Mathematics Department Research/Writing Projects - Fall 2009
Below are some ideas for research/writing projects that you can consider as possibilities for extra credit or honors work in mathematics. Check with your instructor.
We continue to publish the best projects in The Right Angle, monthly newsletter of the Mathematics Department. You should feel free to contact the editor, Randy Schwartz, who can offer advice and sources of information such as those mentioned below (office BTC-510, telephone/voicemail 734-462-4400 ext. 5290, e-mail email@example.com).
In writing your paper, you shouldn’t be satisfied with a general description of the topic. You should actually dig into it, explain what is involved, and get into some actual mathematics using examples that help clarify it. Otherwise, what you and your readers learn will be superficial.
Also, remember that writing a research paper, even a brief one, doesn’t mean splicing together phrases and sentences that you find in other people’s books, articles, and websites. Of course, you’ll need to gather information and to list the sources where you found it, but you’ll also need to:
- analyze the information (break the facts and ideas down into their parts in order to see how they relate to one another)
- synthesize the information (put facts and ideas together in a systematic way in order to get a comprehensive understanding of the topic)
- organize and write the paper (figure out the best way, using your own words and understanding, to unfold what you’ve learned so that the readers get a clear understanding, too).
The Writing Fellows website at http://www.schoolcraft.edu/fellows/ has some useful guidance in how to go about doing this.
MATHEMATICS IN SOUTH ASIA
In conjunction with this year’s Focus South Asia project at Schoolcraft College, we invite you to write about one of the following topics, which help show the historical contributions made to mathematics by people in India, Pakistan, and other countries.
Some of the best general resources are—
- Bose, D. M., and S. N. Sen and B. V. Subbarayappa, eds., A Concise History of Science in India (New Delhi: Indian National Science Academy, 1989). Especially Chapter 2, “Astronomy”; and Chapter 3, “Mathematics”.
- Datta, Bibhutibhusan, and Avadhesh Narayan Singh, History of Hindu Mathematics: A Source Book. Parts I and II (Bombay: Asia Publishing House, 1962).
- Ifrah, Georges, The Universal History of Numbers: From Prehistory to the Invention of the Computer (New York: John Wiley & Sons, Inc., 2000).
- Jaggi, O. P., Science in Modern India (Delhi: Atma Ram & Sons, 1984). Especially Chapter 9, “Mathematics”.
- Joseph, George Gheverghese, The Crest of the Peacock: Non-European Roots of Mathematics, New Edition (Princeton, NJ: Princeton University Press, 2000). Especially Chapter 8, “Ancient Indian Mathematics”; Chapter 9, “Indian Mathematics: The Classical Period and After”; and the final chapter, “Reflections”.
- Katz, Victor J., A History of Mathematics: An Introduction. Third Edition (Reading, MA: Addison-Wesley Publishing Co., 2009). Especially Chapter 8, “Ancient and Medieval India”.
- Plofker, Kim, Mathematics in India (Princeton, NJ: Princeton University Press, 2009).
- Sridharan, R., “Mathematics in Ancient and Medieval India”, in B. V. Subbarayappa, ed., Science in India: Past and Present (Mumbai: Nehru Centre, 2007), pp. 80-116.
Why is our system for writing numerals as strings of digits known as “Hindu” or “Hindu-Arabic” numeration? Write a paper that summarizes the origin and evolution of this numeral system. Try to answer the following questions—
- Why is this way of representing numbers classified not only as a decimal notation but furthermore as a positional (also called place-value or ciphered) notation?
- How did the Hindus invent a concept of “zero”, and why was this symbol crucial for positional notation?
- How did the positional notation make various operations of arithmetic easier to carry out, both on paper and on calculating devices?
- How did the new number system and its arithmetic spread to China, the Arab world, and Europe?
Kaprekar and his numbers
D. R. Kaprekar (1905-1986), a schoolteacher of mathematics in the western Indian state of Maharashtra, discovered some interesting properties of numbers. He was especially fascinated by the decimal representation of numbers, including arithmetical properties involving the sum of the digits of a number. His research was mostly exploratory and recreational, but he was able to publish or self-publish a number of discoveries. Write a paper summarizing the life of Kaprekar, and detail some of his work, such as his investigations into what came to be known as: Kaprekar numbers; the Kaprekar constant; Harshad numbers; Devlali numbers; and Demlo numbers.
Statistics in modern India
Indians have been among the world’s leading statisticians since the early 20th Century. Investigate how Indian proficiency in statistics arose in response to the needs of other fields such as meteorology, food and agriculture, anthropology, and industry. Describe how the study of statistics has also been shaped by government bureaus; by academic institutes such as the Indian Statistical Institute at Kolkata (est. 1932); and by special research journals and awards dedicated to statistics. Summarize the life and contributions of one or more Indian statisticians such as P. C. Mahalanobis (1893-1972), P. V. Sukhatme (1911-1997), C. R. Rao (b. 1920), and S. R. S. Varadhan (b. 1940).
In addition to the book by Jaggi (see above), some online resources include—
- Ghosh, J. K., “Mahalanobis and the Art and Science of Statistics: The Early Days”, Indian Journal of History of Science 29:1 (1994), pp. 89-98. Available at http://www.new.dli.ernet.in/rawdataupload/upload/insa/INSA_1/20005b68_89.pdf
- Gore, A. P., “Statistics in India Today— Past Perfect, Future Tense!” http://stats.unipune.ernet.in/Databook/guestedit.doc.
