Abstract:
The MHB (midbrain-hindbrain boundary) is a key organizing center in the vertebrate brain characterized by highly conserved patterns of gene expression. The evidence for an MHB homolog in protochordates is equivocal, the "neck" region immediately caudal to the sensory vesicle in ascidian larvae being the best accepted candidate. It is argued here that similarities in expression patterns between the MHB and the ascidian neck region are more likely due to the latter being the principal source of neurons in the adult brain, and hence where all the genes involved in patterning the latter will necessarily be expressed. The contrast with amphioxus is exemplified by pax2/5/8, expressed in the neck region in ascidian larvae, but more caudally, along much of the nerve cord in amphioxus. The zone of expression in each case corresponds with that part of the nerve cord ultimately responsible for innervating the adult body, which suggests the spatially restricted MHB-like expression pattern in ascidians is secondarily reduced from a condition more like that in amphioxus. Patterns resembling those of the vertebrate MHB are nevertheless found elsewhere among metazoans. This suggests that, irrespective of its modern function, the MHB marks the site of an organizing center of considerable antiquity. Any explanation for how such a center became incorporated into the chordate brain must take account of the dorsoventral inversion chordates have experienced relative to other metazoans. Especially relevant here is a concept developed by Claus Nielsen, in which the brain is derived from a neural center located behind the ancestral mouth. While this is somewhat counterintuitive, it accords well with emerging molecular data.

Abstract:
Zoology texts typically list four diagnostic features of chordates: pharyngeal (that is, gill) slits or pores, a notochord, a dorsal nerve cord and serial (or segmental) muscles. This last feature, represented by the somite-derived myomere series in the case of cephalochordates (amphioxus) and vertebrates, is the subject of this account, stimulated by the recent description [1] by Simon Conway Morris and Jean-Bernard Caron (here referred to as CMC) of the Middle Cambrian fossil Pikaia gracilens (Figure 1) from the Burgess Shale of British Columbia. The authors interpret Pikaia as a basal chordate and, though this conclusion is provisional, it would be perverse to deny the key similarities between this animal and what would be expected of a basal chordate: much of the body is occupied by a series of vertical bands resembling the septa between segmental muscles, and the authors identify an axial trace that could be either a notochord or a notochord and nerve cord combined. In addition, however, there are peculiar features not known from living chordates: a sausage-shaped dorsal organ running the length of the trunk, and an anterior shield-like structure, the anterior dorsal unit, covering the head region. Pikaia does not, therefore, fit entirely comfortably with modern chordates, suggesting that it is either divergent, if it is a chordate, or is a basal member of the chordate lineage differing in significant ways from surviving members of that lineage.Of the chordate features listed above, the first, the pharyngeal slits, have an evolutionary history that probably predates chordates by a considerable interval, because apparently homologous structures occur in more basal deuterostome phyla, in living hemichordates and fossil echinoderms [2-4]. Pharyngeal pores or slits, where they occur, are assumed to play an ancestral role in deposit- or filter-feeding as a means for disposing of excess water entering the mouth and pharynx with food particles. The remaining three fea

Abstract:
We present the results of a 0.86 square degree CCD photometric survey of the open cluster NGC 2516, which has an age of about 150 Myr and may have a much lower metallicity than the similarly-aged Pleiades. We select a preliminary catalogue of 1254 low mass (between 0.2 and 2.0M_{sun}) cluster candidates, of which about 70--80 percent are expected to be genuine. The mass function is metallicity dependent, but consistent with a Salpeter-like law (dN/dlog M ~ M^{-alpha}, alpha=+1.47+/-0.11 or alpha=+1.67+/-0.11 for solar and half-solar metallicities) between 0.7 and 3.0M_{sun}. At lower masses (between 0.3 and 0.7M_{sun}) there is a sharp fall in the mass function, with alpha=-0.75+/-0.20 (solar metallicity) or alpha=-0.49+/-0.13 (half-solar metallicity), which seems inconsistent with the much flatter mass functions seen in the Pleiades and field populations. We explain this by demonstrating that mass segregation has been at work in NGC 2516 -- more than half the cluster low mass stars are expected to lie outside out survey. The mass of NGC 2516 stars with mass greater than 0.3M_{sun} inside our survey is 950-1200M_{sun}, depending on metallicity and what corrections are applied for unresolved binarity. Correcting for mass segregation increases this to ~1240-1560M_{sun}, about twice the total mass of the Pleiades.

