David Fletcher: Variations on the monotile problem
The monotile problem states “Does there exist a single shape of tile which cannot tile the plane in a repeating manner, but can still tile the plane?”, and is a problem dating back to the origin of the mathematical field of aperiodic tilings. This talk will first give a brief history of this problem, including its links to David Hilbert’s famous problems, attempts to solve the monotile problem, and its links to topology. The second half of the talk will present my work on a variation of the monotile problem, which utilises non-standard ‘atlas’ matching rules. I will illustrate an algorithm for decreasing the number of types of tile (prototiles) needed to tile the plane in a non-repeating manner, by leveraging these atlas matching rules. This algorithm can produce a single prototile that can tile R^3, as will be shown. An example in the plane (reducing the number of prototiles needed from 13 to 2) will also be shown.
Hadi Zare: Bott periodicity elements in homology of loop spaces of spheres
For $k\in Z$, we consider $QS^{-k}$, the infinite loop space associated with $S^{-k}$ defined by $QS^{-k}=colim \Omega^{n+k}S^n$. The Curtis conjecture predicts that the only spherical classes (classes in the image of the Hurewicz homomorphism) in $H_*(Q_0S^0;\Z/2)$ are the Hopf invariant one and the Kervaire invariant one elements. We consider Sullivan's decomposition $Q_0S^0=J\times\cok J$ where $J$ is the fibre of $\psi^3-1$ and observe that the Curtis conjecture holds when we restrict to $J$. We then use the Bott periodicity and the $J$-homomorphism $SO\to Q_0S^0$ to define some generators in $H_*(Q_0S^{-k};\Z/p)$, when $p$ is any prime, and determine the type of subalgebras that they generate. For $p=2$ we determine spherical classes in $H_*(\Omega^k_0J;\Z/2)$.
Jon Woolf: Transversal Homotopy Theory and the Tangle Hypothesis
Transversal homotopy theory assigns invariants to a stratified space X by considering smooth maps from disks into X which are transversal to all strata. As an example, the third transversal homotopy category of the 2-sphere with a marked point is equivalent (essentially by the Pontrjagin-Thom construction) to the category of framed tangles.
Baez-Dolan's Tangle Hypothesis is that `the n-category of codimension k, framed n-tangles is equivalent to the free k-tuply monoidal n-category with duals on one object'. For n=1 and k=2 this is a theorem, due to Shum: usually phrased as `the category of framed tangles is the free ribbon category'.
The aim of this talk is to explain one interpretation of all these terms using a recent, very geometric, definition of n-category due to Morrison and Walker, and then to sketch out the relationship between the transversal homotopy theory of spheres with marked points and the Tangle Hypothesis.
The TTT is partially supported by an LMS scheme 3 grant. Anyone who wishes to participate is welcome: we shall operate the usual arrangements for assistance with travel expenses.
For further information email John Greenlees j.greenlees@sheffield.ac.uk, (or John Hunton jrh7@mcs.le.ac.uk or Nige Ray nige@ma.man.ac.uk) if you are interested, so we can make approximately the right amount of tea and coffee.