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Author Archives: Danny Calegari
Minimal laminations with leaves of different conformal types
The “header image” for this blog is an example of an interesting construction in 2-dimensional conformal geometry, due to Richard Kenyon, that I learned of some time ago; I thought it might be fun to try to explain where it … Continue reading
Bridgeman’s orthospectrum identity
Martin Bridgeman gave a nice talk at Caltech recently on his discovery of a beautiful identity concerning orthospectra of hyperbolic surfaces (and manifolds of higher dimension) with totally geodesic boundary. The -dimensional case is (in my opinion) the most beautiful, … Continue reading
Schwarz-Christoffel transformations, Schwarzian derivatives, and Schwarz’s minimal surface
Hermann Amandus Schwarz (1843-1921) was a student of Kummer and Weierstrass, and made many significant contributions to geometry, especially to the fields of minimal surfaces and complex analysis. His mathematical creations are both highly abstract and flexible, and at the … Continue reading
Quasimorphisms from knot invariants
Last week, Michael Brandenbursky from the Technion gave a talk at Caltech on an interesting connection between knot theory and quasimorphisms. Michael’s paper on this subject may be obtained from the arXiv. Recall that given a group , a quasimorphism … Continue reading
Posted in 3-manifolds, Groups
Tagged 4-ball genus, braids, Cochran-Orr-Teichner, knot concordance, quasimorphisms, ribbon, signature, slice
2 Comments
Harmonic measure
An amenable group acting by homeomorphisms on a compact topological space preserves a probability measure on ; in fact, one can given a definition of amenability in such terms. For example, if is finite, it preserves an atomic measure supported … Continue reading
How to see the genus
Let be a polynomial in two variables; i.e. where each is non-negative, and the coefficients are complex numbers which are nonzero for only finitely many pairs . For a generic choice of coefficients, the equation determines a smooth complex curve … Continue reading
Geometric structures on 1-manifolds
A geometric structure on a manifold is an atlas of charts with values in some kind of “model space”, and transformation functions taken from some pseudogroup of transformations on the model space. If is the model space, and is the … Continue reading
