# REU 1999

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:

## 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).

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