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Wilmington, North Carolina once had a thriving Black middle class and an elected interracial government. In 1898 white supremacists used violence to overthrow that government, kill Black residents and drive many others from the city.
MORE →Reflects the personal views, recollections, and perspective of the author, Mike Davis.
This is a personal recollection on the Move fire on May 13, 1985
In mathematics, Grothendieck's Galois theory is an abstract approach to the Galois theory of fields, developed around 1960 to provide a way to study the fundamental group of algebraic topology in the setting of algebraic geometry. It provides, in the classical setting of field theory, an alternative perspective to that of Emil Artin based on linear algebra, which became standard from about the 1930s.
The approach of Alexander Grothendieck is concerned with the category-theoretic properties that characterise the categories of finite G-sets for a fixed profinite group G. For example, G might be the group denoted (see profinite integer), which is the inverse limit of the cyclic additive groups — or equivalently the completion of the infinite cyclic group for the topology of subgroups of finite index. A finite G-set is then a finite set X on which G acts through a quotient finite cyclic group, so that it is specified by giving some permutation of X.
In the above example, a connection with classical Galois theory can be seen by regarding as the profinite Galois group Gal(F/F) of the algebraic closure F of any finite field F, over F. That is, the automorphisms of F fixing F are described by the inverse limit, as we take larger and larger finite splitting fields over F. The connection with geometry can be seen when we look at covering spaces of the unit disk in the complex plane with the origin removed: the finite covering realised by the zn map of the disk, thought of by means of a complex number variable z, corresponds to the subgroup of the fundamental group of the punctured disk.
The theory of Grothendieck, published in SGA1, shows how to reconstruct the category of G-sets from a fibre functor , which in the geometric setting takes the fibre of a covering above a fixed base point (as a set). In fact there is an isomorphism proved of the type
the latter being the group of automorphisms (self-natural equivalences) of . An abstract classification of categories with a functor to the category of sets is given, by means of which one can recognise categories of G-sets for G profinite.
To see how this applies to the case of fields, one has to study the tensor product of fields. In topos theory this is a part of the study of atomic toposes.
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In mathematics, Grothendieck's Galois theory is an abstract approach to the Galois theory of fields, developed around 1960 to provide a way to study the fundamental group of algebraic topology in the setting of algebraic geometry. It provides, in the classical setting of field theory, an alternative perspective to that of Emil Artin based on linear algebra, which became standard from about the 1930s. The approach of Alexander Grothendieck is concerned with the category-theoretic properties that characterise the categories of finite G-sets for a fixed profinite group G. For example, G might be the group denoted Z ^ {\displaystyle {\hat {\mathbb {Z} }}} (see profinite integer), which is the inverse limit of the cyclic additive groups Z / n Z {\displaystyle \mathbb {Z} /n\mathbb {Z} } — or equivalently the completion of the infinite cyclic group Z {\displaystyle \mathbb {Z} } for the topology of subgroups of finite index. A finite G-set is then a finite set X on which G acts through a quotient finite cyclic group, so that it is specified by giving some permutation of X. In the above example, a connection with classical Galois theory can be seen by regarding Z ^ {\displaystyle {\hat {\mathbb {Z} }}} as the profinite Galois group Gal(F/F) of the algebraic closure F of any finite field F, over F. That is, the automorphisms of F fixing F are described by the inverse limit, as we take larger and larger finite splitting fields over F. The connection with geometry can be seen when we look at covering spaces of the unit disk in the complex plane with the origin removed: the finite covering realised by the zn map of the disk, thought of by means of a complex number variable z, corresponds to the subgroup n Z {\displaystyle n\mathbb {Z} } of the fundamental group of the punctured disk. The theory of Grothendieck, published in SGA1, shows how to reconstruct the category of G-sets from a fibre functor Φ {\displaystyle \Phi } , which in the geometric setting takes the fibre of a covering above a fixed base point (as a set). In fact there is an isomorphism proved of the type G ≅ Aut ( Φ ) {\displaystyle G\cong \operatorname {Aut} (\Phi )} , the latter being the group of automorphisms (self-natural equivalences) of Φ {\displaystyle \Phi } . An abstract classification of categories with a functor to the category of sets is given, by means of which one can recognise categories of G-sets for G profinite. To see how this applies to the case of fields, one has to study the tensor product of fields. In topos theory this is a part of the study of atomic toposes.
In mathematics, the Poincaré metric, named after Henri Poincaré, is the metric tensor describing a two-dimensional surface of constant negative curvature. It is the natural metric commonly used in a variety of calculations in hyperbolic geometry or Riemann surfaces. There are three equivalent representations commonly used in two-dimensional hyperbolic geometry. One is the Poincaré half-plane model, defining a model of hyperbolic space on the upper half-plane. The Poincaré disk model defines a model for hyperbolic space on the unit disk. The disk and the upper half plane are related by a conformal map, and isometries are given by Möbius transformations. A third representation is on the punctured disk, where relations for q-analogues are sometimes expressed. These various forms are reviewed below.
