Relative systoles of relative-essential 2-complexes

We prove a systolic inequality for the phi-relative 1-systole of a phi-essential 2-complex, where phi is a homomorphism from the fundamental group of the complex, to a finitely presented group G. Indeed we show that universally for any phi-essential Riemannian 2-complex, and any G, the area of X is bounded below by 1/8 of sys(X,phi)^2. Combining our results with a method of Larry Guth, we obtain new quantitative results for certain 3-manifolds: in particular for Sigma the Poincare homology sphere, we have sys(Sigma)^3 < 24 vol(Sigma).Comment: 20 pages, to appear in Algebraic and Geometric Topolog

Fermat, Leibniz, Euler, and the gang: The true history of the concepts of limit and shadow

Fermat, Leibniz, Euler, and Cauchy all used one or another form of approximate equality, or the idea of discarding "negligible" terms, so as to obtain a correct analytic answer. Their inferential moves find suitable proxies in the context of modern theories of infinitesimals, and specifically the concept of shadow. We give an application to decreasing rearrangements of real functions.Comment: 35 pages, 2 figures, to appear in Notices of the American Mathematical Society 61 (2014), no.

Projective connections in CR geometry

Holomorphic invariants of an analytic real hypersurface in ℂ n+1 can be computed by several methods, coefficients of the Moser normal form [4], pseudo-con-formal curvature and its covariant derivatives [4], and projective curvature and its covariant derivatives [3]. The relation between these constructions is given in terms of reduction of the complex projective structure to a real form and exponentiation of complex vectorfields to give complex coordinate systems and corresponding Moser normal forms. Although the results hold for hypersurfaces with non-degenerate Levi-form, explicit formulas will be given only for the positive definite case.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/46644/1/229_2005_Article_BF01298334.pd

Determining crystal structures through crowdsourcing and coursework

We show here that computer game players can build high-quality crystal structures. Introduction of a new feature into the computer game Foldit allows players to build and real-space refine structures into electron density maps. To assess the usefulness of this feature, we held a crystallographic model-building competition between trained crystallographers, undergraduate students, Foldit players and automatic model-building algorithms. After removal of disordered residues, a team of Foldit players achieved the most accurate structure. Analysing the target protein of the competition, YPL067C, uncovered a new family of histidine triad proteins apparently involved in the prevention of amyloid toxicity. From this study, we conclude that crystallographers can utilize crowdsourcing to interpret electron density information and to produce structure solutions of the highest quality

Quantum Deformations of Algebras and Their Representations

X, 176 tr.; 24 cm