Algebra

From Wikipedia, the free encyclopedia

The quadratic formula expresses the solution of the degree two equation

in terms of its coefficients , where is not zero.

Algebra (from Arabic al-jebr meaning "reunion of broken parts"^[1]) is one of the broad parts of mathematics, together with number theory, geometry and analysis. In its most general form algebra is the study of symbols and the rules for manipulating symbols^[2] and is a unifying thread of almost all of mathematics.^[3] As such, it includes everything from elementary equation solving to the study of abstractions such as groups, rings, and fields. The more basic parts of algebra are called elementary algebra, the more abstract parts are called abstract algebra or modern algebra. Elementary algebra is essential for any study of mathematics, science, or engineering, as well as such applications as medicine and economics. Abstract algebra is a major area in advanced mathematics, studied primarily by professional mathematicians. Much early work in algebra, as the Arabic origin of its name suggests, was done in the Near East, by such mathematicians as Omar Khayyam (1048–1131).^[4][5]

Elementary algebra differs from arithmetic in the use of abstractions, such as using letters to stand for numbers that are either unknown or allowed to take on many values.^[6] For example, in the letter is unknown, but the law of inverses can be used to discover its value: . In , the letters and are variables, and the letter is a constant. Algebra gives methods for solving equations and expressing formulas that are much easier (for those who know how to use them) than the older method of writing everything out in words.

The word algebra is also used in certain specialized ways. A special kind of mathematical object in abstract algebra is called an "algebra", and the word is used, for example, in the phrases linear algebra and algebraic topology (see below).

A mathematician who does research in algebra is called an algebraist.

Etymology

The word algebra comes from the Arabic language (الجبر al-jabr "restoration") from the title of the book Ilm al-jabr wa'l-muḳābala by al-Khwarizmi. The word entered the English language during Late Middle English from either Spanish, Italian, or Medieval Latin. Algebra originally referred to a surgical procedure, and still is used in that sense in Spanish, while the mathematical meaning was a later development.^[7]

Different meanings of "algebra"

The word "algebra" has several related meanings in mathematics, as a single word or with qualifiers.

• As a single word without article, "algebra" names a broad part of mathematics (see below).
• As a single word with article or in plural, "algebra" denotes a specific mathematical structure. See algebra (ring theory) and algebra over a field. More generally, in universal algebra, it can refer to any structure.
• With a qualifier, there is the same distinction:
  • Without article, it means a part of algebra, such as linear algebra, elementary algebra (the symbol-manipulation rules taught in elementary courses of mathematics as part of primary and secondary education), or abstract algebra (the study of the algebraic structures for themselves).
  • With an article, it means an instance of some abstract structure, like a Lie algebra or an associative algebra.
  • Frequently both meanings exist for the same qualifier, as in the sentence: Commutative algebra is the study of commutative rings, which are commutative algebras over the integers.

Algebra as a branch of mathematics

Algebra began with computations similar to those of arithmetic, with letters standing for numbers.^[6] This allowed proofs of properties that are true no matter which numbers are involved. For example, in the quadratic equation

can be any numbers whatsoever (except that cannot be ), and the quadratic formula can be used to quickly and easily find the value of the unknown quantity .

As it developed, algebra was extended to other non-numerical objects, such as vectors, matrices, and polynomials. Then the structural properties of these non-numerical objects were abstracted to define algebraic structures such as groups, rings, and fields.

Before the 16th century, mathematics was divided into only two subfields, arithmetic and geometry. Even though some methods, which had been developed much earlier, may be considered nowadays as algebra, the emergence of algebra and, soon thereafter, of infinitesimal calculus as subfields of mathematics only dates from 16th or 17th century. From the second half of 19th century on, many new fields of mathematics appeared, most of which made use of both arithmetic and geometry, and almost all of which used algebra.

Today, algebra has grown until it includes many branches of mathematics, as can be seen in the Mathematics Subject Classification^[8] where none of the first level areas (two digit entries) is called algebra. Today algebra includes section 08-General algebraic systems, 12-Field theory and polynomials, 13-Commutative algebra, 15-Linear and multilinear algebra; matrix theory, 16-Associative rings and algebras, 17-Nonassociative rings and algebras, 18-Category theory; homological algebra, 19-K-theory and 20-Group theory. Algebra is also used extensively in 11-Number theory and 14-Algebraic geometry.

History

The start of algebra as an area of mathematics may be dated to the end of 16th century, with François Viète's work. Until the 19th century, algebra consisted essentially of the theory of equations. In the following, "Prehistory of algebra" is about the results of the theory of equations that precede the emergence of algebra as an area of mathematics.

Early history of algebra

A page from Al-Khwārizmī's al-Kitāb al-muḫtaṣar fī ḥisāb al-ğabr wa-l-muqābala

The roots of algebra can be traced to the ancient Babylonians,^[9] who developed an advanced arithmetical system with which they were able to do calculations in an algorithmic fashion. The Babylonians developed formulas to calculate solutions for problems typically solved today by using linear equations, quadratic equations, and indeterminate linear equations. By contrast, most Egyptians of this era, as well as Greek and Chinese mathematics in the 1st millennium BC, usually solved such equations by geometric methods, such as those described in the Rhind Mathematical Papyrus, Euclid's Elements, and The Nine Chapters on the Mathematical Art. The geometric work of the Greeks, typified in the Elements, provided the framework for generalizing formulae beyond the solution of particular problems into more general systems of stating and solving equations, although this would not be realized until mathematics developed in medieval Islam.^[10]

By the time of Plato, Greek mathematics had undergone a drastic change. The Greeks created a geometric algebra where terms were represented by sides of geometric objects, usually lines, that had letters associated with them.^[6] Diophantus (3rd century AD) was an Alexandrian Greek mathematician and the author of a series of books called Arithmetica. These texts deal with solving algebraic equations,^[11] and have led, in number theory to the modern notion of Diophantine equation.

