11 quotes found
"FORTY + TEN + TEN = SIXTY"
"SEND + MORE = MONEY"
"Perhaps no puzzle of the whole collection caused more jollity or was found more entertaining than that produced by the Host of the "Tabard," who accompanied the party all the way. He called the pilgrims together and spoke as follows: "My merry masters all, now that it be my turn to give your brains a twist, I will show ye a little piece of craft that will try your wits to their full bent. And yet methinks it is but a simple matter when the doing of it is made clear. Here be a cask of fine London ale, and in my hands do I hold two measures—one of five pints, and the other of three pints. Pray show how it is possible for me to put a true pint into each of the measures." Of course, no other vessel or article is to be used, and no marking of the measures is allowed. It is a knotty little problem and a fascinating one. A good many persons to-day will find it by no means an easy task. Yet it can be done."
"The problems divided, very roughly, into two algebraic problems, two topological problems, two problems in mathematical physics, and one problem in the theory of computation."
"Posed in 1904 by Henri Poincaré, the leading mathematician of his era and among the most gifted of all time, the Poincaré conjecture is a bold guess about nothing less than the potential shape of our own universe."
"Just as Weil's conjectures were about counting solutions to equations in a situation where the number of solutions is known to be finite, the BSD conjecture concerns the simplest class of polynomial equations—elliptic curves—for which there is no simple way to decide whether the number of solutions is finite or infinite."
"The Hodge conjecture postulates a deep and powerful connection between three of the pillars of modern mathematics: algebra, topology, and analysis. Take any variety. To understand its shape (topology, leading to cohomology classes) pick out special instances of these (analysis, leading to Hodge classes by way of differential equations). These special types of cohomology class can be realised using subvarieties (algebra: throw in some extra equations and look at algebraic cycles). That is, to solve the topology problem 'what shape is this thing?' for a variety, turn the question into analysis and then solve that using algebra. Why is that important? The Hodge conjecture is a proposal to add two new tools to the algebraic geometer's toolbox: topological invariants and Laplace's equation. It's not really a conjecture about a mathematical theorem; it's a conjecture about new kinds of tools."
"Fluids are important and hard to understand. There are many fascinating problems and conjectures about the behavior of solutions of the Euler and Navier–Stokes equations. Since we don’t even know whether these solutions exist, our understanding is at a very primitive level. Standard methods from PDE appear inadequate to settle the problem. Instead, we probably need some deep, new ideas."
"The P versus NP problem was first mentioned in a 1956 letter from Kurt Gödel to John von Neumann, two of the greatest mathematical minds of the twentieth century."
"At the beginning of the new millennium the most famous unsolved problem in complex analysis, if not in all of mathematics, is to determine whether the Riemann hypothesis holds."
"One should mention right at the start that one still does not understand whether quantum mechanics and special relativity are compatible at a fundamental level in our Minkowski four-space world. One generally assumes that this means finding a complete Yang-Mills gauge theory or the interaction of gauge fields with fermionic matter fields, the simplest form being quantum chromodynamics (QCD). Associated with this picture is the belief that the fundamental vector meson excitations are massive (as opposed to photons, which arise in the limiting case of an abelian gauge symmetry. The proof of the existence of a “mass gap” appears a necessary integral part of solving the entire puzzle. This question remains one of the deepest open issues in theoretical physics, as well as in mathematics. Basically the question remains: can one give a mathematical foundation to the theory of fields in four-dimensions? In other words, can do quantum mechanics and special relativity lie on the same footing as the classical physics of Newton, Maxwell, Einstein, or Schrödinger—all of which fits into a mathematical framework that we describe as the language of physics. This glaring gap in our fundamental knowledge even dwarfs questions of whether there are other more complicated and sophisticated approaches to physics—those that incorporate gravity, strings, or branes—for understanding their fundamental significance lies far in the future. In fact, one believes that stringy proposals, if they can be fully implemented, have limiting cases that appear as relativistic quantum fields, just as relativistic quantum fields describe non-relativistic quantum theory and classical physics in various limiting cases."