. • Just as some people may get as much enjoyment out of solving a crossword puzzle or getting a high score in a video game as a mathematician does in discovering something new in Mathematics.

13. Pure Mathematics: Is it beneficial to the society? • Pure Mathematics / Applied Mathematics ? • Generally people think -What is the use of research in pure Mathematics? • It is an abstract pattern or abstract thought which has nothing to do with routine life. • Actually it has the potential to be useful in a variety of different areas: its abstraction is a strength because it maximizes its potential usefulness.

14. We teach kids to count, for which we ask him to count toys, cartoon animals etc

15. Some one may say: In real life will he count toys or animals, once he is grown up ?. • It is merely a means to the key goal of understanding number which is an abstract concept. • Thus counting of toys and cartoon animals is merely a mean to understand the abstract concept of Numbers. • Just like the abstract concept of number, abstract Mathematics seems to be different from real world but tremendous potential is hidden inside.

16. Johann Radon when investigated Radon transform in 1917which was basically discovered and further designed for applications within Pure Mathematics. • But today it is mostly used in Advanced Medical Technology for CT Scan. • These scanners form a three-dimensional image from a collection of two-dimensional images taken from different angles. • In 2003 the paper by V Cong presented a method of medical image reconstruction and the door of real world opened for the number of applications of that abstract concept.

17. Another example of the power of abstraction is provided by Laplace’s equation. • One of the most studied and best understood (non-trivial) partial differential equations in mathematics. • A variety of phenomena in astronomy, electromagnetism and fluid flow are governed by this equation, as is the steady state heat distribution in an object.

18. Thus pure mathematical research for a long time seemed to have little relevance to the real world, but which eventually became of great importance. • Thus research in Pure Mathematics is also as useful to the real life as any other research in applied Mathematics.

19. Starting Research There is a sphere of knowledge. Please start increasing your area of knowledge, start focusing, specialize your reading till you reach the unknown, fix the target and Achieve depth in one area and overall breadth.

20. Things to be done when you start research • Pick up a direction or area of interest based on your background or from network. • Make a list of topics that interest you. • Read the recent work done in these topics. • Solve some of the papers related to your topic of interest. • Write a two page proposal defining, as clearly as possible, the following items: Motivation, Research challenges, Overview of existing work, Limitations of existing work, Potential directions and ideas for improvement Expected results and impact on the field.

21. While doing research -Enjoy mathematics • Bring passion to your study of mathematics. If you enjoy mathematics explorations, you will wishto continue studying and using mathematics throughout your life. Create new problem settings and pose new questions • Develop a new problem: New mathematics is created through the modification of existent questions, statement of new definitions, or the identification of a new area of exploration.

22. Read mathematical works. • Be patient learning how to read & understand primary source mathematics • It is neither a speedy nor a linear process. Strive to read both peer and professional mathematics writings carefully. • Identify the assumptions, test the conditions, and verify the conclusions. For your work. • Develop logical arguments that prove or disprove the conjectures you investigate.

23. Write mathematics clearly. • Write narrations of your explorations and problem-solving efforts clearly. • Use the simplest sentence (in any language) possible to express youridea. Check your reasoning and solutions. • Only accept conclusions if you can verify or estimate the validity of an answer. Use modern technology.

24. Some More Points To Follow while doing research Setting a Goal • Identify an issue, question, or problem. Talk with people who wish or need your study. • Find out what's already known about it. Talk with experts and/orread their reviews and the original research on the topic. • Thus, set your goal. Keep the record of every thing • Keep a record of everything you do and everything you read that relates to this work.

25. PRESENTING YOUR RESEARCH • Prepare for Formal Report, oral presentation and visual display. • Writing process in research forces you to clarify your own thinking and to make sure you really have rigorous arguments.

26. Review of Literature: • This chapter should provide the reader with the background and the context of your topic, demonstrating your expertise in the process. • It is appropriate to identify and discuss books, articles and other source materials that you read for your project. • In short this chapter answers the reader what kind of research has been done before?.

27. Appendices: • In the appendices you should include any data or material that supported your research but that was too long to include in the body of your paper. Materials in an appendix should be referenced at some point in the body of the report. References: • You must give credit to the people whose work you used in writing your report. Include articles in the journals and books that you used also it is appropriate to cite articles and other material that you found on the World Wide Web.

28. What role can technology play in research? • When I have done my Ph.D. work no computer typing was available so the quality of typing was so poor but now the scene is different • There are so many software for typing Mathematics particularly moreover LATEX like software adds to the beauty of the presentation. • Beautiful graphical representation supported by MATLAB or many other programs inspires the reader

29. References • Huynh Quang Linh Vu Cong. 3d medical image reconstruction. 2003. • J. F. Kaiser: Richard Hamming``You and Your Research'‘, Transcription of the Bell Communications Research Colloquium Seminar, 7 March 1986 • Tosiyasu L. Kunii: Technological Impact of Modern Abstract Mathematics, Lecture in Hosei Univ.