Earlier traditions discussed above had a direct influence on Muhammad ibn Mūsā al-Khwārizmī (c. 780–850). He later wrote The Compendious Book on Calculation by Completion and Balancing, which established algebra as a mathematical discipline that is independent of geometry and arithmetic.^[12]

The Hellenistic mathematicians Hero of Alexandria and Diophantus^[13] as well as Indian mathematicians such as Brahmagupta continued the traditions of Egypt and Babylon, though Diophantus' Arithmetica and Brahmagupta's Brahmasphutasiddhanta are on a higher level.^[14] For example, the first complete arithmetic solution (including zero and negative solutions) to quadratic equations was described by Brahmagupta in his book Brahmasphutasiddhanta. Later, Arabic and Muslim mathematicians developed algebraic methods to a much higher degree of sophistication. Although Diophantus and the Babylonians used mostly special ad hoc methods to solve equations, Al-Khwarizmi contribution was fundamental. He solved linear and quadratic equations without algebraic symbolism, negative numbers or zero, thus he has to distinguish several types of equations.^[15]

In the context where algebra is identified with the theory of equations, the Greek mathematician Diophantus has traditionally been known as the "father of algebra" but in more recent times there is much debate over whether al-Khwarizmi, who founded the discipline of al-jabr, deserves that title instead.^[16] Those who support Diophantus point to the fact that the algebra found in Al-Jabr is slightly more elementary than the algebra found in Arithmetica and that Arithmetica is syncopated while Al-Jabr is fully rhetorical.^[17] Those who support Al-Khwarizmi point to the fact that he introduced the methods of "reduction" and "balancing" (the transposition of subtracted terms to the other side of an equation, that is, the cancellation of like terms on opposite sides of the equation) which the term al-jabr originally referred to,^[18] and that he gave an exhaustive explanation of solving quadratic equations,^[19] supported by geometric proofs, while treating algebra as an independent discipline in its own right.^[20] His algebra was also no longer concerned "with a series of problems to be resolved, but an exposition which starts with primitive terms in which the combinations must give all possible prototypes for equations, which henceforward explicitly constitute the true object of study". He also studied an equation for its own sake and "in a generic manner, insofar as it does not simply emerge in the course of solving a problem, but is specifically called on to define an infinite class of problems".^[21]

The Persian mathematician Omar Khayyam is credited with identifying the foundations of algebraic geometry and found the general geometric solution of the cubic equation. Another Persian mathematician, Sharaf al-Dīn al-Tūsī, found algebraic and numerical solutions to various cases of cubic equations.^[22] He also developed the concept of a function.^[23] The Indian mathematicians Mahavira and Bhaskara II, the Persian mathematician Al-Karaji,^[24] and the Chinese mathematician Zhu Shijie, solved various cases of cubic, quartic, quintic and higher-order polynomial equations using numerical methods. In the 13th century, the solution of a cubic equation by Fibonacci is representative of the beginning of a revival in European algebra. As the Islamic world was declining, the European world was ascending. And it is here that algebra was further developed.

History of algebra

Italian mathematician Girolamo Cardano published the solutions to the cubic and quartic equations in his 1545 book Ars magna.

François Viète's work on new algebra at the close of the 16th century was an important step towards modern algebra. In 1637, René Descartes published La Géométrie, inventing analytic geometry and introducing modern algebraic notation. Another key event in the further development of algebra was the general algebraic solution of the cubic and quartic equations, developed in the mid-16th century. The idea of a determinant was developed by Japanese mathematician Kowa Seki in the 17th century, followed independently by Gottfried Leibniz ten years later, for the purpose of solving systems of simultaneous linear equations using matrices. Gabriel Cramer also did some work on matrices and determinants in the 18th century. Permutations were studied by Joseph-Louis Lagrange in his 1770 paper Réflexions sur la résolution algébrique des équations devoted to solutions of algebraic equations, in which he introduced Lagrange resolvents. Paolo Ruffini was the first person to develop the theory of permutation groups, and like his predecessors, also in the context of solving algebraic equations.

Abstract algebra was developed in the 19th century, deriving from the interest in solving equations, initially focusing on what is now called Galois theory, and on constructibility issues.^[25] George Peacock was the founder of axiomatic thinking in arithmetic and algebra. Augustus De Morgan discovered relation algebra in his Syllabus of a Proposed System of Logic. Josiah Willard Gibbs developed an algebra of vectors in three-dimensional space, and Arthur Cayley developed an algebra of matrices (this is a noncommutative algebra).^[26]

Areas of mathematics with the word algebra in their name

Some areas of mathematics that fall under the classification abstract algebra have the word algebra in their name; linear algebra is one example. Others do not: group theory, ring theory, and field theory are examples. In this section, we list some areas of mathematics with the word "algebra" in the name.

• Elementary algebra, the part of algebra that is usually taught in elementary courses of mathematics.
• Abstract algebra, in which algebraic structures such as groups, rings and fields are axiomatically defined and investigated.
• Linear algebra, in which the specific properties of linear equations, vector spaces and matrices are studied.
• Commutative algebra, the study of commutative rings.
• Computer algebra, the implementation of algebraic methods as algorithms and computer programs.
• Homological algebra, the study of algebraic structures that are fundamental to study topological spaces.
• Universal algebra, in which properties common to all algebraic structures are studied.
• Algebraic number theory, in which the properties of numbers are studied from an algebraic point of view.
• Algebraic geometry, a branch of geometry, in its primitive form specifying curves and surfaces as solutions of polynomial equations.
• Algebraic combinatorics, in which algebraic methods are used to study combinatorial questions.

Many mathematical structures are called algebras:

• Algebra over a field or more generally algebra over a ring.
  Many classes of algebras over a field or over a ring have a specific name:
  • Associative algebra
  • Non-associative algebra
  • Lie algebra
  • Hopf algebra
  • C*-algebra
  • Symmetric algebra
  • Exterior algebra
  • Tensor algebra
• In measure theory,
  • Sigma-algebra
  • Algebra over a set
• In category theory
  • F-algebra and F-coalgebra
  • T-algebra
• In logic,
  • Relational algebra: a set of finitary relations that is closed under certain operators.
  • Boolean algebra, a structure abstracting the computation with the truth values false and true. See also Boolean algebra (structure).
  • Heyting algebra

Elementary algebra

Algebraic expression notation:
  1 – power (exponent)
  2 – coefficient
  3 – term
  4 – operator
  5 – constant term
  x y c – variables/constants

Elementary algebra is the most basic form of algebra. It is taught to students who are presumed to have no knowledge of mathematics beyond the basic principles of arithmetic. In arithmetic, only numbers and their arithmetical operations (such as +, −, ×, ÷) occur. In algebra, numbers are often represented by symbols called variables (such as a, n, x, y or z). This is useful because:

• It allows the general formulation of arithmetical laws (such as a + b = b + a for all a and b), and thus is the first step to a systematic exploration of the properties of the real number system.
• It allows the reference to "unknown" numbers, the formulation of equations and the study of how to solve these. (For instance, "Find a number x such that 3x + 1 = 10" or going a bit further "Find a number x such that ax + b = c". This step leads to the conclusion that it is not the nature of the specific numbers that allows us to solve it, but that of the operations involved.)
• It allows the formulation of functional relationships. (For instance, "If you sell x tickets, then your profit will be 3x − 10 dollars, or f(x) = 3x − 10, where f is the function, and x is the number to which the function is applied".)