In mathematics, the exponential sheaf sequence is a fundamental short exact sequence of sheaves used in complex geometry. Let M be a complex manifold, and write OM for the sheaf of holomorphic functions on M. Let OM* be the subsheaf consisting of the non-vanishing holomorphic functions. These are both sheaves of abelian groups. The exponential function gives a sheaf homomorphism exp : O M → O M ∗ , {\displaystyle \exp :{\mathcal {O}}_{M}\to {\mathcal {O}}_{M}^{*},} because for a holomorphic function f, exp(f) is a non-vanishing holomorphic function, and exp(f + g) = exp(f)exp(g). Its kernel is the sheaf 2πiZ of locally constant functions on M taking the values 2πin, with n an integer. The exponential sheaf sequence is therefore 0 → 2 π i Z → O M → O M ∗ → 0. {\displaystyle 0\to 2\pi i\,\mathbb {Z} \to {\mathcal {O}}_{M}\to {\mathcal {O}}_{M}^{*}\to 0.} The exponential mapping here is not always a surjective map on sections; this can be seen for example when M is a punctured disk in the complex plane. The exponential map is surjective on the stalks: Given a germ g of an holomorphic function at a point P such that g(P) ≠ 0, one can take the logarithm of g in a neighborhood of P. The long exact sequence of sheaf cohomology shows that we have an exact sequence ⋯ → H 0 ( O U ) → H 0 ( O U ∗ ) → H 1 ( 2 π i Z | U ) → ⋯ {\displaystyle \cdots \to H^{0}({\mathcal {O}}_{U})\to H^{0}({\mathcal {O}}_{U}^{*})\to H^{1}(2\pi i\,\mathbb {Z} |_{U})\to \cdots } for any open set U of M. Here H0 means simply the sections over U, and the sheaf cohomology H1(2πiZ|U) is the singular cohomology of U. One can think of H1(2πiZ|U) as associating an integer to each loop in U. For each section of OM*, the connecting homomorphism to H1(2πiZ|U) gives the winding number for each loop. So this homomorphism is therefore a generalized winding number and measures the failure of U to be contractible. In other words, there is a potential topological obstruction to taking a global logarithm of a non-vanishing holomorphic function, something that is always locally possible. A further consequence of the sequence is the exactness of ⋯ → H 1 ( O M ) → H 1 ( O M ∗ ) → H 2 ( 2 π i Z ) → ⋯ . {\displaystyle \cdots \to H^{1}({\mathcal {O}}_{M})\to H^{1}({\mathcal {O}}_{M}^{*})\to H^{2}(2\pi i\,\mathbb {Z} )\to \cdots .} Here H1(OM*) can be identified with the Picard group of holomorphic line bundles on M. The connecting homomorphism sends a line bundle to its first Chern class.
In geometry, the orthocentroidal circle of a non-equilateral triangle is the circle that has the triangle's orthocenter and centroid at opposite ends of its diameter. This diameter also contains the triangle's nine-point center and is a subset of the Euler line, which also contains the circumcenter outside the orthocentroidal circle. Andrew Guinand showed in 1984 that the triangle's incenter must lie in the interior of the orthocentroidal circle, but not coinciding with the nine-point center; that is, it must fall in the open orthocentroidal disk punctured at the nine-point center. The incenter could be any such point, depending on the specific triangle having that particular orthocentroidal disk. Furthermore, the Fermat point, the Gergonne point, and the symmedian point are in the open orthocentroidal disk punctured at its own center (and could be at any point therein), while the second Fermat point and Feuerbach point are in the exterior of the orthocentroidal circle. The set of potential locations of one or the other of the Brocard points is also the open orthocentroidal disk. The square of the diameter of the orthocentroidal circle is D 2 − 4 9 ( a 2 + b 2 + c 2 ) , {\displaystyle D^{2}-{\tfrac {4}{9}}(a^{2}+b^{2}+c^{2}),} where a, b, and c are the triangle's side lengths and D is the diameter of its circumcircle. The orthocentroidal circle, circumcircle, nine-point circle, polar circle and circumcircle of the tangential triangle are coaxal, and all meet on the common radical axis if the triangle is not acute. The radical axis is the orthic axis.
Before the 1921 destruction of Tulsa’s Greenwood District, Black residents had created a remarkable center of business and community life. The district included stores, professional offices, entertainment venues and homes owned by Black citizens. Understanding Greenwood means learning what was built—not only what was burned.
MORE →Shirley Chisholm, elected in 1968.
