Mathematics Senior Seminar Project Ideas
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You may choose from one of these ideas, or design a project idea of your own. In choosing or designing a topic, you should have a professional mathematics journal article or textbook to support your work. However original work extending previous work found in one of these sources is encouraged. To guide you in your search to find an article, go to the electronic resources of the Hilton Briggs Library: http://lib.sdstate.edu/Lib17/guidemath.html
1. Conic Sections. How are they related to “sections” of a cone? Derivations, diagrams, etc. How are rotations and translations related to linear algebra?
2. We all "know" that sin(x)/x does not have an elementary antiderivative. What does this mean and how can it be proved? See "An Invitation to Integration in Finite Terms" by Elena Marchisotto & Gholam-Ali Zakeri, The College Mathematics Journal, September, 1994.
3. What are "simple groups"? Just what did John Thompson prove about them that was worthy of a Field's Medal?
4. We have all heard of and perhaps seen general solutions to cubic and quartic equations. How are these derived? Proofs, examples, history, etc.
5. We "know" that general solutions of 5th degree and higher equations are not generally possible in terms of radicals. What does this mean and how do we know it's true? When can higher order polynomial equations be solved? Theorems, proofs, history, etc.
6. A proof of a non-elementary Euclidean Geometry Theorem such as Feuerbach’s Theorem. See Dr. Kemp
7. Solutions to systems of linear differential equations. How are matrices involved? See most any DE book.
8. We "know" that Newton "proved" that his law of universal gravitation implied that the planets orbit the sun in an elliptical path. How can this be proven today using vectors? Proof, background, history, etc.
9. What is a quaternion? – linear algebra. See Dr. Abraham.
10. You know quite a bit about quadratic polynomials. What about cubic polynomials? See Cubic Polynomials by Alan Lipp in the December, 2000 issue of the Mathematics Teacher.
11. The Generalized Riemann Integral. The Riemann integral, as you know it, was first formulated by Bernhard Riemann in 1854. Generalizations of this integral are relatively modern, dating from the 1960’s. This would be a technical, but accessible, topic. See Dr. Flint
12. What is an Elliptic Curve? The study of such curves is an important part of some branches of modern mathematics.
13. Using the Krylov method to find eigenvalues of a matrix without using determinants. Initial source: The Algebraic Eigenvalue Problem by J.H. Wilkinson. See Dr. Schmidt.
14. Using Newton’s method to generate terms in a power series for the reciprocal of a function. Initial source: Topics in Advanced Scientific Computation b Richard Crandall, p. 3. See Dr. Schmidt.
15. What is the probability that a random nXn matrix will have k, 0?k<n, REAL eigenvalues. Use a computer to simulate the probabilities for low values of n and then try and prove a conjecture. See Dr. Kemp.
16. Available in Briggs Library, the book EULER the Master of Us All by William Dunham (Dolciani Mathematical Expositions No. 22, MAA ã1999, ISBN 0-88385-328-0) contains a wealth of topics and is very accessible. See the online review at http://www.maa.org/reviews/master.html.
17. Available at Briggs Library, the book Exploring the Real Numbers by Frederick W. Stevenson (Prentice-Hall ã2000, ISBN 0-13-040261-3) is a very readable account of basic information about number systems leading up to the real numbers. It concludes with 21 PROJECTS, any one of which could be the basis for a senior thesis.
18. Recent issues of The American Mathematical Monthly, The College Mathematics Journal, Mathematics Magazine, The Mathematics Teacher, Journal of Undergraduate Mathematics and its Applications and other mathematical periodicals found in Briggs Library are good sources of subjects. Many such articles would contain too much information for a one-credit paper and a 30 minute presentation, but parts could make a very nice project.
20. The book Mathematics Galore! by C.J. Budd and C.J. Sangwin (Oxford University Press, ISBN 0-19-850770-4) has some interesting projects designed as mathematical workshops. Possible topics: Mazes, Castle Defense, Dancing, Sundials and others. (Available in Dr. Flint’s office)
21. Number Theory topics: Farey Series, Perfect Numbers, Amicable Numbers and others
22. Early computing devices, how they work, and mathematically, why they work. (Abacus, Napiers Bones, etc.)
23. Optimization using Linear Programming and the Simplex Method
24. The book The Heart of Mathematics by Edward Burger and Michael Starbird (Key Publishing ISBN 1-55953-407-9) has a variety of interesting topics each of which might make a nice project. (Available in Dr. Kemp or Dr. Flint’s office)
25. Number theory topics (see Dr. D. Vestal) Partition Theory, Pythagorean Triples, Sums of Squares, Diophantine Approximation
26. Game Theory (e.g. Two-person zero-sum games) (see Dr. D. Vestal)
27. Cryptography (see Dr. D. Vestal)
28. The “No-7” Series (see Dr. D. Vestal)
29. Operations Research (see Dr. D. Vestal)
30. Quadratic Residues/Reciprocity (see Dr. D. Vestal)
31. (See Dr. D. Vestal) The book UMAP Modules found here:
32. Any idea you have which is original or a topic you touched on in a regular class, but want to investigate further.