Polynomials

The graph of a polynomial function of degree 3.

A polynomial is an expression that is the sum of a finite number of non-zero terms, each term consisting of the product of a constant and a finite number of variables raised to whole number powers. For example, x² + 2x − 3 is a polynomial in the single variable x. A polynomial expression is an expression that may be rewritten as a polynomial, by using commutativity, associativity and distributivity of addition and multiplication. For example, (x − 1)(x + 3) is a polynomial expression, that, properly speaking, is not a polynomial. A polynomial function is a function that is defined by a polynomial, or, equivalently, by a polynomial expression. The two preceding examples define the same polynomial function.

Two important and related problems in algebra are the factorization of polynomials, that is, expressing a given polynomial as a product of other polynomials that can not be factored any further, and the computation of polynomial greatest common divisors. The example polynomial above can be factored as (x − 1)(x + 3). A related class of problems is finding algebraic expressions for the roots of a polynomial in a single variable.

Teaching algebra

It has been suggested that elementary algebra should be taught as young as eleven years old,^[27] though in recent years it is more common for public lessons to begin at the eighth grade level (≈ 13 y.o. ±) in the United States.^[28]Since 1997, Virginia Tech and some other universities have begun using a personalized model of teaching algebra that combines instant feedback from specialized computer software with one-on-one and small group tutoring, which has reduced costs and increased student achievement.^[29]

Abstract algebra

Abstract algebra extends the familiar concepts found in elementary algebra and arithmetic of numbers to more general concepts. Here are listed fundamental concepts in abstract algebra.
Sets: Rather than just considering the different types of numbers, abstract algebra deals with the more general concept of sets: a collection of all objects (called elements) selected by property specific for the set. All collections of the familiar types of numbers are sets. Other examples of sets include the set of all two-by-two matrices, the set of all second-degree polynomials (ax² + bx + c), the set of all two dimensional vectors in the plane, and the various finite groups such as the cyclic groups, which are the groups of integers modulo n. Set theory is a branch of logic and not technically a branch of algebra.

Binary operations: The notion of addition (+) is abstracted to give a binary operation, ∗ say. The notion of binary operation is meaningless without the set on which the operation is defined. For two elements a and b in a set S, a ∗ b is another element in the set; this condition is called closure. Addition (+), subtraction (-), multiplication (×), and division (÷) can be binary operations when defined on different sets, as are addition and multiplication of matrices, vectors, and polynomials.

Identity elements: The numbers zero and one are abstracted to give the notion of an identity element for an operation. Zero is the identity element for addition and one is the identity element for multiplication. For a general binary operator ∗ the identity element e must satisfy a ∗ e = a and e ∗ a = a. This holds for addition as a + 0 = a and 0 + a = a and multiplication a × 1 = a and 1 × a = a. Not all sets and operator combinations have an identity element; for example, the set of positive natural numbers (1, 2, 3, ...) has no identity element for addition.

Inverse elements: The negative numbers give rise to the concept of inverse elements. For addition, the inverse of a is written −a, and for multiplication the inverse is written a⁻¹. A general two-sided inverse element a⁻¹ satisfies the property that a ∗ a⁻¹ = 1 and a⁻¹ ∗ a = 1 .

Associativity: Addition of integers has a property called associativity. That is, the grouping of the numbers to be added does not affect the sum. For example: (2 + 3) + 4 = 2 + (3 + 4). In general, this becomes (a ∗ b) ∗ c = a ∗ (b ∗ c). This property is shared by most binary operations, but not subtraction or division or octonion multiplication.

Commutativity: Addition and multiplication of real numbers are both commutative. That is, the order of the numbers does not affect the result. For example: 2 + 3 = 3 + 2. In general, this becomes a ∗ b = b ∗ a. This property does not hold for all binary operations. For example, matrix multiplication and quaternion multiplication are both non-commutative.

Groups

Combining the above concepts gives one of the most important structures in mathematics: a group. A group is a combination of a set S and a single binary operation ∗, defined in any way you choose, but with the following properties:

• An identity element e exists, such that for every member a of S, e ∗ a and a ∗ e are both identical to a.
• Every element has an inverse: for every member a of S, there exists a member a⁻¹ such that a ∗ a⁻¹ and a⁻¹ ∗ a are both identical to the identity element.
• The operation is associative: if a, b and c are members of S, then (a ∗ b) ∗ c is identical to a ∗ (b ∗ c).

If a group is also commutative—that is, for any two members a and b of S, a ∗ b is identical to b ∗ a—then the group is said to be abelian.

For example, the set of integers under the operation of addition is a group. In this group, the identity element is 0 and the inverse of any element a is its negation, −a. The associativity requirement is met, because for any integers a, b and c, (a + b) + c = a + (b + c)

The nonzero rational numbers form a group under multiplication. Here, the identity element is 1, since 1 × a = a × 1 = a for any rational number a. The inverse of a is 1/a, since a × 1/a = 1.

The integers under the multiplication operation, however, do not form a group. This is because, in general, the multiplicative inverse of an integer is not an integer. For example, 4 is an integer, but its multiplicative inverse is ¼, which is not an integer.

The theory of groups is studied in group theory. A major result in this theory is the classification of finite simple groups, mostly published between about 1955 and 1983, which separates the finite simple groups into roughly 30 basic types.

Semigroups, quasigroups, and monoids are structures similar to groups, but more general. They comprise a set and a closed binary operation, but do not necessarily satisfy the other conditions. A semigroup has an associative binary operation, but might not have an identity element. A monoid is a semigroup which does have an identity but might not have an inverse for every element. A quasigroup satisfies a requirement that any element can be turned into any other by either a unique left-multiplication or right-multiplication; however the binary operation might not be associative.

All groups are monoids, and all monoids are semigroups.