Abstract:
We show that a smooth unknotted curve in R^3 satisfies an isoperimetric inequality that bounds the area of an embedded disk spanning the curve in terms of two parameters: the length L of the curve and the thickness r (maximal radius of an embedded tubular neighborhood) of the curve. For fixed length, the expression giving the upper bound on the area grows exponentially in 1/r^2. In the direction of lower bounds, we give a sequence of length one curves with r approaching 0 for which the area of any spanning disk is bounded from below by a function that grows exponentially with 1/r. In particular, given any constant A, there is a smooth, unknotted length one curve for which the area of a smallest embedded spanning disk is greater than A.

Abstract:
This paper completely classifies which numbers arise as the topological entropy associated to postcritically finite self-maps of the unit interval. Specifically, a positive real number h is the topological entropy of a postcritically finite self-map of the unit interval if and only if exp(h) is an algebraic integer that is at least as large as the absolute value of any of the conjugates of exp(h); that is, if exp(h) is a weak Perron number. The postcritically finite map may be chosen to be a polynomial all of whose critical points are in the interval (0,1). This paper also proves that the weak Perron numbers are precisely the numbers that arise as exp(h), where h is the topological entropy associated to ergodic train track representatives of outer automorphisms of a free group.

Abstract:
Knotted trivalent graphs (KTGs) form a rich algebra with a few simple operations: connected sum, unzip, and bubbling. With these operations, KTGs are generated by the unknotted tetrahedron and Moebius strips. Many previously known representations of knots, including knot diagrams and non-associative tangles, can be turned into KTG presentations in a natural way. Often two sequences of KTG operations produce the same output on all inputs. These `elementary' relations can be subtle: for instance, there is a planar algebra of KTGs with a distinguished cycle. Studying these relations naturally leads us to Turaev's shadow surfaces, a combinatorial representation of 3-manifolds based on simple 2-spines of 4-manifolds. We consider the knotted trivalent graphs as the boundary of a such a simple spine of the 4-ball, and to consider a Morse-theoretic sweepout of the spine as a `movie' of the knotted graph as it evolves according to the KTG operations. For every KTG presentation of a knot we can construct such a movie. Two sequences of KTG operations that yield the same surface are topologically equivalent, although the converse is not quite true.

Abstract:
For any cluster algebra whose underlying combinatorial data can be encoded by a bordered surface with marked points, we construct a geometric realization in terms of suitable decorated Teichmueller space of the surface. On the geometric side, this requires opening the surface at each interior marked point into an additional geodesic boundary component. On the algebraic side, it relies on the notion of a non-normalized cluster algebra and the machinery of tropical lambda lengths. Our model allows for an arbitrary choice of coefficients which translates into a choice of a family of integral laminations on the surface. It provides an intrinsic interpretation of cluster variables as renormalized lambda lengths of arcs on the surface. Exchange relations are written in terms of the shear coordinates of the laminations, and are interpreted as generalized Ptolemy relations for lambda lengths. This approach gives alternative proofs for the main structural results from our previous paper, removing unnecessary assumptions on the surface.

Abstract:
In response to Jaffe and Quinn [math.HO/9307227], the author discusses forms of progress in mathematics that are not captured by formal proofs of theorems, especially in his own work in the theory of foliations and geometrization of 3-manifolds and dynamical systems.

Abstract:
We show that for links with at most 5 components, the only finite type homotopy invariants are products of the linking numbers. In contrast, we show that for links with at least 9 components, there must exist finite type homotopy invariants which are not products of the linking numbers. This corrects previous errors of the first author.