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 $n-$dimensional
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).