Examples
Set:	Natural numbers N		Integers Z		Rational numbers Q (also real R and complex C numbers)				Integers modulo 3: Z3 = {0, 1, 2}
Operation	+	× (w/o zero)	+	× (w/o zero)	+	−	× (w/o zero)	÷ (w/o zero)	+	× (w/o zero)
Closed	Yes	Yes	Yes	Yes	Yes	Yes	Yes	Yes	Yes	Yes
Identity	0	1	0	1	0	N/A	1	N/A	0	1
Inverse	N/A	N/A	−a	N/A	−a	N/A	1/a	N/A	0, 2, 1, respectively	N/A, 1, 2, respectively
Associative	Yes	Yes	Yes	Yes	Yes	No	Yes	No	Yes	Yes
Commutative	Yes	Yes	Yes	Yes	Yes	No	Yes	No	Yes	Yes
Structure	monoid	monoid	abelian group	monoid	abelian group	quasigroup	abelian group	quasigroup	abelian group	abelian group (Z2)

Rings and fields

Groups just have one binary operation. To fully explain the behaviour of the different types of numbers, structures with two operators need to be studied. The most important of these are rings, and fields.
A ring has two binary operations (+) and (×), with × distributive over +. Under the first operator (+) it forms an abelian group. Under the second operator (×) it is associative, but it does not need to have identity, or inverse, so division is not required. The additive (+) identity element is written as 0 and the additive inverse of a is written as −a.

Distributivity generalises the distributive law for numbers. For the integers (a + b) × c = a × c + b × c and c × (a + b) = c × a + c × b, and × is said to be distributive over +.

The integers are an example of a ring. The integers have additional properties which make it an integral domain.

A field is a ring with the additional property that all the elements excluding 0 form an abelian group under ×. The multiplicative (×) identity is written as 1 and the multiplicative inverse of a is written as a⁻¹.

The rational numbers, the real numbers and the complex numbers are all examples of fields.

The Quest for Everlasting Agriculture

By Brooke Borel on Wed, 05 Nov 2014

Original link:   http://www.pbs.org/wgbh/nova/next/nature/perennial-agriculture/

It’s a cycle nearly as old as human history. Plow, plant, harvest, and repeat. It worked for our ancestors, and it’s working for us now, though with ever more problems, from obliterating soil nutrients to encouraging erosion. And things may get worse in the future, too, when climate change threatens—whether through drowning or drought—to topple our food production system at the moment we’ll need it most: In less than 90 years, the world’s population could crest between 9 and 12 billion, and that will test the limits of farming.

“Soil quality around the world has become degraded,” says Sieglinde Snapp, an agroecologist at Michigan State University. “So how are we going to feed more people with higher quality food? How will we provide more protein? And the big question is: how are we going to feed 9 billion in a sustainable way with degraded soil?”

Modifying crops like wheat could help boost yields by extending the

growing season.

There are myriad possibilities. Among the options is raising the output of current farming techniques using genetic modification, specialized fungi, or precision agriculture. But another ambitious idea is to extend the growing season, which will involve rewriting much of the book of agriculture. In other words, if we were to redo the agricultural revolution today, what would it look like?

For untold generations, farmers have plowed their fields and planted their crops in the spring, harvested them in the fall, and done it all over again the next year. But in the last several decades, agricultural experts have been experimenting with eliminating the spring planting by developing perennial crops, essentially revising thousands of years of selective breeding.

To see how perennials could help, just visit a farm in the Midwest in the dead of winter. You’ll likely find fallow fields scattered with dead plants. Some of them may be covered in snow. But under the surface, the frozen soil has locked in key nutrients and water. When the spring thaw begins—but before it’s dry and pliable enough for planting the next season’s crops—the warming fields will begin to lose some of their moisture and nutrients, which end up draining into ditches, rivers, and streams or seeping into the atmosphere. From a farmer’s perspective, the combination of these soil changes, the longer days, and the springtime rains add up to a lost opportunity. If only they could start growing sooner.

Which is why a relatively small group of scientists are developing year-round cereals and oilseeds, both key ingredients of the modern human diet. Most of these grains today are annuals, which complete a lifecycle once every year and must be replanted the next growing season. Depending on the farming method, the cycle may include tilling, sowing, and harvesting, which, when done regularly on the same plot of land, leeches nutrients from the soil and contributes to erosion. This system also requires more energy-intensive machines and materials, from fossil-fuel-burning farming equipment to synthetic fertilizers that push nitrogen back into over-taxed soil.

 

“The way we currently grow grains is very similar to how we started growing grains a long, long time ago”

Perennial crops, on the other hand, could survive for many seasons, axing the annual cycle and lessening farming’s wear-and-tear on the environment. Some varieties could also have longer, lusher root systems that would sink deeper into the ground, helping maintain soil health and curbing erosion. They could even help the plants survive a drought.

Such a system would allow for longer growing seasons, too, taking advantage of the late autumn and early spring months when fields usually lay bare. Assuming that perennial crops produced the same amount as their annual counterparts—a big assumption—this would provide additional food each year from the same plot of land. A shift from annuals to perennials, or a mixture of both, could benefit both the environment and food security.

“The way in which we currently grow grains is very similar to how we started growing grains a long, long time ago, and the ecosystem of agriculture has not changed much over that period of time,” says Timothy Crews, an ecologist and the research director at the Land Institute in Salina, Kansas, where scientists have been studying perennial crops since the mid-1970s and actively breeding them since 2000. “We need to supplant purchased, high-energy inputs and mechanization inputs with ecological processes that achieve comparable or superior outcomes, which could build slow organic matter in cropping systems instead of maintaining or depleting it, which is what current agriculture does.”

The trick, however, will be coaxing crops into simultaneously surviving year-round and growing plump and harvestable seeds. Plants, as we’ve discovered over the millennia, tend to prefer one or the other, not both. Though thanks to the work of Crews and a handful of enterprising scientists, that may be changing.

Perennial Advances

Agronomists and botanists have been trying to create perennial crops since at least the 1920s, when Russian scientists started a program to breed perennial wheat. But over the past ten to 15 years or so, the field has grown significantly, says Lee DeHaan, a plant geneticist at the Land Institute. This is partly because more research groups have taken up the idea and those labs have had a chance to mature. In October, one of the first dedicated scientific meetings on the topic attracted around 50 such researchers to Estes Park, Colorado to discuss breeding and management strategies.

For perennial crop researchers, advances in genetics have given them an unprecedented level of understanding and control over their subjects. “If you consider the early Russian work, they were working very blindly,” DeHaan says. “They could make crosses and observe the plants, but they had no way to know the genes or chromosomes involved.”

