Starting in Summer 1999 the Mathematics Department of the Pennsylvania State University will host a Research Experience for Undergraduates site. Support for the site is provided by the National Science Foundation VIGRE grant. Eleven participants are selected from qualified applicants for this year. Seven of them are participants of the MASS program.
The program run from June 20 to July 31, 1999 (6 weeks). It will combine learning with research and include:
- Program schedule
- Two two-week mini-courses
- Symmetry and Representations of Finite Groups
Instructor: Adrian Ocneanu, Professor of Mathematics - The Theory of Partitions
Instructor: George Andrews, Evan Pugh Professor of Mathematics
- Symmetry and Representations of Finite Groups
- Weekly seminar run by Misha Guysinsky
- Research projects
- A geometric spectral theory for n-tuples of self-adjoint matrices and their generalizations. Advisor: Joel Anderson
- Problems from the Partition Theory. Advisor: George Andrews
- Polygons with only one integer point. Advisor: Alexander Borisov
- The growth rate of the number of generalized diagonals for a polygonal billiard. Advisor: Anatoly Katok and Misha Guysinsky
- Matrices with non-conjugate centralizers. Advisor: Svetlana Katok
- Symmetry and Representations of Finite Groups. Advisor: Adrian Ocneanu
- New Congruences for the Partition Function. Advisor: Ken Ono
Courses
SYMMETRY AND REPRESENTATIONS OF FINITE GROUPS
June 21 - July 2
TIME: MWF 12:30 pm - 2:00 pm
INSTRUCTOR: ADRIAN OCNEANU
THE THEORY OF PARTITIONS
July 7 - July 16
TIME: MWF 12:30 pm - 2:00 pm
INSTRUCTOR: GEORGE ANDREWS
At the beginning of the twentieth century, P.A. MacMahon developed an operator method to study problems in the theory of partitions. While obviously powerful, the method was extremely cumbersome algebraically. Now in the age of computer algebra systems, it is being revived. There is now an implementation in Mathematica in a program called Omega. The topic of partitions begins with the fundamental question: How many ways can you write a given integer as sums of positive integers?
In trying to answer this question we will be led to a variety of intriguing discoveries including some of those found by India's great, enigmatic genius, Srinivasa Ramanujan. The basics of this method will be covered in the course, and the students will be given access to the relevant software. Students will be encouraged to use Omega for course-related projects.
Schedule
JUNE 20, 1999
WELCOMING PIZZA PARTY
TIME: 5:30PM
PLACE: 212 MCALLISTER BUILDING
JUNE 21, 1999
INITIAL MEETING
TIME: 10:00AM
PLACE: 223 THOMAS BUILDING
JUNE 21, 1999 - JULY 2, 1999 MINI COURSE I
"SYMMETRY AND REPRESENTATIONS OF FINITE GROUPS"
INSTRUCTOR: ADRIAN OCNEANU
TIME: MWF 12:30PM-2:00PM
PLACE: 104 MCALLISTER BUILDING
JUNE 22, 1999 - JULY 20, 1999 SEMINAR SESSIONS
INSTRUCTOR: MISHA GUYSINSKY
TIME: T 9:35AM-11:35AM
PLACE: 269 WILLARD BUILDING
JULY 7, 1999 - JULY 16, 1999 MINI COURSE II
"THE THEORY OF PARTITIONS"
INSTRUCTOR: GEORGE ANDREWS
TIME: MWF 12:30PM-2:00PM
PLACE: 104 MCALLISTER BUILDING
(SPECIAL CLASS SESSION: T-7/13/99- 210 THOMAS - 12:30PM-2:00PM)
JULY 19, 1999 - JULY 23, 1999
COMPLETION OF PROJECTS AND PREPARATION FOR CONFERENCE
PRESENTATIONS
JULY 25, 26, 27, 1999
MASS FEST CONFERENCE
TIME: 10:00AM-12:00PM, 2:00PM-6:00PM
PLACE: 115 OSMOND LABORATORY
Research
Problems from the Partition Theory
My course on the Theory of Partitions will focus on introducing the students to the Omega computer package (implemented in Mathematica) that has been developed by Peter Paule, Axel Riese and me. I believe that this package can be used with a minimal knowledge of Mathematica. It will be my intention to instruct the students early on in what they need to know. It is my hope to pose some of the problems that occupied MacMahon when he developed the theory early in the century. The problems are still interesting; however, MacMahon's method (called Partition Analysis by him) was sufficiently unwiledy that he was able to make only minimal progress with his projects. Paule, Riese and I have established that Partition Analysis is, in fact, an algorithm which we have implemented. Thus we are able to do calculations that took MacMahon days or weeks.
