Algebraic geometry

"The twin conjectures of Hodge and Tate have a status in algebraic and arithmetic geometry similar to that of the Riemann hypothesis in analytic number theory."

"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."

"In his 1950 Congress address, Hodge reported on the topological and differential-geometric methods in studying algebraic varieties and complex manifolds which had been initiated by Lefschetz and developed by Hodge himself. He raised there many problems, and most of them were settled in 1950's by extensive works due to Kodaira and others. One notable exception to this is the so-called Hodge Conjecture which, if true, will give a characterization of cohomology classes of algebraic cycles on a nonsingular projective variety, generalizing the Lefschetz criterion for the case of divisors. This conjecture has an arithmetic flavour, as is common to most problems concerning algebraic cycles, which makes the problem interesting and difficult at the same time."

"While the move from dimension 2 to dimension 3 appears to be the obvious step there is a sense in which one should move from 2 to 4. This comes from the consideration of complex algebraic geometry. For complex dimension 1 this theory was started by Abel and continued by Riemann. For algebraic varieties of complex dimension n the real dimension is 2n, so the case n = 2 leads to 4-dimensional real manifolds. The key figures in the topology of higher-dimensional algebraic varieties were Lefschetz, Hodge, Cartan and Serre. While general algebraic geometry was one of the major developments of the second half of the 20th century, the topology of real 4-manifolds had a great surprise in store when Simon Donaldson made spectacular discoveries opening up an entirely new area."

"Should you just be an algebraist or a geometer? is like saying Would you rather be deaf or blind? If you are blind, you do not see space: if you are deaf, you do not hear, and hearing takes place in time. On the whole, we prefer to have both faculties."

"In the various forms of geometry (differential, metric, affine, algebraic), the central object is the variety, considered as a set of points."

"Riemann's theory of Abelian integrals was far ahead of its time in dealing with the fundamental properties of Riemann surfaces and in introducing theta functions. His viewpoint was to consider plane curves as one-dimensional complex manifolds, namely, Riemann surfaces, and it led to very important results in algebraic geometry as well. One of his basic discoveries was that a Riemann surface is uniquely determined by the set of all rational functions on it (called the field of rational functions). This was the birth of birational geometry."

"Enumerative geometry is an old subject that has been revisited extensively over the past 150 years. Enumerative geometry was an active field in the 19th century. … Unfortunately, many fundamental enumerative problems eluded the best mathematicians for most of the 20th century. Progress came from a seemingly unlikely source: string theory in physics."

"... when non-perturbative phenomena are included, there is no problem from the string theory point of view in effecting continuous transitions between Calabi-Yau spaces of different topology. This shows that stringy ideas about geometry are really more general than those found in classical Riemannian geometry. The moduli space of Calabi-Yau manifolds should thus be regarded as a continuously connected whole, rather than a series of different ones individually associated with different topological objects ... Thus, questions about the topology of Calabi-Yau spaces must be treated on the same footing as questions about the metric on the spaces. That is, the issue of topology is another aspect of the the moduli fields. These considerations are relevant to understanding the ground state of the universe."

"Calabi-Yau manifolds admit Kähler metrics with vanishing Ricci curvatures. They are solutions of the Einstein field equation with no matter. The theory of motions of circles inside of a Calabi-Yau manifold provide a model of a conformal field theory. (It is called a σ-model in physics.) Because of this, Calabi-Yau manifolds are pivotal in superstring theory. ... It has long been argued that, in order to solve certain classic problems of unified gauge theories such as the gauge hierarchy problem, the 4-dimensional effective theory should admit an N = 1 supersymmetry. In a fundamental paper, Candelas-Horowitz-Strominger-Witten ... analyzed what the constraint of that N = 1 supersymmetry would mean for the geometry of the internal space X. They found that, for the most basic product models with N = 1 supersymmetry, the space X must be a Calabi-Yau manifold of complex dimension 3. Shortly afterwards, Strominger ... considered slightly more general models, allowing warped products. For these models, the N = 1 supersymmetry constraint results in a modification of the Ricci-flat equation of the earlier model."

"Even though the Weil's Conjectures have been proved by Deligne without appealing to the theory of motives, an enlarged and in part still conjectural theory of mixed motives has in the meanwhile proved its usefulness in explaining conceptually, some intriguing phenomena arising in several areas of pure mathematics, such as Hodge theory, K-theory, algebraic cycles, polylogarithms, L-functions, Galois representations etc."

"Grothendieck's dream was to produce, for any system of polynomial equations, the essential nugget that would remain after everything apart from the shared topological flavour of the system was washed away. Perhaps borrowing the French musical term for a recurring theme, Grothendieck dubbed this the motif of the system."

"The unlikely interplay between motives and quantum field theory has recently become an area of growing interest at the interface of algebraic geometry, number theory, and theoretical physics. The first substantial indications of a relation between these two subjects came from extensive computations of Feynman diagrams carried out by Broadhurst and Kreimer ..., which showed the presence of multiple zeta values as results of Feynman integral calculations. From the number theoretic viewpoint, multiple zeta values are a prototype case of those very interesting classes of numbers which, although not themselves algebraic, can be realized by integrating algebraic differential forms on algebraic cycles in arithmetic varieties. Such numbers are called periods, ... and there are precise conjectures on the kind of operations (changes of variables, Stokes formula) one can perform at the level of the algebraic data that will correspond to relations in the algebra of periods. As one can consider periods of algebraic varieties, one can also consider periods of motives. In fact, the nature of the numbers one obtains is very much related to the motivic complexity of the part of the cohomology of the variety that is involved in the evaluation of the period."

"In Euclid's Elements we meet the concept which later plays a significant role in the development of science. The concept is called the "division of a line in extreme and mean ratio" (DEMR). ...the concept occurs in two forms. The first is formulated in Proposition 11 of Book II. ...why did Euclid introduce different forms... which we can find in Books II, VI and XIII? ...Only three types of regular polygons can be faces of the s: the equilateral triangle... the square... and the regular pentagon. In order to construct the Platonic solids... we must build the two-dimensional faces... It is for this purpose that Euclid introduced the ... (Proposition II.11)... By using the "golden" isosceles triangle...we can construct the regular pentagon... Then only one step remains to construct the ... which for Plato is one of the most important regular polyhedra symbolizing the universal harmony in his cosmology."

"In Western Culture, starting from Phidias and the Parthenon, the Golden Section and the Golden Number are present, consciously or unconsciously, in very famous works. In the Renaissance, after the rediscovery of Fibonacci, it was a symbol of aesthetic perfection to be used in architecture and art with, among others, Leonardo da Vinci (1542-1519) and Albrecht Dürer (1471-1528). The Golden Number is in many geometric figures making them Golden. We have it among other things in the octagonal architecture of Castel del Monte. The Golden Ratio enters the pentagon which is Golden because the side of the star and the side of the pentagon are in the ratio of 38% and 62%, as required by the Golden Number."