Today, there are two main approaches to breeding perennial crops, both of which require genetic tinkering. Both, too, are numbers games. The first is domestication, where plant breeders try to tame a wild perennial plant. This requires planting and observing thousands or tens of thousands of individual plants and then selecting those with the most promising characteristics—large seeds that hang onto the plant long enough for a harvest, for example, or ears that contain many seeds. The next step is to crossbreed these winners in an attempt to capture their positive traits in the next generation. Software that tracks which individual plants possess which traits—a tool that was unavailable to the Russian scientists in the 1920s—helps guide decisions on which offspring make it to the next round.

Shuwen Wang, a perennial wheat breeder at the Land Institute, inspects

a crossbred plant.

Still, domestication is numbingly slow and difficult work. Wild perennials tend to drop their ripe seeds earlier than tame ones, a trait plant scientists call “shattering.” It’s advantageous for wild varieties to shatter because it allows their seeds to germinate when they’re ripest, but it’s useless to a farmer who wants those ripe seeds to stay on the plant until harvest. And while many wild perennials can easily survive multiple seasons, Crews says, it has been difficult to increase their yields and grain sizes to anywhere near those of annual crops.

The second approach, and the more common one, is hybridization. Here, an annual crop is crossbred with a wild perennial counterpart in hopes that they will eventually produce a perennial crop. Hybridization offers a shortcut: annual varieties already contain the genetic recipe for high yields and big, harvestable seeds, and the wild perennials host the genetic code for longevity. Researchers can quickly identify genetic information that is linked to specific physical characteristics by using known stretches of genetic code called DNA markers. By snipping some tissue from a plant and extracting its DNA, breeders can see which genetic variations it inherited from its parents rather than waiting for the plant to grow and observing its traits, accelerating the breeding process.

Unfortunately, hybrids are often sterile, and even if they do produce offspring, they don’t always pass on the desired traits. While a few offspring will be fertile, they can also be fragile. Sometimes hybrid embryos must be coddled in a lab in a process called “embryo rescue,” which involves growing them in special nutrients to ensure their growth. Even the most successful hybrids may then need to be cross-bred to bring them up to par.

A wheat embryo, sitting on the tip of the scalpel, is rescued from an

immature hybrid seed.

Researchers across the country are trying both approaches and testing their results in the field. The Land Institute, for example, is domesticating wild sunflowers and wheatgrass—a variety they’ve named “kernza”—and their scientists are also developing a hybrid perennial wheat. Snapp, the Michigan State agroecologist, and her team conduct field research on perennial wheat in Michigan, as well as on the naturally-occurring perennial pigeon pea, a legume and a source of high protein, in Tanzania and Malawi. Still others across the world are working on rice, sorghum, corn, and mustard plants that don’t have to be planted every year.

Andrew Paterson, a plant geneticist at the University of Georgia, is among the researchers working on hybrid sorghum perennials, including a project to cross an annual domesticated variety with a wild weedy sorghum called Johnson grass, which produces an extensive underground stem system called a rhizome. When the stems of a plant with rhizomes are cut, more grow in their place. Paterson points out that there are around nine or ten genetic variations known to be responsible for perennial characteristics like this in wild sorghum, and some of the same genes are also found in rice. That these two grains diverged from a common ancestor around 50 million years ago yet still retain such similar perennial genetic traits suggests that a wide range of crops which have distant common ancestors may also share the same features.

“It looks like genetic control of perenniality is pretty similar in very different grain crops,” Paterson says. “So as we learn more about one, we learn more about all of them.”

New Plantings

Should researchers successfully develop perennial crops, it’ll be up to farmers to put them to work. Currently, philosophies differ on how crops should be planted in the future perennial landscape. The Land Institute envisions farmers sowing prairie-like fields with a mixture of perennials. And while this approach may help ward off pest insects and weeds, which more easily infiltrate a conventional field of a single crop, it complicates matters come harvest time. Currently, when a farmer cuts wheat, he knows he’s not going to be accidentally including corn in the harvest because each field grows a separate crop. Crews doesn’t think this will pose a significant challenge since the machinery to sort different seeds already exists, though it would likely need to be modified for this scenario.
Paterson and others take the opposite view, suggesting that future perennial crops will simply be plugged into the current monoculture system, which would be easier to harvest with existing technology.

Still others suggest a mosaic approach, which would include both monocultures and mixed fields as well as a combination of annuals and perennials. The idea is, in part, a practical one since large-scale monocultures already exist and probably aren’t going anywhere. “There’s corn and soybeans out there on millions of acres of Midwestern landscape. They’re not going away,” says Donald Wyse, an agroecologist at the University of Minnesota. Wyse oversees several projects that aim for year-round field coverage, including both perennials and annual winter crops, such as hazel nuts that could be swapped out with an annual summer crop. “It’s going to come down to what we can put in mixtures or in perennial monocultures,” he adds.

Workers harvest crossbred wheat for analysis.

To a degree, mosaics already exist, it just depends on the scale at which you look for them. Look out of the window the next time you’re in or flying over the Midwest. There, the farmland is a patchwork quilt interrupted by occasional stands of trees and shrubs. A mosaic that includes perennials could break farmland up even more, with smaller patches containing a wider variety of plants. These perennials might be grown on parts of a farmer’s field that are usually left unplanted, where they would provide both environmental and economic benefits. For example, Wyse says, some perennials could be not only food crops, but also oilseeds for making biofuels, forage for animal feed, and raw materials for other commercial products like cosmetics. If these perennials were planted around the edges of larger single-crop field, they could recreate, on a small scale, the region’s once extensive prairies while also giving farmers something to sell.

“Deliberately planned landscapes like mosaics are the future, and perennial grains offers an option for mosaics that we don’t have now,” Snapp says. “Not all farmers can afford to have strips of prairie in their fields to provide sustainable grasses. It’s better to have practical options that also provide something they can sell or eat.”

Making Space

Regardless of how perennial fields will look—whether they’re blankets of monoculture, edible prairies, or a patchwork of both—we’ll still have to determine how they fit into our current food system. After all, we’ve been cooking and baking with many of the same grains for generations.

In some places, we can glimpse this future. Kernza, the wheatgrass from the Land Institute, is already on limited menus. It’s in the pancake mix at Birchwood Café in Minneapolis, for example, which has been cooking around 50 pounds of the grain each season over the past two years. The Free State Brewery in Lawrence, Kansas made a pilot batch of a kernza saison beer several years ago, and WheatFields, a nearby bakery, has been experimenting with kernza bread for the past five or so years.