Polygons with only one integer point
Let G be a grid of points in the plane with all integer coordinates (i.e. containing points (m,n) where both m and n are integers). For a higher-dimensional space we define G in the same way (i.e. set of points with all integer coordinates). Consider a polygon P in the plane (or a polyhedra in three or higher dimensional space) with vertices in G. Problems: Describe all such polygons for which the origin 0 is the only point of G which lies inside P. An exapmle of such polygon would be a square with vertices (0,1), (1,0), (0, -1) and (-1,0). There are MANY similar problems, e.g. describe all convex polygons P with vertices in G such that $\frac{1}{2}P$ does not have points of G other than the origin. Another class of problems stems from the convex rational cones such that the convex hull of the origin and the Z-generators of the extremal rays contains no other integer points. Those problems are very easy to describe and require very little background to study. What is known: In any fixed dimension there are finitely many classes of those polygons/polyhedra, so it should be possible to give a complete list of them. There are some algorithms which should describe all such polygons/polyhedra. There are also finitely many "series" of cones as above. One can implement those algorithms in a computer language of choice (or develop different approaches) to calculate how many such classes are there and maybe to obtain a complete list of them. Note: If solved, those questions would help describe some structures studied in algebraic geometry (the so called toric varieties). Note: if you are comfortable with abstract linear algebra (which is not a requirement for working with those problems and which one can study, if interested, while working on this problem), "classes" here means sets of polygons which can be transferred into each other by linear transformations from SL(n,Z). SUGGESTED READING. GENERAL READING ON TORIC VARIETIES. (Formally speaking not necessary for the project, but helpful if you wish to understand the place of the project in mathematics as a whole.) Danilov, V.I.: Geometry of toric varieties. {\it Russ. Math. Surv.} {\bf 33} (1978), No. 2, 97--154; translation from {\it Usp. Mat. Nauk} {\bf 33} (1978), No. 2(200), 85--134. Reid, M.: Decomposition of toric morphisms. Arithmetic and geometry. Pap. dedic. I. R. Shafarevich, Vol II, Progr. Math., {\bf 36} (1983), 395--418. There are also excellent books of Fulton and Oda on the toric varieties. More references are available from A. Borisov, if you are interested. SOME PAPERS THAT ARE MORE DIRECTLY RELATED TO THE PROJECT. Note: There are also some texts and computer codes that are only available directly from A. Borisov. \bibitem{BB} Borisov, A. A.; Borisov, L. A.: Singular toric Fano varieties. Math. USSR. Sb.{\bf 75}, No. 1, 277-283 (1993); translation from Mat. Sb. {\bf 183}, No. 2, 134-141 (1992). Borisov, A: On classification of toric singularities, http://www.math.psu.edu/borisov Borisov, A: Minimal discrepancies of toric singularities. {\it Manuscripta Math.} {\bf 92} (1997), no. 1, 33--45. Lawrence, Jim: Finite Unions of closed subgroups of the - torus. {\it Applied geometry and discrete mathematics}, 433--441, DIMACS Ser. Discrete Math. Theoret. Comput. Sci. {\bf 4}, {\it Amer. Math. Soc., Providence, RI,} 1991. Mori, S.; Morrison, D.R.; Morrison, I.:On four-dimensional terminal quotient singularities. Math. Comput. {\bf 51} (1988), no. 184, 769--786. Morrison, D.R.; Stevens, G.: Terminal quotient singularities in dimensions three and four. {\it Pro. Amer. Math. Soc.} {\bf 90} (1984), no.1, 15--20. Sankaran, G.K: Stable quintiples and terminal quotient singularities, {\it Math. Proc. Cambridge Philos. Soc.} {\bf 107} (1990), no. 1, 91--101. Kantor, J.-M.: On the width of lattice-free simplices, preprint. {\it Duke math. server,} alg-geom/9709026 (1997).