“Mirror symmetry is concerned with counting the number of holomorphic curves on Calabi-Yau manifolds, i.e. compact Kähler manifolds X with trivial canonical bundle KX.”

— Robbert Dijkgraaf

[Mirror symmetry and elliptic curves by Robert Dijkgraaf, The moduli space of curves, 149–163, Progress in Mathematics, vol. 129, Birkhäuser Boston, 1995, 10.1007/978-1-4612-4264-2_5]

Adopted from Wikiquote. Last update June 3, 2021. History

Help us to complete the source, original and additional information

Do you have more details about the quote "Mirror symmetry is concerned with counting the number of holomorphic curves on Calabi-Yau manifolds, i.e. compact Kähle…" by Robbert Dijkgraaf?

Robbert Dijkgraaf 3

Dutch mathematical physicist and string theorist 1960

Related quotes

“In a mathematical sense, space is manifoldness, or combination of numbers. Physical space is known as the 3-dimension system. There is the 4-dimension system, there is the 10-dimension system.”

Charles Proteus Steinmetz (1865–1923) Mathematician and electrical engineer

New York Times interview (1911)

“Supergravity theories generically contain non-compact global symmetry groups. The general rule is that the scalar fields of the theory in question parametrize a symmetric space. Thus, if the non-compact symmetry group is G, and its maximal compact subgroup is H, the scalar fields map the space-time into the symmetric space G/H, and the number of scalar fields is dim G – dim H. The first supergravity example of this type to be found, N = 4 supergravity is one of the most interesting. In this case there are two scalar fields and the symmetric space is SL(2,R)/SO(2).”

John Henry Schwarz (1941) American theoretical physicist

p. 1

“All political arrangements, in that they have to bring a variety of widely-discordant interests into unity and harmony, necessarily occasion manifold collisions. From these collisions spring misproportions between men’s desires and their powers; and from these, transgressions. The more active the State is, the greater is the number of these.”

Wilhelm Von Humboldt (1767–1835) German (Prussian) philosopher, government functionary, diplomat, and founder of the University of Berlin

Source: The Limits of State Action (1792), Ch. 8

“Definite portions of a manifoldness, distinguished by a mark or by a boundary, are called Quanta. Their comparison with regard to quantity is accomplished in the case of discrete magnitudes by counting, in the case of continuous magnitudes by measuring. Measure consists in the superposition of the magnitudes to be compared; it therefore requires a means of using one magnitude as the standard for another. In the absence of this, two magnitudes can only be compared when one is a part of the other; in which case also we can only determine the more or less and not the how much. The researches which can in this case be instituted about them form a general division of the science of magnitude in which magnitudes are regarded not as existing independently of position and not as expressible in terms of a unit, but as regions in a manifoldness.”

Bernhard Riemann (1826–1866) German mathematician

On the Hypotheses which lie at the Bases of Geometry (1873)

“The use of canon raised numerous questions concerning the paths of projectiles. …One might determine… what type of curve a projectile follows and…. prove some geometrical facts about this curve, but geometry could never answer such questions as how high the projectile would go or how far from the starting point it would land. The seventeenth century sought the quantitative or numerical information needed for practical applications, and such information is provided by algebra.”

Morris Kline (1908–1992) American mathematician

Source: Mathematics and the Physical World (1959), p. 148

“Suppose we a certain Number of things exposed, different each from other, as a, b, c, d, e, &c.; The question is, how many ways the order of these may be varied? as, for instance, how many changes may be Rung upon a certain Number of Bells; or, how many ways (by way of Anagram) a certain Number of (different) Letters may be differently ordered?
Alt.1,21) If the thing exposed be but One, as a, it is certain, that the order can be but one. That is 1.
2) If Two be exposed, as a, b, it is also manifest, that they may be taken in a double order, as ab, ba, and no more. That is 1 x 2 = 2. Alt.3
3) If Three be exposed; as a, b, c: Then, beginning with a, the other two b, c, may (by art. 2,) be disposed according to Two different orders, as bc, cb; whence arise Two Changes (or varieties of order) beginning with a as abc, acb: And, in like manner it may be shewed, that there be as many beginning with b; because the other two, a, c, may be so varied, as bac, bca. And again as many beginning with c as cab, cba. And therefore, in all, Three times Two. That is 1 x 2, x 3 = 6.
Alt.34) If Four be exposed as a, b, c, d; Then, beginning with a, the other Three may (by art. preceeding) be disposed six several ways. And (by the same reason) as many beginning with b, and as many beginning with c, and as many beginning with d. And therefore, in all, Four times six, or 24. That is, the Number answering to the case next foregoing, so many times taken as is the Number of things here exposed. That is 1 x 2 x 3, x 4 = 6 x 4 = 24.
5) And in like manner it may be shewed, that this Number 24 Multiplied by 5, that is 120 = 24 x 5 = 1 x 2 x 3 x 4 x 5, is the number of alternations (or changes of order) of Five things exposed. (Or, the Number of Changes on Five Bells.) For each of these five being put in the first place, the other four will (by art. preceeding) admit of 24 varieties, that is, in all, five times 24. And in like manner, this Number 120 Multiplied by 6, shews the Number of Alternations of 6 things exposed; and so onward, by continual Multiplication by the conse quent Numbers 7, 8, 9, &c.;
6) That is, how many so ever of Numbers, in their natural Consecution, beginning from 1, being continually Multiplied, give us the Number of Alternations (or Change of order) of which so many things are capable as is the last of the Numbers so Multiplied. As for instance, the Number of Changes in Ringing Five Bells, is 1 x 2 x 3 x 4 x 5 = 120. In Six Bells, 1 x 2 x 3 x 4 x 5 x 6 = 120 x 6 = 720. In Seven Bells, 720 x 7 = 5040. In Eight Bells, 5040 x 8 = 40320, And so onward, as far as we please.”

John Wallis (1616–1703) English mathematician

Source: A Discourse of Combinations, Alterations, and Aliquot Parts (1685), Ch.II Of Alternations, or the different Change of Order, in any Number of Things proposed.