Wheatgrass heads are placed in paper bags during breeding to ensure

pollen is only transferred between selected plants.

The grain’s largest commercial debut is forthcoming from Patagonia Provisions, a cousin of the clothing company, which is rolling out a line of environmentally-conscious foods. According to director Birgit Cameron, the company is experimenting with kernza both as a whole grain and ground into flour, and they plan to have a product out within the next two years.

So how does it taste? Most people who have cooked or brewed with it want to try more, as long as they can get their hands on supplies. (The grain has a much lower yield than annual crops.) “The kernza is popular—it’s very nutritious, it cooks well, and it has good protein content,” says Marshall Paulsen, the head chef at Birchwood Café. “I hope to keep cooking with it.”

To move the crop, and others like it, from farms to more restaurants, bakeries, and breweries, however, will require a herculean shift not only in plant genetics and farming practices, but also in how grains are bought and sold. Our most common grains and oilseeds—wheat, corn, soybeans, oats, rice, and canola—are traded on the global commodities market, where prices fluctuate based on activity on various futures exchanges. Bringing a new crop into that system is virtually impossible because there is literally no button or bin for it in the grain elevator, says plant geneticist Stephen Jones, who heads, among other things, research on perennials at Washington State University.

To explore another model, Jones opened the Bread Lab at WSU around three years ago to facilitate smaller local markets that will support new perennials and other crops that don’t fit into the current system. The lab has a resident baker as well as visiting chefs, brewers, and millers to experiment with new crop varieties. It also helps pair up different businesses to speed along commercialization.

The Bread Lab model allows for more leeway in the distribution and use of perennial crops. The lab is growing, for example, a perennial wheat that has a blue-green tint inherited from its wild relatives. “It’s really pretty, but there’s no commodity stream for that,” says Colin Curwen-McAdam, a graduate student who works on plant breeding and genetics in Jones’s department. “If you’re working in the traditional commodity sense trying to make these commodity grains, now we have to put them in the commodity box, which makes your job even harder.”

In a way, the Bread Lab is a microcosm of the perennial crop world. Everything there is up for grabs, from commodities markets to the crops themselves. “It’s a brand new crop type,” Curwen-McAdam says, “and it’s completely unwritten as to what that can be.”

Geometry

From Wikipedia, the free encyclopedia

An illustration of Desargues' theorem, an important result in Euclidean and projective geometry

Geometry (from the Ancient Greek: γεωμετρία; geo- "earth", -metron "measurement") is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. A mathematician who works in the field of geometry is called a geometer. Geometry arose independently in a number of early cultures as a body of practical knowledge concerning lengths, areas, and volumes, with elements of formal mathematical science emerging in the West as early as Thales (6th century BC). By the 3rd century BC, geometry was put into an axiomatic form by Euclid, whose treatment—Euclidean geometry—set a standard for many centuries to follow.^[1] Archimedes developed ingenious techniques for calculating areas and volumes, in many ways anticipating modern integral calculus. The field of astronomy, especially as it relates to mapping the positions of stars and planets on the celestial sphere and describing the relationship between movements of celestial bodies, served as an important source of geometric problems during the next one and a half millennia. In the classical world, both geometry and astronomy were considered to be part of the Quadrivium, a subset of the seven liberal arts considered essential for a free citizen to master.

The introduction of coordinates by René Descartes and the concurrent developments of algebra marked a new stage for geometry, since geometric figures such as plane curves could now be represented analytically in the form of functions and equations. This played a key role in the emergence of infinitesimal calculus in the 17th century. Furthermore, the theory of perspective showed that there is more to geometry than just the metric properties of figures: perspective is the origin of projective geometry. The subject of geometry was further enriched by the study of the intrinsic structure of geometric objects that originated with Euler and Gauss and led to the creation of topology and differential geometry.

In Euclid's time, there was no clear distinction between physical and geometrical space. Since the 19th-century discovery of non-Euclidean geometry, the concept of space has undergone a radical transformation and raised the question of which geometrical space best fits physical space. With the rise of formal mathematics in the 20th century, 'space' (whether 'point', 'line', or 'plane') lost its intuitive contents, so today one has to distinguish between physical space, geometrical spaces (in which 'space', 'point' etc. still have their intuitive meanings) and abstract spaces. Contemporary geometry considers manifolds, spaces that are considerably more abstract than the familiar Euclidean space, which they only approximately resemble at small scales. These spaces may be endowed with additional structure which allow one to speak about length. Modern geometry has many ties to physics as is exemplified by the links between pseudo-Riemannian geometry and general relativity. One of the youngest physical theories, string theory, is also very geometric in flavour.

While the visual nature of geometry makes it initially more accessible than other mathematical areas such as algebra or number theory, geometric language is also used in contexts far removed from its traditional, Euclidean provenance (for example, in fractal geometry and algebraic geometry).^[2]

Overview

Visual checking of the Pythagorean theorem for the (3, 4, 5) triangle as in the Chou Pei Suan Ching 500–200 BC.

Because the recorded development of geometry spans more than two millennia, it is hardly surprising that perceptions of what constitutes geometry have evolved throughout the ages:

Practical geometry

Geometry originated as a practical science concerned with surveys, measurements, areas, and volumes. Among other highlights, notable accomplishments include formulas for lengths, areas and volumes, such as the Pythagorean theorem, circumference and area of a circle, area of a triangle, volume of a cylinder, sphere, and a pyramid. A method of computing certain inaccessible distances or heights based on similarity of geometric figures is attributed to Thales. The development of astronomy led to the emergence of trigonometry and spherical trigonometry, together with the attendant computational techniques.

Axiomatic geometry

An illustration of Euclid's parallel postulate

Euclid took a more abstract approach in his Elements, one of the most influential books ever written. Euclid introduced certain axioms, or postulates, expressing primary or self-evident properties of points, lines, and planes. He proceeded to rigorously deduce other properties by mathematical reasoning. The characteristic feature of Euclid's approach to geometry was its rigor, and it has come to be known as axiomatic or synthetic geometry. At the start of the 19th century, the discovery of non-Euclidean geometries by Nikolai Ivanovich Lobachevsky (1792–1856), János Bolyai (1802–1860) and Carl Friedrich Gauss (1777–1855) and others led to a revival of interest in this discipline, and in the 20th century, David Hilbert (1862–1943) employed axiomatic reasoning in an attempt to provide a modern foundation of geometry.

