(1) An algorithm for finding parameters of tunnels by Kai Ishihara
Cho and McCullough gave a numerical parameterization of the collection of all tunnels of all tunnel number 1 knots and links in the 3-sphere. Here we give an algorithm for finding the parameter of a given tunnel by using its Heegaard diagram.
(2) Quantum invariants of random 3-manifolds by Nathan M Dunfield and Helen Wong
We consider the SO(3) Witten-Reshetikhin-Turaev quantum invariants of random 3-manifolds. When the level r is prime, we show that the asymptotic distribution of the absolute value of these invariants is given by a Rayleigh distribution which is independent of the choice of level. Hence the probability that the quantum invariant certifies the Heegaard genus of a random 3-manifold of a fixed Heegaard genus g is positive but very small, less than 10^-7 except when g<=3. We also examine random surface bundles over the circle and find the same distribution for quantum invariants there.
(3) Flat structures on surface bundles by Jonathan Bowden
We show that there exist flat surface bundles with closed leaves having nontrivial normal bundles. This leads us to compute the abelianisation of surface diffeomorphism groups with marked points. We also extend a formula of Tsuboi that expresses the Euler class of a flat circle bundle in terms of the Calabi invariant of certain Hamiltonian diffeomorphisms to surfaces of higher genus and derive a similar formula for the first MMM-class of surface bundles with punctured fibre.
(4) Connected components of the compactification of representation spaces of surface groups by Maxime Wolff
The Thurston compactification of Teichmuller spaces has been generalised to many different representation spaces by Morgan, Shalen, Bestvina, Paulin, Parreau and others. In the simplest case of representations of fundamental groups of closed hyperbolic surfaces in PSL(2,R), we prove that this compactification behaves very badly: the nice behaviour of the Thurston compactification of the Teichmuller space contrasts with wild phenomena happening on the boundary of the other connected components of these representation spaces. We prove that it is more natural to consider a refinement of this compactification, which remembers the orientation of the hyperbolic plane. The ideal points of this compactification are oriented R-trees, ie, R-trees equipped with a planar structure.
(5) Minimal pseudo-Anosov translation lengths on the complex of curves by Vaibhav Gadre and Chia-Yen Tsai
We establish bounds on the minimal asymptotic pseudo-Anosov translation lengths on the complex of curves of orientable surfaces. In particular, for a closed surface with genus g at least 2, we show that there are positive constants a_1 < a_2 such that the minimal translation length is bounded below and above by a_1/g^2 and a_2/g^2.
(6) Deformed Hamiltonian Floer theory, capacity estimates and Calabi quasimorphisms by Michael Usher
We develop a family of deformations of the differential and of the pair-of-pants product on the Hamiltonian Floer complex of a symplectic manifold (M,omega) which upon passing to homology yields ring isomorphisms with the *big* quantum homology of M. By studying the properties of the resulting deformed version of the Oh-Schwarz spectral invariants, we obtain a Floer-theoretic interpretation of a result of Lu which bounds the Hofer-Zehnder capacity of M when M has a nonzero Gromov-Witten invariant with two point constraints, and we produce a new algebraic criterion for (M,omega) to admit a Calabi quasimorphism and a symplectic quasistate. This latter criterion is found to hold whenever M has generically semisimple quantum homology in the sense considered by Dubrovin and Manin (this includes all compact toric M), and also whenever M is a point blowup of an arbitrary closed symplectic manifold.
(7) Line patterns in free groups by Christopher H Cashen and Natasa Macura
We study line patterns in a free group by considering the topology of the decomposition space, a quotient of the boundary at infinity of the free group related to the line pattern. We show that the group of quasi-isometries preserving a line pattern in a free group acts by isometries on a related space if and only if there are no cut pairs in the decomposition space. We also give an algorithm to detect such cut pairs.
(8) Isosystolic genus three surfaces critical for slow metric variations by Stephane Sabourau
We show that the two piecewise flat surfaces with conical singularities conjectured by E Calabi as extremal surfaces for the isosystolic problem in genus 3 are critical with respect to some metric variations. The proof relies on a new approach to study isosystolic extremal surfaces.
(9) Non-commutative Donaldson-Thomas theory and vertex operators by Kentaro Nagao
In [K Nagao, Refined open non-commutative Donaldson--Thomas theory for small toric Calabi-Yau 3-folds, Pacific J. Math. (to appear), arXiv:0907.3784], we introduced a variant of non-commutative Donaldson-Thomas theory in a combinatorial way, which is related to the topological vertex by a wall-crossing phenomenon. In this paper, we (1) provide an alternative definition in a geometric way, (2) show that the two definitions agree with each other and (3) compute the invariants using the vertex operator method, following [A Okounkov, N Reshetikhin, C Vafa, Quantum Calabi-Yau and classical crystals, from: "The unity of mathematics", Progr. Math., Birkhauser (2006) 597--618] and [B Young, Generating functions for colored 3D Young diagrams and the Donaldson-Thomas invariants of orbifolds, Duke Math. J. 152 (2010) 115--153]. The stability parameter in the geometric definition determines the order of the vertex operators and hence we can understand the wall-crossing formula in non-commutative Donaldson-Thomas theory as the commutator relation of the vertex operators.
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