Appendix A Appendix B Modular Symbol Algorithms, List of American Autobiographical List of Suggested Writing Prompts Literature How does where you’re from or your family’s heritage influence Computational Number Theory, who you are? Reflect on your geographic, ethnic, cultural From The Best American Essays of the Century. Ed. Joyce Carol or religious background. and the Millennium Problems Oates and Robert What are some unsaid rules in your life (as a woman or man, as Atwan. New York: Houghton Mifflin Company, 2000. a student or teacher, as a young person, as a Hungarian, as Angelou, Maya. “I Know Why the Caged Bird Sings.” 342-357. a member of whichever group with which you identify)? Bourne, Randolph. “The Handicapped.” 57-70. What do you want outsiders to know about your culture? Write Ehrlich, Gretel. “Solace of Open Spaces.” 467-76. about your culture. Stephanie Jakus Herr, Michael. “Illumination Rounds.” 327-341. Write about a time you had to stand up for a belief when most Kingston, Maxine Hong. “No Name Woman.” 383-94. people didn’t agree with you. Momaday, N. Scott. “The Way to Rainy Mountain.” 313-18. Whose voices are part of multicultural Hungary? Who is a Oates, Joyce Carol. “They All Just Went Away.” 553-63. Hungarian? Rodriguez, Richard. “Aria: A Memoir of a Bilingual ............................................................................................... Childhood.” 447-466. Bibliography Twain, Mark. “Corn-pone Opinions.” 1-5. Hideg, Éva and Erzsébet Nováky. “The Future Orientation of The University of Michigan, Ann Arbor Eötvös Loránd University Wright, Richard. “The Ethics of Living Jim Crow: An Hungarian Youth in the Years 2074 East Hall, 530 Church Street, 1117 Budapest, Pázmány Péter sétány 1/C Autobiographical Sketch.” 159 -70. of the Transformation.”World Futures Studies Federation. Ann Arbor, MI 48109-1043 www.elte.hu From Literature and Society: An Introduction to Fiction, 1997. http://www.wfsf.org/pub http://www.math.lsa.umich.edu/ Adviser: Árpád Tóth Poetry, Drama, Nonfiction. Ed. /publications/Brisbane_97/HIDEGNOV.pdf [email protected] Pamela Annas and Robert Rosen. New Jersey: Prentice-Hall, Raphael, Taffy, Laura Pardo and Kathy Highfield. Book Club: 1994. A Literature-Based ............................................................................................... Kovic, Ron. From “Born on the Fourth of July.” 1059-69. Curriculum. Massachusetts: Small Planet Communications, Whitecloud, Thomas. “Blue Winds Dancing.” 1336-41 1997. As one of the Millennium Problems, the solutions to which carry a prize of one million dollars Wright, Richard. “The Man Who Went to Chicago.” 858-82. ---- “Book Club Workshop: Learning About Language and a piece, the Birch Swinnerton-Dyer conjecture, since its introduction in the early 1960’s, has Kingsolver, Barbara. High Tide in Tucson: Essays from Now or Literacy through Culture.” Ciera. remained both a fundamental unsolved problem in algebraic number theory and one of the Never. USA: Harper April 1999. http://www.ciera.org/library/archive/1999-04/ most challenging problems of the twenty first century. My work on the Fulbright fellowship Perennial, 1996. 1-16. abs-online-99-04.html In J. in computing cohomology groups using the modular symbol method is not in the direction of Michie, Gregory. Holler If You Hear Me. New York: Teachers Many (Ed.), Instructional practices for literacy teacher- proving the Millennium Problem, but in implementing the tools of computational number College Press, 1999. 1- 12. educators. Mahwah, NJ: Erlbaum. 39-49. theory that have developed in the past twenty years of progress on the conjecture, and in the Sandburg, Carl. “Chicago.” http://www.carl-sandburg.com/ direction of expanding these tools for new uses. The mathematics behind the algorithms in this chicago.htm. 7 April 2006. project lies in the intersection of algebraic number theory, homology and cohomology theory, Sedaris, David. Me Talk Pretty One Day. USA: Little Brown, complex analysis, and algebraic geometry, and is well explained in the order of its development 2000. 153-65. using the history of the Birch Swinnerton-Dyer conjecture. Selzer, Richard. From “Confessions of a Knife” Modern American Memoirs. Ed. Annie One of the seven Millennium Problems of the Clay Mathematics Institute in Dillard and Cort Conley. USA: Harper Perennial, 1996. 