Geometry lessons in the 20th century

Geometric constructions

Classical geometers paid special attention to constructing geometric objects that had been described in some other way. Classically, the only instruments allowed in geometric constructions are the compass and straightedge. Also, every construction had to be complete in a finite number of steps. However, some problems turned out to be difficult or impossible to solve by these means alone, and ingenious constructions using parabolas and other curves, as well as mechanical devices, were found.

Numbers in geometry

The Pythagoreans discovered that the sides of a triangle could have incommensurable lengths.

In ancient Greece the Pythagoreans considered the role of numbers in geometry. However, the discovery of incommensurable lengths, which contradicted their philosophical views, made them abandon abstract numbers in favor of concrete geometric quantities, such as length and area of figures. Numbers were reintroduced into geometry in the form of coordinates by Descartes, who realized that the study of geometric shapes can be facilitated by their algebraic representation, and for whom the Cartesian plane is named. Analytic geometry applies methods of algebra to geometric questions, typically by relating geometric curves to algebraic equations. These ideas played a key role in the development of calculus in the 17th century and led to the discovery of many new properties of plane curves. Modern algebraic geometry considers similar questions on a vastly more abstract level.

Geometry of position

Even in ancient times, geometers considered questions of relative position or spatial relationship of geometric figures and shapes. Some examples are given by inscribed and circumscribed circles of polygons, lines intersecting and tangent to conic sections, the Pappus and Menelaus configurations of points and lines. In the Middle Ages, new and more complicated questions of this type were considered: What is the maximum number of spheres simultaneously touching a given sphere of the same radius (kissing number problem)? What is the densest packing of spheres of equal size in space (Kepler conjecture)? Most of these questions involved 'rigid' geometrical shapes, such as lines or spheres. Projective, convex, and discrete geometry are three sub-disciplines within present day geometry that deal with these types of questions.
Leonhard Euler, in studying problems like the Seven Bridges of Königsberg, considered the most fundamental properties of geometric figures based solely on shape, independent of their metric properties. Euler called this new branch of geometry geometria situs (geometry of place), but it is now known as topology. Topology grew out of geometry, but turned into a large independent discipline. It does not differentiate between objects that can be continuously deformed into each other. The objects may nevertheless retain some geometry, as in the case of hyperbolic knots.

Geometry beyond Euclid

Differential geometry uses tools from calculus to study problems involving curvature.

In the nearly two thousand years since Euclid, while the range of geometrical questions asked and answered inevitably expanded, the basic understanding of space remained essentially the same. Immanuel Kant argued that there is only one, absolute, geometry, which is known to be true a priori by an inner faculty of mind: Euclidean geometry was synthetic a priori.^[3] This dominant view was overturned by the revolutionary discovery of non-Euclidean geometry in the works of Bolyai, Lobachevsky, and Gauss (who never published his theory). They demonstrated that ordinary Euclidean space is only one possibility for development of geometry. A broad vision of the subject of geometry was then expressed by Riemann in his 1867 inauguration lecture Über die Hypothesen, welche der Geometrie zu Grunde liegen (On the hypotheses on which geometry is based),^[4] published only after his death. Riemann's new idea of space proved crucial in Einstein's general relativity theory, and Riemannian geometry, that considers very general spaces in which the notion of length is defined, is a mainstay of modern geometry.

Dimension

The Koch snowflake, with fractal dimension=log4/log3 and topological dimension=1

Where the traditional geometry allowed dimensions 1 (a line), 2 (a plane) and 3 (our ambient world conceived of as three-dimensional space), mathematicians have used higher dimensions for nearly two centuries. Dimension has gone through stages of being any natural number n, possibly infinite with the introduction of Hilbert space, and any positive real number in fractal geometry. Dimension theory is a technical area, initially within general topology, that discusses definitions; in common with most mathematical ideas, dimension is now defined rather than an intuition. Connected topological manifolds have a well-defined dimension; this is a theorem (invariance of domain) rather than anything a priori.

The issue of dimension still matters to geometry, in the absence of complete answers to classic questions. Dimensions 3 of space and 4 of space-time are special cases in geometric topology. Dimension 10 or 11 is a key number in string theory. Research may bring a satisfactory geometric reason for the significance of 10 and 11 dimensions.

Symmetry

A tiling of the hyperbolic plane

The theme of symmetry in geometry is nearly as old as the science of geometry itself. Symmetric shapes such as the circle, regular polygons and platonic solids held deep significance for many ancient philosophers and were investigated in detail before the time of Euclid. Symmetric patterns occur in nature and were artistically rendered in a multitude of forms, including the graphics of M. C. Escher. Nonetheless, it was not until the second half of 19th century that the unifying role of symmetry in foundations of geometry was recognized. Felix Klein's Erlangen program proclaimed that, in a very precise sense, symmetry, expressed via the notion of a transformation group, determines what geometry is. Symmetry in classical Euclidean geometry is represented by congruences and rigid motions, whereas in projective geometry an analogous role is played by collineations, geometric transformations that take straight lines into straight lines. However it was in the new geometries of Bolyai and Lobachevsky, Riemann, Clifford and Klein, and Sophus Lie that Klein's idea to 'define a geometry via its symmetry group' proved most influential. Both discrete and continuous symmetries play prominent roles in geometry, the former in topology and geometric group theory, the latter in Lie theory and Riemannian geometry.

A different type of symmetry is the principle of duality in projective geometry (see Duality (projective geometry)) among other fields. This meta-phenomenon can roughly be described as follows: in any theorem, exchange point with plane, join with meet, lies in with contains, and you will get an equally true theorem. A similar and closely related form of duality exists between a vector space and its dual space.

History of geometry

A European and an Arab practicing geometry in the 15th century.

