shapes, figures, and forms.
“Ramified” means that around the point in question the projection map has the same behaviour as does the projection from the parabola to its abscissa. C. Herbert Clemens, A Scrapbook of ℂ Curves
Test functions and [tempered] distributions require the notion of topological vector space … distributions can be traced back to Green’s functions in the 1830’s to solve ordinary differential equations … the 1936 work of Sergei Sobolev on hyperbolic PDE’s. Laurent Schwartz introduced the term “distribution” by analogy with a distribution of electrical charge, possibly including…
“the Yang-Mills equations are nonlinear, therefore there is little hope of finding a closed-form solution.” Such a statement seems plausible. Linear differential equations with constant coefficients are the only differential equations for which a general solution is given in closed form. As often occurs in life, however, the exceptions to the rule are sometimes more interesting than the rules…
Hadamard knew in 1898 that negative curvature and simply connectedness for surfaces embedded in 3-space force uniqueness of geodesics joining two points—implying that any segment of geodesic is also a shortest path. But there is a long way toward the modern statement: “on any complete abstract Riemannian manifold of ≥0 curvature of any dimension, curvature is the quotient of its universal covering…
Hamiltonian mechanics is the feminine side of classical physics. Its masculine side is Lagrangian mechanics, formulated in terms of velocities (tangent vectors) rather than momenta (cotangent vectors). Lagrangian mechanics focusses on the difference of kinetic – potential energies; Hamiltonian mechanics focusses on their sum. Richard Montgomery, reviewing a book by Stephanie Frank Singer and…
the cotangent bundle (differential forms) is the feminine side of calculus-on-manifolds; the tangent bundle (vector-fields) is the masculine side. Shing-Shen Chern, via Richard Montgomery
isomorphismes : “Michael Jordan has always got to be beating someone at something.” — Kareem Abdul-Jabar, as interviewed by Claudia Dreifus
isomorphismes : It’s impossible to get far in reading 20th-century mathematics without encountering the word cohomology . Cohomology & schemes are the subject of Hartshorne ’s classic, where you can find out (Appendix C) that the Weil conjectures were resolved by defining a thing called l -adic cohomology. (Cohomology even showed up in economics , information theory and computer theory — although…
isomorphismes : “the most important part of a principal components analysis is naming the axes” — William Kruskal (source: I heard this second-hand but I don’t know if it’s written down anywhere) (For those not in-the-know, Principal components are composite dimensions, like 5×faculty pay + 3×library size + … ÷ 10 or 3×vote on bill 3 + 8×vote on bill 12 + … ÷ 100 The hope is that, by using linear…
isomorphismes : Notice how some of the historical error bars do not contain the future “right answer”. These historical data (thank you to C. Amsler et al. for compiling them from across many articles!) of particle physics measurements show not only the epistemic nature of probability estimates and confidence intervals, but also the difference between probability as computed within one experiment…
isomorphismes : Carl Crow’s map of Shanghai I came to the story of Crow through Hua Hsu story about expertise about “China” . Hsu was lecturing as well about the origins of pleasure —how marketing shapes desire, and (defensively) how criticism — to praise and to blame —can serve as a counterweight to the mind-share that corporations vie for. This image—of the city of Shanghai commissioning a rich…
Cartesian functions send {A}→{B} with exactly one tail a↦ per a∈{A} connecting to each head ↦b∈{B}. In other words B has to be equal size or smaller than A. This is true mapping rings to rings, groups to groups, sets to sets, vector spaces to vector spaces, … it’s just a property of arrows really. When mathematicians want to talk about “one-to-many” (using the database…
isomorphismes : What are sites and sheaves? Internal Sieve is a downward-closed collection of open subsets of a topological space “Downward-closed” means all smaller subsets wholly contained in any part of the sieve, also are in the sieve–and smaller subsets wholly contained in those are also in the sieve, and so on. External Sieve is the following: Map the category of open sets on 𝒳𝐗𝒪𝓞𝑿…
isomorphismes : the inventor of Swype went to graduate school to try to communicate with dolphins → help alleviate various human conditons that impair communication → more usable phone interfaces for everyone else lives in Nevada City, Calif. (home of Joanna Newsom and Terry Rile) https://www.youtube.com/watch?v=2OnHlC7zi_8 invention
isomorphismes : “The … conception of the person as a bounded, unique, more or less integrated motivational and cognitive universe, a dynamic center of awareness, emotion, judgment, and action organized into a distinctive whole and set contrastively both against other such wholes and against its social and natural background, is, however incorrigible it may seem to us, a rather peculiar idea within…
elanormcinerney : Star Trek Voyager
isomorphismes : “At the turn of the century, the Swiss historian Jakob Burckhardt, who, unlike most historians, was fond of guessing the future, once confided to his friend Friedrich Nietzsche the prediction that the twentieth century would be “the age of oversimplification”. Burckhardt’s prediction has proved frighteningly accurate. Promising a life of bread and bliss, just after the war to end…
isomorphismes : “As telling examples of the views Sokal satirized, one might quote some other statements. Consider the following extrapolation of Heisenberg’s uncertainty and Bohr’s complementarity into the political realm: “The thesis ‘light consists of particles’ and the antithesis ‘light consists of waves’ fought with one another until they were united in the synthesis of quantum mechanics.…
Besides John Baez ’s explanation , I like the one in Coxeter’s Regular Polytopes chapter 5. He calls the phenomenon we are describing The Dihedral Kaleidoscope . Take an image in the plane Joan Miró, Women & Birds at Sunrise and reflect it across any of the ( half-open ) semicircle’s worth of options, of lines-thru-the-origin, that you could reflect it across. Call the action of…
“Tensor products are the things you need to form a basis for multilinear maps.” - (As distinct from Cartesian products , whose components operate independently from each other, if you adjust any of the sliders α, β, γ on αa + βb + γc , the single output slides proportionally as well. So tensoring a⊗b⊗c would put a,b,c together that way .)