The Monotone Upper Bound Problem asks for the maximal number {\small $M(d,n)$} of vertices on a strictly increasing edge-path on a simple {\small $d$}-polytope with {\small $n$} facets. More specifically, it asks whether the upper bound

{\small \[ M(d,n)\ \le\ M_{\rm ubt}(d,n) \]}\!\!

provided by McMullen's [McMullen 70] Upper Bound Theorem is tight, where {\small $M_{\rm ubt}(d,n)$} is the number of vertices of a dual-to-cyclic d-polytope with n facets.

It was recently shown that the upper bound {\small $M(d,n)\le M_{\rm ubt}(d,n)$} holds with equality for small dimensions ({\small $d\le 4$} [Pfeifle 04]) and for small corank ({\small $n\le d+2$} [Gärtner et al. 01]). Here we prove that it is not tight in general: in dimension {\small $d=6$}, a polytope with {\small $n=9$} facets can have {\small $ M_{\rm ubt}(6,9)=30$} vertices, but not more than {\small $M(6,9)\le29$} vertices can lie on a strictly increasing edge-path.

The proof involves classification results about neighborly polytopes of small corank, Kalai's [Kalai 88] concept of abstract objective functions, the Holt-Klee conditions [Holt and Klee 98], explicit enumeration, Welzl's extended Gale diagrams [Welzl 01], and randomized generation of instances, as well as nonrealizability proofs via a version of the Farkas lemma.

We organize the nilpotent orbits in the exceptional complex Lie algebras into series and show that within each series the dimension of the orbit is a linear function of the natural parameter {\small $a=1, 2, 4, 8$}, respectively for {\small $\ff_4,\fe_6,\fe_7,\fe_8$}. We observe similar regularities for the centralizers of nilpotent elements in a series and grade components in the associated grading of the ambient Lie algebra. More strikingly, we observe that for {\small $a\geq 2$} the numbers of {\small $\FF_q$}-rational points on the nilpotent orbits of a given series are given by polynomials that have uniform expressions in terms of a. This even remains true for the degrees of the unipotent characters associated to these series through the Springer correspondence. We make similar observations for the series arising from the other rows of Freudenthal's magic chart and make some observations about the general organization of nilpotent orbits, including the description of and dimension formulas for several universal nilpotent orbits (universal in the sense that they occur in almost every simple Lie algebra).

We study the space {\small $\nua{m}{d}$} of clouds in {\small $\bbr^d$} (ordered sets of m points modulo the action of the group of affine isometries). We show that {\small $\nua{m}{d}$} is a smooth space, stratified over a certain hyperplane arrangement in {\small $\bbr^m$}. We give an algorithm to list all the chambers and other strata (this is independent of d). With the help of a computer, we obtain the list of all the chambers for {\small $m\leq 9$} and all the strata when {\small $m\leq 8$}. As the strata are the product of a polygon space with a disk, this gives a classification of {\small $m$}-gon spaces for {\small $m\leq 9$}. When {\small $d=2,3\,$}, {\small $m=5,6,7\,$}, and modulo reordering, we show that the chambers (and so the different generic polygon spaces) are distinguished by the ring structure of their {\small ${\rm mod}\, 2$}-cohomology.

We present an explicit sequence of pseudo-Anosov maps $\phi_k: S_{2k}\rightarrow S_{2k}$ of surfaces of genus $2k$ whose growth rates converge to one.

The operation of "mating'' two suitable complex polynomial maps {\small $f_1$} and {\small $f_2$} constructs a new dynamical system by carefully pasting together the boundaries of their filled Julia sets so as to obtain a copy of the Riemann sphere, together with a rational map {\small $f_1 \mat f_2$} from this sphere to itself. This construction is particularly hard to visualize when the filled Julia sets {\small $K(f_i)$} are dendrites, with no interior. This note will work out an explicit example of this type, with effectively computable maps from {\small $K(f_1)$} and {\small $K(f_2)$} onto the Riemann sphere.

Let s be any integer between 1 and 5. We determine necessary and sufficient conditions that a ternary quartic be expressible as a (possibly degenerate) sum of fourth powers of s linear forms.

A polynomial of the form {\small $x^\alpha - p(x)$}, where the degree of p is less than the total degree of {\small $x^\alpha$}, is said to be least deviation from zero if it has the smallest uniform norm among all such polynomials. We study polynomials of least deviation from zero over the unit ball, the unit sphere, and the standard simplex. For {\small $d=3$}, extremal polynomial for {\small $(x_1x_2x_3)^k$} on the ball and the sphere is found for {\small $k=2$} and 4. For {\small $d \ge 3$}, a family of polynomials of the form {\small $(x_1\cdots x_d)^2 - p(x)$} is explicitly given and proved to be the least deviation from zero for {\small $d =3,4,5$}, and it is conjectured to be the least deviation for all d.

We study the "approximate squaring'' map {\small $f(x) := x \lceil x \rceil$} and its behavior when iterated. We conjecture that if f is repeatedly applied to a rational number {\small $r = l/d > 1$} then eventually an integer will be reached. We prove this when {\small $d=2$}, and provide evidence that it is true in general by giving an upper bound on the density of the "exceptional set'' of numbers which fail to reach an integer. We give similar results for a p-adic analogue of f, when the exceptional set is nonempty, and for iterating the "approximate multiplication'' map {\small $f_r(x) := r \lceil x \rceil$}, where r is a fixed rational number. We briefly discuss what happens when "ceiling'' is replaced by "floor'' in the definitions.