Christopher Tuffley

Finite subset spaces of graphs and punctured surfaces

The kth nite subset space of a topological space X is the space expkX of non-empty nite subsets of X of size at most k, topologised as aq uotient of X k . The construction is a homotopy functor and may be regarded as a union of conguration spaces of distinct unordered points in X. We calculate the homology of the nite subset spaces of a connected graph , and study the maps (exp k) induced by a map : ! 0 between two such graphs. By homotopy functoriality the results apply to punctured surfaces also. The braid group Bn may be regarded as the mapping class group of an n{punctured disc Dn , and as such it acts on H(exp k Dn). We prove a structure theorem for this action, showing that the image of the pure braid group is nilpotent of class at mostb(n 1)=2c. AMS Classication 54B20; 05C10, 20F36, 55Q52

NTRFinder: a software tool to find nested tandem repeats

We introduce the software tool NTRFinder to search for a complex repetitive structure in DNA we call a nested tandem repeat (NTR). An NTR is a recurrence of two or more distinct tandem motifs interspersed with each other. We propose that NTRs can be used as phylogenetic and population markers. We have tested our algorithm on both real and simulated data, and present some real NTRs of interest. NTRFinder can be downloaded from http://www.maths.otago.ac.nz/~aamatroud/.

Subsumed Homoclinic Connections and Infinitely Many Coexisting Attractors in Piecewise-Linear Maps

We establish an equivalence between infinitely many asymptotically stable periodic solutions and subsumed homoclinic connections for

Some Ramsey-type results on intrinsic linking of n-complexes

N

A Comparison of Three Heuristic Methods for Solving the Parsing Problem for Tandem Repeats

-dimensional piecewise-linear continuous maps. These features arise as a codimension-three phenomenon. The periodic solutions are single-round: they each involve one excursion away from a central saddle-type periodic solution. The homoclinic connection is subsumed in the sense that one branch of the unstable manifold of the saddle solution is contained entirely within its stable manifold. The results are proved by using exact expressions for the periodic solutions and components of the stable and unstable manifolds which are available because the maps are piecewise-linear. We also describe a practical approach for finding this phenomenon in the parameter space of a map and illustrate the results with the three-dimensional border-collision normal form.

Level Structures on the Weierstrass Family of Cubics

Define the complete n-complex on N vertices to be the n-skeleton of an (N-1)-simplex. We show that embeddings of sufficiently large complete n-complexes in R^{2n+1} necessarily exhibit complicated linking behaviour, thereby extending known results on embeddings of large complete graphs in R^3 (the case n=1) to higher dimensions. In particular, we prove the existence of links of the following types: r-component links, with the linking pattern of a chain, necklace or keyring; 2-component links with linking number at least lambda in absolute value; and 2-component links with linking number a non-zero multiple of a given integer q. For fixed n the number of vertices required for each of our results grows at most polynomially with respect to the parameter r, lambda or q.

Modeling the covarion hypothesis of nucleotide substitution

In many applications of tandem repeats the outcome depends critically on the choice of boundaries (beginning and end) of the repeated motif: for example, different choices of pattern boundaries can lead to different duplication history trees. However, the best choice of boundaries or parsing of the tandem repeat is often ambiguous, as the flanking regions before and after the tandem repeat often contain partial approximate copies of the motif, making it difficult to determine where the tandem repeat (and hence the motif) begins and ends. We define the parsing problem for tandem repeats to be the problem of discriminating among the possible choices of parsing.

Correcting the Apparent Mutation Rate Acceleration at Shorter Time Scales under a Jukes–Cantor Model

Let W → 𝔸 2 be the universal Weierstrass family of cubic curves over ℂ. For each N ≥ 2, we construct surfaces parameterizing the three standard kinds of level N structures on the smooth fibers of W. We then complete these surfaces to finite covers of 𝔸 2. Since W → 𝔸 2 is the versal deformation space of a cusp singularity, these surfaces convey information about the level structure on any family of curves of genus g degenerating to a cuspidal curve. Our goal in this note is to determine for which values of N these surfaces are smooth over (0, 0). From a topological perspective, the results determine the homeomorphism type of certain branched covers of S 3 with monodromy in SL2 (ℤ/N).