109 research outputs found
No embedding of the automorphisms of a topological space into a compact metric space endows them with a composition that passes to the limit
The Hausdorff distance, the Gromov-Hausdorff, the Fr\'echet and the natural
pseudo-distances are instances of dissimilarity measures widely used in shape
comparison. We show that they share the property of being defined as where is a suitable functional and varies in a set of
correspondences containing the set of homeomorphisms. Our main result states
that the set of homeomorphisms cannot be enlarged to a metric space
, in such a way that the composition in (extending
the composition of homeomorphisms) passes to the limit and, at the same time,
is compact.Comment: 6 pages, no figure
Topologie
The Oberwolfach conference “Topologie” is one of the few occasions where researchers from many different areas in algebraic and geometric topology are able to meet and exchange ideas. Accordingly, the program covered a wide range of new developments in such fields as classification of manifolds, isomorphism conjectures, geometric topology, and homotopy theory. More specifically, we discussed progress on problems such as the Farrell-Jones conjecture, higher dimensional analogues of Harer’s homological stability of automorphism groups of manifolds and new algebraic concepts for equivariant spectra, to mention just a few subjects. One of the highlights was a series of four talks on new methods and results about the Farrell-Jones conjecture by Arthur Bartels and Wolfgang Lück
The tropicalization of the moduli space of curves
We show that the skeleton of the Deligne-Mumford-Knudsen moduli stack of
stable curves is naturally identified with the moduli space of extended
tropical curves, and that this is compatible with the "naive" set-theoretic
tropicalization map. The proof passes through general structure results on the
skeleton of a toroidal Deligne-Mumford stack. Furthermore, we construct
tautological forgetful, clutching, and gluing maps between moduli spaces of
extended tropical curves and show that they are compatible with the analogous
tautological maps in the algebraic setting.Comment: v2: 55 pages. Expanded Section 2 with improved treatment of the
category of generalized cone complexes. Clarified the role of the coarse
moduli space and its analytification in the construction of the skeleton for
a toroidal DM stac
Symmetry defects and orbifolds of two-dimensional Yang-Mills theory
We describe discrete symmetries of two-dimensional Yang-Mills theory with gauge group associated to outer automorphisms of , and their corresponding defects. We show that the gauge theory partition function with defects can be computed as a path integral over the space of twisted -bundles, and calculate it exactly. We argue that its weak-coupling limit computes the symplectic volume of the moduli space of flat twisted -bundles on a surface. Using the defect network approach to generalised orbifolds, we gauge the discrete symmetry and construct the corresponding orbifold theory, which is again two-dimensional Yang-Mills theory but with gauge group given by an extension of by outer automorphisms. With the help of the orbifold completion of the topological defect bicategory of two-dimensional Yang-Mills theory, we describe the reverse orbifold using a Wilson line defect for the discrete gauge symmetry. We present our results using two complementary approaches: in the lattice regularisation of the path integral, and in the functorial approach to area-dependent quantum field theories with defects via regularised Frobenius algebras
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