100-08. Cambridge, Massachusetts is the Birch Swinnerton-Dyer conjecture, named after two British mathematicians, Bryan Birch and Peter Swinnerton-Dyer, who first formulated the conjecture. The conjecture relates the number of infinite order basis elements of 204 205 AY 2005-2006 Stephanie Jakus: Symbol Algorithms, Computational Number Theory the group of rational points on an elliptic is, “all preface from beginning to end,” In the early part of the nineteenth sequence formed by counting the number curve to what is called the L-function and is aimed at a general audience century, Louis Joel Mordell (1888-1972), of solutions to the equation of an elliptic of the curve. As one of the Millennium without omitting technical terminology, an American born mathematician who curve modulus a prime for each prime Problems, the solutions to which carry a but also without developing it to any worked for most of his life in England, number. Taniyama worked with Goro prize of one million dollars a piece, the extent. The reader is referred to [3],[8], made contributions to the theory of Shimura through 1957 on refining the Birch Swinnerton-Dyer conjecture, since or [12] for details about the mathematics. modular forms by using what is now conjecture, but in 1958, for reasons that its introduction in the early 1960’s, has The history of the relationship between know as Hecke operators to prove one were allusive even to Taniyama himself, remained both a fundamental unsolved modular forms and elliptic curves that of Ramanujan’s conjectures [9]. This he committed suicide. Shortly after, his problem in algebraic number theory and lead to modular symbol algorithms was followed by the introduction of the fiancee also committed suicide because one of the most challenging problems of and to the partial solution of the Birch L-function by Erich Hecke (1887-1947), she had vowed to Taniyama never to the twenty first century. The history of the Swinnerton-Dyer conjecture, begins with and his research on the properties of part from him [14]. For a long time the problem is noteworthy, as the predecessor such mathematical luminaries as Srinivasa the algebra of Hecke operators, two conjecture was not even recognized as to the Birch Swinnerton-Dyer conjecture, Ramanujan, Felix Klein, and Henri crucial steps in the direction of the being correct, let alone of significance. the Taniyama Shimura theorem led to Poincare. Srinivasa Aiyangar Ramanujan Birch Swinnerton-Dyer conjecture [5]. It was not until 1980 that the German one of the most well-publicized results in (1887-1920), the self-taught Indian Hecke operators play a role as averaging mathematician Gerhard Frey suggested number theory, the solution of Fermat’s number theorist who is credited with operators on the space of modular forms, that the Taniyama Shimura conjecture Last Theorem, while the development over three thousand theorems and had and L-functions are functions with two implied Fermat’s Last Theorem. Fermat’s of the mathematics instigated by incredible insight into the relevance of inputs, an elliptic curve and a number Last Theorem is the statement that it is work on the conjecture has led to new modular forms in number theory, studied from the complex plane. An elliptic curve impossible to find integer solutions to an error-correcting codes from the algebraic a particular modular form, the cusp form is a polynomial of the form y2=f(x), where equation of the form of the Pythagorean geometry of elliptic curves and an during the beginning of the twentieth f(x) is a polynomial with degree three, or theorem, but with exponents, or degree, improved method for factoring integers century [10]. At the same time Felix the highest power of x being x3. It wasn’t greater than two. In the seventeenth based on elliptic curves over finite fields. Klein (1849-1925) and Henri Poincare until the end of the 1950’s, however, that century, Pierre de Fermat had scribbled a My work on the Fulbright fellowship (1854-1912) studied automorphic the extent of the relationship between note in the margin of a math book he was in computing cohomology groups using functions and Klein summarized his work modular forms and the theory of elliptic studying, saying that he had discovered the modular symbol method is not in on automorphic and elliptic modular curves was hypothesized. a marvelous proof of this theorem, but the direction of proving the Millennium functions in a four volume treatise [6],[7]. The connection between modular the margin did not contain enough room Problem, but in implementing the tools A modular function is a meromorphic forms and elliptic curves appeared in for the proof. Many speculate that he of computational number theory that function on the complex upper-half 1955 as Yutaka Taniyama’s conjecture could not have found such a proof, and, have developed in the past twenty years plane that is invariant under the action that all elliptic curves over the rationals in any case, he never wrote it down. of progress on the conjecture, and in of certain groups of matrices on the are modular. By modular elliptic curve, The conjecture remained unproven for the direction of expanding these tools upper-half plane, while a modular form Taniyama meant that the elliptic curve three hundred fifty seven years. In the for new uses. The mathematics behind is holomorphic on the upper-half plane could be associated with a modular early nineteen nineties, Ken Ribet of the algorithms in this project lies in the union a point at infinity with the same form in the following manner: being the University of California, Berkeley intersection of algebraic number theory, invariance.