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American Mathematical Society(On the other hand, he also stated: If all mathematics disappeared today, physics would be set back exactly one week, to which a mathematician had the clever riposte: This was the week that God created the world.)(2009)In 1986, geometer Sheldon Katz proved that the number of curves, such as circles, that are defined by polynomials of degree two and lie entirely in the quintic is 609,250.[38]In each case, the space B {displaystyle B} provides a kind of blueprint that describes how these tori are assembled into a CalabiYau manifold.[63]Lian, Bong; Liu, Kefeng; Yau, Shing-Tung (2000)575 ^ a b c Yau and Nadis 2010, pOne crucial point about D-branes is that the dynamics on the D-brane worldvolume is described by a gauge theory, a kind of highly symmetric physical theory which is also used to describe the behavior of elementary particles in the standard model of particle physicsdoi:10.4310/ajm.1999.v3.n4.a4Some gauge theories which are not part of the standard model, but which are nevertheless important for theoretical reasons, arise from strings propagating on a nearly singular backgroundThe derived category of coherent sheaves is constructed using tools from complex geometry, a branch of mathematics that describes geometric curves in algebraic terms and solves geometric problems using algebraic equations.[14] On the other hand, the Fukaya category is constructed using symplectic geometry, a branch of mathematics that arose from studies of classical physicsISBN978-0-226-87033-5It is comforting to see how mathematics has been able to absorb so much of the intuitive, often imprecise reasoning of quantum physics and string theory, and to transform many of these ideas into rigorous statements and proofsRoughly speaking, they are what mathematicians call special Lagrangian submanifolds.[52] This means among other things that they have half the dimension of the space in which they sit, and they are length-, area-, or volume-minimizing.[53] The category having these branes as its objects is called the Fukaya category.[52]

Wolfgang Pauli wittily summarized this duality in a letter to Heisenberg dated October 19, 1926, just a few weeks after the discovery: One can see the world with the p-eye, and one can see it with the q-eye, but if one opens both eyes, then one becomes crazy.The story starts in enumerative geometry, a well-established, but not very exciting branch of algebraic geometry that counts objects479 (1): 243259Arnold Lecture Notes for Complex Analysis PDF Complex Analysis Richard F2009, pThis auxiliary circle is said to parametrize the circles of the decomposition, meaning there is a correspondence between them and points of B {displaystyle B} 616 ^ Yau and Nadis 2010, pdoi:10.1016/0550-3213(90)90185-GAsian Journal of MathematicsarXiv:math/0007104Verlinde Witten Yau Zaslow History Glossary v t e Through Einsteins equation we learn that mass can be transformed into energy, and vice versaMirror Symmetry (PDF)Mirror symmetry illustrates a powerful property of quantum theory called duality: Two classical models can become equivalent when considered as quantum systems, as if a magic wand is waved and all the differences suddenly disappearOne can choose an auxiliary circle B {displaystyle B} (the pink circle in the figure) such that each of the infinitely many circles decomposing the torus passes through a point of B {displaystyle B} b7dc4c5754

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