On Algebra (2000-2009)

"Algebraic topology can be roughly defined as the study of techniques for forming algebraic images of topological spaces. Most often these algebraic images are groups, but more elaborate structures such as rings, modules, and algebras also arise. The mechanisms that create these images - the ‘lanterns’ of algebraic topology, one might say - are known formally as functors and have the characteristic feature that they form images not only of spaces but also of maps. Thus, continuous maps between spaces are projected onto homomorphisms between their algebraic images, so topologically related spaces have algebraically related images." (Allen Hatcher, "Algebraic Topology", 2001)

"Algebraic topology is most often concerned with properties of spaces that depend only on homotopy type, so local topological properties do not play much of a role." (Allen Hatcher, "Algebraic Topology", 2001)

"The geometry of Algebraic Topology is so pretty, it would seem a pity to slight it and miss all the intuition which it provides." (Allen Hatcher, "Algebraic Topology", 2002)

"Combinatorics could be described as the study of arrangements of objects according to specified rules. We want to know, first, whether a particular arrangement is possible at all, and, if so, in how many different ways it can be done. Algebraic and even probabilistic methods play an increasingly important role in answering these questions. If we have two sets of arrangements with the same cardinality, we might want to construct a natural bijection between them. We might also want to have an algorithm for constructing a particular arrangement or all arrangements, as well as for computing numerical characteristics of them; in particular, we can consider optimization problems related to such arrangements. Finally, we might be interested in an even deeper study, by investigating the structural properties of the arrangements. Methods from areas such as group theory and topology are useful here, by enabling us to study symmetries of the arrangements, as well as topological properties of certain spaces associated with them, which translate into combinatorial properties." (Cristian Lenart, "The Many Faces of Modern Combinatorics", 2003)

"Some areas of human knowledge ever since its origin had shaken our understanding of the universe from time to time. While this is more true about physics, it is true about mathematics as well. The birth of topology as analysis situs meaning rubbersheet geometry had a similar impact on our traditional knowledge of analysis. Indeed, topology had enough energy and vigour to give birth to a new culture of mathematical approach. Algebraic topology added a new dimension to that. Because quantum physicists and applied mathematicians had noted wonderful interpretations of many physical phenomena through algebraic topology, they took immense interest in the study of topology in the twentieth century." (D Chatterjee, "Topology: General & Algebraic", 2003) 

"The exquisite world of algebraic topology came into existence out of our attempts to solve topological problems by the use of algebraic tools and this revealed to us the nice interplay between algebra and topology which causes each to reinforce interpretations of the other there by breaking down the often artificial subdivision of mathematics into different branches and emphasizing the essential unity of all mathematics. The homology theory is the main branch of algebraic topology and plays the main role in the classification problems of topological spaces. There are various approaches to the study of this theory such as geometric approach, abstract approach and axiomatic approach. Since geometric approach appeals easily to our intuition, we shall start with geometric approach, the relevant development being called simplicial homology theory." (D Chatterjee, "Topology: General & Algebraic", 2003)

"Algebra is the offer made by the devil to the mathematician. The devil says: I will give you this powerful machine, it will answer any question you like. All you need to do is give me your soul: give up geometry and you will have this marvelous machine." (Michael F Atiyah, 2004)

"We divide math up into separate areas (analysis, mechanics, algebra, geometry, electromagnetism, number theory, quantum mechanics, etc.) to clarify the study of each part; but the equally valuable activity of integrating the components into a working whole is all too often neglected. Without it, the stated aim of ‘taking something apart to see how it ticks’ degenerates imperceptibly into ‘taking it apart to ensure it never ticks again’." (Miles Reid & Balazs Szendröi, "Geometry and Topology", 2005)

"Algebra is what most people associate with mathematics. In a sense, this is justified. Mathematics is the study of abstract objects, numerical, logical, or geometrical, that follow a set of several carefully chosen axioms. And basic algebra is about the simplest meaningful thing that can satisfy the above definition of mathematics. There are only a dozen or so postulates, but that is enough to make the system beautifully symmetric." (Terence Tao, "Solving Mathematical Problems: A Personal Perspective", 2006)

"Analysis is also a heavily explored subject, and it is just as general as algebra: essentially, analysis is the study of functions and their properties. The more complicated the properties, the higher the analysis." (Terence Tao, "Solving Mathematical Problems: A Personal Perspective", 2006)

"Number theory may not necessarily be divine, but it still has an aura of mystique about it. Unlike algebra, which has as its backbone the laws of manipulating equations, number theory seems to derive its results from a source unknown." (Terence Tao, "Solving Mathematical Problems: A Personal Perspective", 2006)