- Lindstrøm, Tom Louis, “S. R. S. Varadhan”, http://www.abelprisen.no/nedlastning/2007/varadhan_en.pdf
- Raussen, Martin, and Christian Skau, “Interview with Srinivasa Varadhan”, Notices of the AMS 55:2 (February 2008), pp. 238-246. Available at http://www.ams.org/notices/200802/tx080200238p.pdf
- Sukhatme, Shashikala, “Pandurang V. Sukhatme 1911–1997”, Journal of Statistical Planning and Inference 102:1 (1 March 2002), pp. 3-24. Available at http://linkinghub.elsevier.com/retrieve/pii/S0378375801001604
Followers of the Jaina religion were among the earliest and most ardent investigators of mathematics in India. Some of the most famous included Bhadrabahu (c. 300 BCE), Umasvati (c. 50 CE), Siddhasena (c. 250 CE), Mahavira (c. 850 CE), and Silanka (c. 850 CE). Write a paper that summarizes these scholars’ discoveries about numbers, calculation, and various other topics such as the infinite, exponents, logarithms, magic squares, permutations and combinations, number sequences, and number theory. Why did these Jaina philosophers feel drawn to investigate such topics?
In South Asia’s ancient Vedic period of history, a key source for the development of mathematics was the study of shapes used in constructing and reconfiguring religious altars. For example, to help determine which altar designs could be reconfigured into other shapes, scholars developed all sorts of formulae for calculating the area of such shapes. Their study of geometry led in turn to the exploration of such algebraic topics as quadratic equations and irrational numbers. Write a paper that summarizes some of the mathematics found in these sulba-sutras, or ritual geometry manuscripts. In addition to the above-listed books by Bose et al., Joseph, Katz, Plofker, and Sridharan, you might want to check out this online resource—
- Drew, Pippa, and Dorothy Wallace, “Pattern”, Mathematics Across the Curriculum project, Dartmouth College, http://math.dartmouth.edu/~matc/math5.pattern/index.html. Especially Lesson 1 “Ritual Geometry, the Mandala, and its Symmetries”, and Lesson 2, “Mandala Symmetries, Group Elements, Lusona”.
From poetry to mathematics
Chandas is the Indian word for the study of poetic meter, the formal patterns of long and short syllables that make up the lines of Sanskrit and other classical Indian verse. This study was an early stimulus for mathematics, as scholars became interested in counting the number of different possible metrical patterns of various types. Write a paper that explains the methods that were developed to do this, and the results that were found. Some good sources include the works by Plofker and Sridharan (see above), and the recent article by Rachel Wells Hall, “Math for Poets and Drummers”, Math Horizons 15:3 (Feb. 2008), pp. 10-11, 24.
Trigonometry: the mathematics of heaven and earth
Jyotisa is the branch of ancient and medieval science in South Asia that investigated astronomy and astrology. The study of the sky and its movements was of central importance to traditional societies in this region because it laid the basis for calendrics, i.e., the calculation of the correct times to perform various Hindu religious rites. As a result, jyotisa was a very fertile breeding ground for the development of trigonometry, calculation algorithms, and other subjects in mathematics. For example, very early on, one such scholar figured out how to accurately construct a table of sines of various angles by using an iterative second-order interpolation procedure. In your paper, aim to—
- describe the astronomical-mathematical treatises called siddhantas
- detail the trigonometric functions that were used in ancient and medieval South Asia, and the names that were used for them
- explain how trigonometric tables were constructed and used.
The Pell equation
The mathematicians Brahmagupta (c. 650 CE) and Bhaskara Carya (c. 1150 CE) made the first important breakthroughs in finding rational-number solutions for 92x² + 1 = y² and similar equations, and more generally those of the form ax² + b = y². For this purpose, they perfected an ingenious method that they called the kuttaka (“pulverizer”). Centuries later in Europe, when Leonhard Euler attached the label “Pell” to equations such as 92x² + 1 = y², he rediscovered the method, unaware that he was retracing steps that had already been taken in medieval India. Write a paper summarizing the kuttaka method. Show how it is used to solve equations such as 92x² + 1 = y², and explain how this is part of a more general field of “indeterminate equations”. Some good sources include the above-listed works by Bose et al., Joseph, Katz, Plofker, and Sridharan.
How have earth climates changed over geologic time? Are greenhouse gas emissions and other human industrial activities responsible for an overall warming of the earth’s atmosphere in modern times? Answering this controversial question isn’t easy, partly because climate varies naturally over time— even without human interference— and partly because the underlying mechanisms of climate change are very complex. Write a paper discussing how mathematical models are used as an important part of coming to grips with the challenge of global warming.
- A number of recent essays on climate modeling and prediction are available at http://www.mathaware.org/mam/09/essays.html
- Dana Mackenzie, Mathematics of Climate Change: A New Discipline for an Uncertain Century (Berkeley, CA: Mathematical Sciences Research Institute, 2007) is a 28-page report available free at http://www.msri.org/forms/ccbook/ccbookrequest.
- Videos from the April 2007 symposium on “Climate Change: From Global Models to Local Action” can be viewed at http://www.msri.org/specials/climatechange/workshop.
University of Michigan professors of geological sciences Henry N. Pollack and Chris J. Poulsen are nationally known experts in this field. They and their colleagues have gathered geothermal temperatures at sites around the world and have developed theoretical climate models to reconstruct the history of Earth’s surface temperatures and to help predict future trends. For information, see:
- Univ. of Michigan Paleoclimate Research Group publications page, http://www.geo.lsa.umich.edu/paleoclimate/publications.htm
- Henry N. Pollack and Shaopeng Huang, “Underground Temperatures Reveal Changing Climate”, Geotimes 43:8 (August 1998).
- Univ. of Michigan press release, “New Evidence of Global Warming”, San Diego Earth Times, Jan. 1998, available at http://www.sdearthtimes.com/et0198/et0198s2.html.
- Henry N. Pollack and David S. Chapman, “Underground Records of Changing Climate”, Scientific American June 1993, pp. 16-22.
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