Boris and Michael Shapiro have a conjecture concerning the Schubert calculus and real enumerative geometry and which would give infinitely many families of zero-dimensional systems of real polynomials (including families of overdetermined systems--all of whose solutions are real. It has connections to the pole placement problem in linear systems theory and to totally positive matrices. We give compelling computational evidence for its validity, prove it for infinitely many families of enumerative problems, show how a simple version implies more general versions, and present a counterexample to a general version of their conjecture.

Homotopies provide computational evidence for a challenging instance of a conjecture about whether all solutions are real. By a homotopy we mean a family of polynomial systems that describes algebraically the geometric transition from an easier configuration in special position into the general configuration for the problem we want to solve. The solutions to our problem lie at the end of the solution paths we trace with numerical continuation methods starting at the solutions of the easier, special problem. The numerical difficulties are overcome if we work in the true synthetic spirit of the Schubert calculus, selecting the numerically most favorable equations to represent the geometric problem. Since a well-conditioned polynomial system allows perturbations on the input data without destroying the reality of the solutions we obtain not just one instance, but a whole manifold of systems that satisfy the conjecture. Also an instance that involves totally positive matrices has been verified. The optimality of the solving procedure is a promising first step towards the development of numerically stable algorithms for the pole placement problem in linear systems theory.

We consider the question of existence of embedded doubly periodic minimal surfaces in $\bfR^3$ with Scherk-type ends, surfaces that topologically are Scherk's doubly periodic surface with handles added in various ways. We extend the existence results of H. Karcher and F. Wei to more cases, and we find other cases where existence does not hold.

We define the regular immersed projective solution space for a triangulated 3-manifold, and completely determine it for the figure-8 knot complement. We show that there is a close relationship between vertices of this polytope and normal surfaces of a particularly simple form immersed in the manifold.

Bestvina and Handel have introduced an effective algorithm that determines whether a given homeomorphism of an orientable, possibly punctured surface is pseudo-Anosov. We present a Java software package that realizes this algorithm for surfaces with one puncture. It allows the user to define homeomorphisms in terms of Dehn twists, and in the pseudo-Anosov case it generates images of train tracks in the sense of Bestvina-Handel.

This paper presents a computational method to find good, conjecturally optimal coverings of an equilateral triangle with up to 36 equal circles. The algorithm consists of two nested levels: on the inner level the uncovered area of the triangle is minimized by a local optimization routine while the radius of the circles is kept constant. The radius is adapted on the outer level to find a locally optimal covering. Good coverings are obtained by applying the algorithm repeatedly to random initial configurations.

The structures of the coverings are determined and the coordinates of each circle are calculated with high precision using a mathematical model for an idealized physical structure consisting of tensioned bars and frictionless pin joints. Best found coverings of an equilateral triangle with up to 36 circles are displayed, 19 of which are either new or improve on earlier published coverings.

We give a constructive proof of a theorem of Tate, which states that (under Stark's Conjecture) the field generated over a totally real field K by the Stark units contains the maximal real abelian extension of K. As a direct application of this proof, we show how one can compute explicitly real abelian extensions of K. We give two examples.

We show that the mapping class group of a closed surface of genus 2 does not satisfy the Kazhdan property by constructing subgroups of finite index having a nonvanishing first cohomology group. We also construct some subgroups of finite index in the mapping class group of a genus 3 surface and calculate their first cohomology groups, which all turn out to be trivial. Most of the calculations have been carried out by computer using GAP.

We present a computer program based on bistellar operations that provides a useful tool for the construction of simplicial manifolds with few vertices. As an example, we obtain a 16-vertex triangulation of the Poincaré homology 3-sphere; we construct an infinite series of non-PL d-dimensional spheres with d+13 vertices for $d\geq 5$; and we show that if a d-manifold, with $d\ge 5$, admits any triangulation on n vertices, it admits a noncombinatorial triangulation on n+12 vertices.

We consider the solvable Baumslag--Solitar group

$$\BS_{n}=\<a,b\mid aba^{-1}=b^{n}\>$$,

for $n\geq 2$, and try to compute the spectrum of the associated Markov operators $M_{S}$, either for the oriented Cayley graph ($S=\{a,b\}$), or for the usual Cayley graph ($S=\{a^{\pm1},b^{\pm1}\}$). We show in both cases that $\Sp M_{S}$ is connected.

For S={a,b} (nonsymmetric case), we show that the intersection of $\Sp M_{S}$ with the unit circle is the set $C_{n-1}$ of $(n{-}1)$-st roots of 1, and that $\Sp M_{S}$ contains the $n-1$ circles

$$\{z\in\bbC:|z-\half{\omega}|=\half\},\quad\hbox{for $\omega\in C_{n-1}$}$$,

together with the $n+1$ curves given by

$$\bigl(\half{w^k}-\lambda\bigr)\bigl(\half{w^{-k}}-\lambda\bigr)-\tfrac14{\exp{4\pi i\theta}}=0,$$ where $w\in C_{n+1}$, $\theta\in [ 0,1]$.

Conditional on the Generalized Riemann Hypothesis (actually on Artin's conjecture), we show that $\Sp M_{S}$ also contains the circle $\{z\in\bbC:|z|=\frac{1}{2}\}$. This is confirmed by numerical computations for n=2,3,5.

For $S=\{a^{\pm1},b^{\pm1}\}$ (symmetric case), we show that $\Sp M_{S}=[-1,1]$ for n odd, and $\Sp M_{S}=[-\frac{3}{4},1]$ for n=2. For n even, at least 4, we only get $\Sp M_{S}=[r_{n},1]$, with

$$-1<r_{n}\leq -\sin^{2}\frac{\pi n}{2(n+1)}\hbox{.}$$

We also give a potential application of our computations to the theory of wavelets.

We discuss scaling in the parameter space of a family of maps arising from the iteration of a map of the two-torus defined in terms of a Jacobian elliptic function. This map appears to show a complex analog of the Feigenbaum-Kadanoff-Shenker scaling found in bifurcation sequences of circle maps.

We consider the doubling map $T:z\mapsto z^2$ of the circle. For each T-invariant probability measure $\mu$ we define its barycentre $b(\mu)=\int_{S^1}z\, d\mu(z)$, which describes its average weight around the circle. We study the set $\Omega$ of all such barycentres, a compact convex set with nonempty interior. Its boundary \box9\ has a countable dense set of points of nondifferentiability, the worst possible regularity for the boundary of a convex set. We explain this behaviour in terms of the frequency locking of rotation numbers for a certain class of invariant measures, each supported on the closure of a Sturmian orbit.