"Geometric algebra provides the most powerful artefact for dealing with rotations and dilations. It generalizes the role of complex numbers in two dimensions, and quaternions in three dimensions, to a wider scheme for dealing with rotations in arbitrary dimensions in a simple and comprehensive manner." (Venzo de Sabbata & Bidyut K Datta, "Geometric Algebra and Applications to Physics", 2007)

"Poetry and code - and mathematics - make us read differently from other forms of writing. Written poetry makes the silent reader read three kinds of pattern at once; code moves the reader from a static to an active, interactive and looped domain; while algebraic topology allows us to read qualitative forms and their transformations." (Stephanie Strickland & Cynthia L Jaramillo, "Dovetailing Details Fly Apart - All over, again, in code, in poetry, in chreods", 2007)

"The role of conceptual modelling in information systems development during all these decades is seen as an approach for capturing fuzzy, ill-defined, informal 'real-world' descriptions and user requirements, and then transforming them to formal, in some sense complete, and consistent conceptual specifications." (Janis A Burbenko jr., "From Information Algebra to Enterprise Modelling and Ontologies", Conceptual Modelling in Information Systems Engineering, 2007)

"[...] transcendental numbers, those numbers that lie beyond those that arise through euclidean geometry and ordinary algebraic equations. [...] The transcendentals are the numbers that fill the huge void between the more familiar algebraic numbers and the collection of all decimal expansions: to use an astronomical comparison, the transcendentals are the dark matter of the number world." (Peter M Higgins, "Number Story: From Counting to Cryptography", 2008)

"Algebraic symbols carry a universality of interpretation that allows them to be manipulated in a way that words cannot. Indeed, this was the key breakthrough that allowed mathematics to flourish in a way that was not possible until the advent of algebra. All higher mathematics relies on constant use of algebraic manipulation and would be impossible without it." (Peter M Higgins, "Number Story: From Counting to Cryptography", 2008)

"An algebra is a mathematical structure consisting of a set of elements and a collection of operators on those elements - though the term is variously defined in different contexts in mathematics, so as to narrow the meaning" (e.g. an algebra is a vector space with a bilinear multiplication operation). " (Bas C van Fraassen, "Scientific Representation: Paradoxes of Perspective", 2008)

"For mathematics to be applicable in any sense at all we need to be able to do something with it. In practice this nearly always means developing forms of calculation, and this imperative channels its practitioners into algebraic manipulations of one form or another and ultimately into producing numbers. To the modern mind, this might seem natural and inevitable." (Peter M Higgins, "Number Story: From Counting to Cryptography", 2008)

"The curious switch, from initially perceiving an obstruction to a problem to eventually embodying this obstruction as a number or an algebraic object of some sort that we can effectively study, is repeated over and over again, in different contexts, throughout mathematics." (Timothy Gowers, "The Princeton Companion to Mathematics", 2008)

"The rationals therefore also form a countable set, as do the euclidean numbers, and indeed if we consider the set of all numbers that arise from the rationals through taking roots of any order, the collection produced is still countable. We can even go beyond this: the collection of all algebraic numbers, which are those that are solutions of ordinary polynomial equations∗ form a collection that can, in principle, be arrayed in an infinite list: that is to say it is possible, with a little more crafty argument, to describe a systematic listing that sweeps them all out." (Peter M Higgins, "Number Story: From Counting to Cryptography", 2008)

"The simple algebraic numbers, like √2, seem closest in nature to the rationals, while we might expect that non-algebraic numbers, the transcedentals, to live apart and not to have close rational neighbors. Surprisingly, the opposite is true. On the one hand, it can be proved that any irrational number that can be well-approximated by rationals" (in a sense that can be made precise) must be transcendental. Indeed this affords one of the standard techniques for showing that a number is transcendental." (Peter M Higgins, "Number Story: From Counting to Cryptography", 2008)

"What was clearly useful was the use of diagrams to prove certain results either in algebraic topology, homological algebra or algebraic geometry. It is clear that doing category theory, or simply applying category theory, implies manipulating diagrams: constructing the relevant diagrams, chasing arrows by going via various paths in diagrams and showing they are equal, etc. This practice suggests that diagram manipulation, or more generally diagrams, constitutes the natural syntax of category theory and the category-theoretic way of thinking. Thus, if one could develop a formal language based on diagrams and diagrams manipulation, one would have a natural syntactical framework for category theory. However, moving from the informal language of categories which includes diagrams and diagrammatic manipulations to a formal language based on diagrams and diagrammatic manipulations is not entirely obvious." (Jean-Pierre Marquis, "From a Geometrical Point of View: A Study of the History and Philosophy of Category Theory", 2009)