Topics in Mathematics and Computer Science

1. Topics in Mathematics and Computer Science CSUCI Masters Seminars August 31, 2005 Dr. AJ Bieszczad Dr. Cindy Wyels

2. What is Mathematics? • Mathematics is the study of quantity, structure, space, and change. Historically, mathematics developed from counting, calculation, measurement, and the study of the shapes and motions of physical objects, through the use of abstraction and deductive reasoning. (Wikipedia) • Mathematics: the abstract science of number, quantity, and space studied in its own right (pure mathematics) or as applied to other disciplines such as physics, engineering, etc. (applied mathematics). (Oxford) • Mathematics: the science of numbers and their operations, interrelations, combinations, generalizations, and abstractions and of space configurations and their structure, measurement, transformations, and generalizations (Merriam Webster)

3. What do some experts say? • Mathematics is the tool specially suited for dealing with abstract concepts of any kind and there is no limit to its power in this field. Paul Dirac (1902 – 1984) • Mathematics is the science of patterns. Keith Devlin • Mathematics is a language. Josiah Willard Gibbs (1839 – 1903) • Mathematics is not a deductive science -- that's a cliche. When you try to prove a theorem, you don't just list the hypotheses, and then start to reason. What you do is trial and error, experimentation, guesswork. Paul Halmos

4. Some perspectives • There is no branch of mathematics, however abstract, which may not some day be applied to phenomena of the real world. Nikolai Lobatchevsky (1792 – 1856) • Mathematics knows no races or geographic boundaries; for mathematics, the cultural world is one country. David Hilbert (1862 – 1943) • Anyone who cannot cope with mathematics is not fully human. At best he is a tolerable subhuman who has learned to wear shoes, bathe, and not make messes in the house. Robert A. Heinlein • Life is good for only two things, discovering mathematics and teaching mathematics. Simeon Poisson (1781 – 1840)

5. One scheme for classifying mathematics • Foundations considers questions in logic or set theory -- the language of mathematics • Algebra symmetry, patterns, discrete sets, and the rules for manipulating arithmetic operations • Geometry shapes and sets, properties of shapes and sets that are preserved under various kinds of motions • Analysis functions, the real number line, and the ideas of continuity and limit • ProbabilityandStatistics can be mathematical and/or experimental • Computersciences algorithms, information handling, etc. • Mathematics Education; History of Mathematics • Mathematics used to express ideas in physicalsciences, engineering, etc.

6. The NSF Organizational Scheme • Algebra and Number Theory • Topology and Foundations • Geometric Analysis • Analysis • Statistics and Probability • Computational Mathematics • Applied Mathematics

7. An AMS Scheme for tracking new Ph.D.s • Logic/Discrete Math/Combinatorics/Computer Science (90) • Algebra and Number Theory (169) • Geometry and Topology (132) • Differential, Integral, and Difference Equations (98) • Real, Complex, Functional, and Harmonic Analysis (105) • Numerical Analysis, Approximations (79) • Probability (51) • Statistics (269) • Applied Mathematics (100) • Linear, Nonlinear Optimization and Control (23) • Mathematics Education (14) • Other/Unknown (3) Numbers from 1999

8. Mathematical areas of some local experts • Logic/Philosophy:Dennis Slivinski • Discrete Math and Combinatorics:Cindy Wyels, Ron Rieger, Nathaniel Emerson • Computer Science:Bill Wolfe, AJ Bieszczad, Anna Bieszczad • Algebra:Jesse Elliott, Ivona Grzegorczyk, Morgan Sherman • Number Theory:  Jesse Elliott • Geometry:Ivona Grzegorczyk, Morgan Sherman • Topology: Mohamed Ait Nouh, Peter Yi • Differential Equations and Dynamical Systems:Nathaniel Emerson • Real and Complex Analysis:  Roger Roybal, Nathaniel Reid, Nathaniel Emerson, Jorge Garcia • Probability: Jorge Garcia • Statistics: James Sayre, Matthew Wiers, Jorge Garcia • Applied Mathematics:Aemiro Beyene, Greg Woods (Phys), Tabitha Swan-Wood (Phys), Jerry Clifford (Phys), Nick Bosco (Phys) • Imaging/ Pattern Recognition:  Geoff Dougherty (Phys) • Linear, Nonlinear Optimization and Control:Ron Rieger • Mathematics Education:  Ivona Grzegorczyk, Marguerite George, Steven Thomassin, Cindy Wyels, Merilyn Buchanan

9. Open Problem: Geometry and Logic Give criteria that may be used to determine whether the plane may be tiled with a random tile. (Suggested here by Ivona Grzegorczyk)

10. Open Problem: Operator theory Consider polynomials of one variable that take on only non-negative values. It has been shown that such polynomials are each a sum of the squares of at most two polynomials. But, if we consider polynomials in two or more variables, there exist positive polynomials that are not the sum of any number of squares. An important problem is to find a complete algebraic characterization of all such polynomials. A solution would have important ramifications in real algebraic geometry, operator theory, medical imaging, and general mathematical awesomeness. (Suggested here by Roger Roybal)

11. Open Problem: Complex Dynamics Is there a Julia set of a polynomial with positive area (Lebesgue measure)?  For all polynomials where the area is known, it is 0. (Suggested here by Nathaniel Emerson) pictures borrowed from Wikipedia

12. Open Problem: Geometry and Number Theory How many points can you find on the (half) parabola y = x2, x > 0, so that the distance between any pair of these points is rational? This sounds like geometry, but it is likely to require techniques in number theory… we don’t really know! Source: Nate Dean at DIMACS http://dimacs.rutgers.edu/~hochberg/undopen/

13. Open Problem: Number Theory The Twin Prime Conjecture There are infinitely many twin primes. Or you might prefer: Are there infinitely many primes of the form n2+1?

14. Open Problem: Combinatorial Game Theory Gale’s Vingt-et-un game: Cards numbered 1 through 10 are laid on the table. L chooses a card. Then R chooses cards until his total of chosen cards exceeds the card chosen by L. Then L chooses until her cumulative total exceeds that of R, etc. The first player to get 21 wins. Who is it?

15. Open Problem: Graph Theory What is the crossing number of the complete bipartite graph K(9, 9)? Equivalently: Place 9 red points in the plane and 9 blue points in the plane, and then connect each red point to each blue point with curves (81 curves in all). What is the minimum number of crossing points that must appear in your drawing? It is conjectured to be 256, but nobody knows. An example for K(4,4) is shown to the right, with 8 crossings. Actually, this graph can be drawn with just 4 crossings...can you find it? Source: Robert Hochberg at DIMACS http://dimacs.rutgers.edu/~hochberg/undopen/ K4,4

16. Is any of this stuff practical?

17. References for further investigation • The Mathematical Atlas – a fantastic source for learning more about various subfields of mathematics. Very comprehensive. http://www.math.niu.edu/~rusin/known-math/index/tour.html • Math on the Web (by the AMS) – web material organized by mathematical topics. Extensive. http://www.ams.org/mathweb/mi-mathbytopic.html • The Math Archives – web resources grouped by mathematical topic. http://archives.math.utk.edu/topics/ • MathPages – a series of well-written articles outlining interesting subtopics/ problems, organized by mathematical topic. Very extensive. http://www.mathpages.com/home/index.htm • SIAM, Mathematics in Industry report – examines applications of mathematics to problems arising in industry, government, and business. Thought-provoking. http://www.siam.org/mii/ • Unsolved problems – this site gives links to many pages giving unsolved problems. http://www.mathsoft.com/mathsoft_resources/unsolved_problems/1999.asp