The monomial initial ideals of a graded polynomial ideal are in bijection with the vertices of a convex polytope known as the state polytope of the ideal. The Gröbner fan of the ideal is the normal fan of its state polytope. In this paper we present a software system called TiGERS (Toric Gröbner bases Enumeration by Reverse Search) for computing the Gröbner fan of a toric ideal by enumerating the edge graph of its state polytope. The key contributions are an inexpensive algorithm for local change of Gröbner bases in toric ideals and the identification of a reverse search tree on the vertices of the state polytope. Using these ideas we obtain a combinatorial Gröbner walk procedure for toric ideals. TiGERS has been used to compute state polytopes with over 200,000 vertices.

Using an algorithm developed by the author, two new presentations for R. Lyons' sporadic simple group $\Ly$ are established, which contain fewer relations and are shorter than previously known ones.

We study the average number of intersecting points of a given curve with random hyperplanes in an $n$-dimensional Euclidean space. As noticed by A. Edelman and E. Kostlan, this problem is closely linked to finding the average number of real zeros of random polynomials. They showed that a real polynomial of degree $n$ has on average $\frac{2}{\pi}\log n +O(1)$ real zeros (M. Kac's theorem).

This result leads us to the following problem: given a real sequence $(\alpha_k)_{k\in\N }$, study the average $$\frac{1}{N}\sum_{n=0}^{N-1} \rho(f_{n}),$$ where $\rho(f_n)$ is the number of real zeros of $f_n(X)=\alpha_0+\alpha_1X+\cdots+\alpha_nX^n$. We give theoretical results for the Thue-Morse polynomials and numerical evidence for other polynomials.

Let $\funnyGamma$ be a torsion-free finite-index subgroup of $\SL_{n} (\Z )$ or $\GL_{n} (\Z )$, and let $\nu $ be the cohomological dimension of $\funnyGamma $. We present an algorithm to compute the eigenvalues of the Hecke operators on $H^{\nu -1} (\funnyGamma ;\Z )$, for n= 2, 3, and 4. In addition, we describe a modification of the modular symbol algorithm of Ash and Rudolph for computing Hecke eigenvalues on $H^{\nu } (\funnyGamma ;\Z )$.

The identity $$\medmuskip 0mu minus 2mu \bigl((x^2-85)^2@-@@4176\bigr)^2-2880^2=(x^2-@ 1^2)\*(x^2-@ 7^2)\*(x^2-@ 11^2)\*(x^2-@ 13^2),$$ discovered by R. E. Crandall, allows the evaluation of a product of 8 integers by a succession of 3 squares and 3 subtractions. The question arises whether there exist formulas like Crandall's with more than 3 nested squares. It will be shown that this is not the case; however, there are infinitely many formulas of length 3.

We show that the Holt-Rees extension of the standard MeatAxe procedure finds submodules of modules over finite algebras with positive probability in more cases than originally claimed. For the case when the Holt-Rees method fails we propose a further, but still simple and efficient extension.

We present tables of conjugacy classes of the hyperelliptic mapping class group of genus 2 and 3, and some theorems on the Sp representation, the Jones representation, and Meyer's function.

The well-known dual pair of Napoleon equilateral triangles intrinsic to each triangle is extended to infinite sequences of them, shown to be special cases of infinite regular hexagon sequences on each triangle. A set of hexagon-to-hexagon transformations, the hex operators, is defined for this purpose, a set forming an abelian monoid under function composition. The sequences result from arbitrary strings of hex operators applied to a particular truncation of a given triangle to a hexagon. The deep structure of the sequence constructions reveals surprising infinite sequences of nonconcentric, symmetric equilateral triangle pairs parallel to one of the sequences of hexagons and provides the most visually striking contribution. Extensive experimentation with a plane geometry educational program inspired all theorems, proofs of which utilize eigenvector analysis of polygons in the complex plane.

We present the results of our computations concerning the space groups of dimension 5 and 6. We find 222 018 and 28 927 922 isomorphism types of these groups, respectively. Some overall statistics on the number of $\funnyQ$-classes and $\funnyZ$-classes in dimensions up to six are provided. The computations were done with the package CARAT, which can parametrize, construct and identify all crystallographic groups up to dimension 6.

It is known that the Hausdorff dimension of the invariant set $\Lambda_t$ of an iterated function system ${\cal F}_t$ on $\R^n$ depending smoothly on a parameter $t$ varies lower-semicontinuously, but not necessarily continuously. For a specific family of systems we investigate numerically the conjecture that discontinuities in the dimension only arise when in some iterate of the iterated function system two or more branches coincide. This happens in a dense set of codimension one. All other points are conjectured to be points of continuity.

We study the minimal degree d(m) of a polynomial with all coefficients in $\{-1,0,1\}$ and a zero of order m at 1. We determine d(m) for $m\leq10$ and compute all the extremal polynomials. We also determine the minimal degree for $m=11$ and $m=12$ among certain symmetric polynomials, and we find explicit examples with small degree for $m\leq21$. Each of the extremal examples is a pure product polynomial. The method uses algebraic number theory and combinatorial computations and relies on showing that a polynomial with bounded degree, restricted coefficients, and a zero of high order at 1 automatically vanishes at several roots of unity.

Finding optimal packings of a symplectic manifold with symplectic embeddings of balls is a well known problem. In the following, an alternate symplectic packing problem is explored where the target and domains are 2n-dimensional manifolds which have first homology group equal to $\funnyZ^n$ and the embeddings induce isomorphisms of first homology. When the target and domains are $\funnyT^n \times V$ and $\funnyT^n \times U$ in the cotangent bundle of the torus, all such symplectic packings give rise to packings of $V$ by copies of $U$ under $\GL(n,\funnyZ)$ and translations. For arbitrary dimensions, symplectic packing invariants are computed when packing a small number of objects. In dimensions 4 and 6, computer algorithms are used to calculate the invariants associated to packing a larger number of objects. These alternate and classic symplectic packing invariants have interesting similarities and differences.

Denote by $\Phi_2$ the automorphism group of the free group $F_2$ on two generators. We classify the irreducible 6-dimensional complex representations of $\Phi_2$ whose restriction to $F_2$ is nontrivial. J. Dyer, E. Formanek, and E. Grossman have shown how the Bürau representation of the braid group $B_4$ gives rise to a one-parameter family of irreducible 6-dimensional representations of $\Phi_2$. The faithfulness question for these and some other closely related representations of $\Phi_2$ is open. Our classification shows that all other 6-dimensional representations of $\Phi_2$ are not faithful.

We perform an experiment concerning certain sets of 2-colorings and the existence of monochromatic arithmetic progressions of odd step. We use the results of the experiment to prove more general statements.

We give examples where Kanenobu's necessary condition for the rationality of a knot is not sufficient, and show that such examples are atypical.