The earliest recorded beginnings of geometry can be traced to ancient Mesopotamia and Egypt in the 2nd millennium BC.^[5][6] Early geometry was a collection of empirically discovered principles concerning lengths, angles, areas, and volumes, which were developed to meet some practical need in surveying, construction, astronomy, and various crafts. The earliest known texts on geometry are the Egyptian Rhind Papyrus (2000–1800 BC) and Moscow Papyrus (c. 1890 BC), the Babylonian clay tablets such as Plimpton 322 (1900 BC). For example, the Moscow Papyrus gives a formula for calculating the volume of a truncated pyramid, or frustum.^[7] South of Egypt the ancient Nubians established a system of geometry including early versions of sun clocks.^[8][9]

In the 7th century BC, the Greek mathematician Thales of Miletus used geometry to solve problems such as calculating the height of pyramids and the distance of ships from the shore. He is credited with the first use of deductive reasoning applied to geometry, by deriving four corollaries to Thales' Theorem.^[10] Pythagoras established the Pythagorean School, which is credited with the first proof of the Pythagorean theorem,^[11] though the statement of the theorem has a long history^[12][13] Eudoxus (408–c. 355 BC) developed the method of exhaustion, which allowed the calculation of areas and volumes of curvilinear figures,^[14] as well as a theory of ratios that avoided the problem of incommensurable magnitudes, which enabled subsequent geometers to make significant advances. Around 300 BC, geometry was revolutionized by Euclid, whose Elements, widely considered the most successful and influential textbook of all time,^[15] introduced mathematical rigor through the axiomatic method and is the earliest example of the format still used in mathematics today, that of definition, axiom, theorem, and proof.
Although most of the contents of the Elements were already known, Euclid arranged them into a single, coherent logical framework.^[16] The Elements was known to all educated people in the West until the middle of the 20th century and its contents are still taught in geometry classes today.^[17] Archimedes (c. 287–212 BC) of Syracuse used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, and gave remarkably accurate approximations of Pi.^[18] He also studied the spiral bearing his name and obtained formulas for the volumes of surfaces of revolution.

Woman teaching geometry. Illustration at the beginning of a medieval translation of Euclid's Elements, (c. 1310)

Indian mathematicians also made many important contributions in geometry. The Satapatha Brahmana (ninth century BC) contains rules for ritual geometric constructions that are similar to the Sulba Sutras.^[19] According to (Hayashi 2005, p. 363), the Śulba Sūtras contain "the earliest extant verbal expression of the Pythagorean Theorem in the world, although it had already been known to the Old Babylonians. They contain lists of Pythagorean triples,^[20] which are particular cases of Diophantine equations.^[21] In the Bakhshali manuscript, there is a handful of geometric problems (including problems about volumes of irregular solids). The Bakhshali manuscript also "employs a decimal place value system with a dot for zero."^[22] Aryabhata's Aryabhatiya (499) includes the computation of areas and volumes. Brahmagupta wrote his astronomical work Brāhma Sphuṭa Siddhānta in 628. Chapter 12, containing 66 Sanskrit verses, was divided into two sections: "basic operations" (including cube roots, fractions, ratio and proportion, and barter) and "practical mathematics" (including mixture, mathematical series, plane figures, stacking bricks, sawing of timber, and piling of grain).^[23] In the latter section, he stated his famous theorem on the diagonals of a cyclic quadrilateral. Chapter 12 also included a formula for the area of a cyclic quadrilateral (a generalization of Heron's formula), as well as a complete description of rational triangles (i.e. triangles with rational sides and rational areas).^[23]

In the Middle Ages, mathematics in medieval Islam contributed to the development of geometry, especially algebraic geometry^[24] and geometric algebra.^[25] Al-Mahani (b. 853) conceived the idea of reducing geometrical problems such as duplicating the cube to problems in algebra.^[26] Thābit ibn Qurra (known as Thebit in Latin) (836–901) dealt with arithmetic operations applied to ratios of geometrical quantities, and contributed to the development of analytic geometry.^[27] Omar Khayyám (1048–1131) found geometric solutions to cubic equations.^[28] The theorems of Ibn al-Haytham (Alhazen), Omar Khayyam and Nasir al-Din al-Tusi on quadrilaterals, including the Lambert quadrilateral and Saccheri quadrilateral, were early results in hyperbolic geometry, and along with their alternative postulates, such as Playfair's axiom, these works had a considerable influence on the development of non-Euclidean geometry among later European geometers, including Witelo (c. 1230–c. 1314), Gersonides (1288–1344), Alfonso, John Wallis, and Giovanni Girolamo Saccheri.^[29]

In the early 17th century, there were two important developments in geometry. The first was the creation of analytic geometry, or geometry with coordinates and equations, by René Descartes (1596–1650) and Pierre de Fermat (1601–1665). This was a necessary precursor to the development of calculus and a precise quantitative science of physics. The second geometric development of this period was the systematic study of projective geometry by Girard Desargues (1591–1661). Projective geometry is a geometry without measurement or parallel lines, just the study of how points are related to each other.

Two developments in geometry in the 19th century changed the way it had been studied previously. These were the discovery of non-Euclidean geometries by Nikolai Ivanovich Lobachevsky, János Bolyai and Carl Friedrich Gauss and of the formulation of symmetry as the central consideration in the Erlangen Programme of Felix Klein (which generalized the Euclidean and non-Euclidean geometries). Two of the master geometers of the time were Bernhard Riemann (1826–1866), working primarily with tools from mathematical analysis, and introducing the Riemann surface, and Henri Poincaré, the founder of algebraic topology and the geometric theory of dynamical systems. As a consequence of these major changes in the conception of geometry, the concept of "space" became something rich and varied, and the natural background for theories as different as complex analysis and classical mechanics.

Contemporary geometry

Euclidean geometry

The 4₂₁polytope, orthogonally projected into the E₈ Lie group Coxeter plane

Euclidean geometry has become closely connected with computational geometry, computer graphics, convex geometry, incidence geometry, finite geometry, discrete geometry, and some areas of combinatorics. Attention was given to further work on Euclidean geometry and the Euclidean groups by crystallography and the work of H. S. M. Coxeter, and can be seen in theories of Coxeter groups and polytopes. Geometric group theory is an expanding area of the theory of more general discrete groups, drawing on geometric models and algebraic techniques.

Differential geometry

Differential geometry has been of increasing importance to mathematical physics due to Einstein's general relativity postulation that the universe is curved. Contemporary differential geometry is intrinsic, meaning that the spaces it considers are smooth manifolds whose geometric structure is governed by a Riemannian metric, which determines how distances are measured near each point, and not a priori parts of some ambient flat Euclidean space.

Topology and geometry

A thickening of the trefoil knot

The field of topology, which saw massive development in the 20th century, is in a technical sense a type of transformation geometry, in which transformations are homeomorphisms. This has often been expressed in the form of the dictum 'topology is rubber-sheet geometry'. Contemporary geometric topology and differential topology, and particular subfields such as Morse theory, would be counted by most mathematicians as part of geometry. Algebraic topology and general topology have gone their own ways.

Algebraic geometry

Quintic Calabi–Yau threefold

The field of algebraic geometry is the modern incarnation of the Cartesian geometry of co-ordinates. From late 1950s through mid-1970s it had undergone major foundational development, largely due to work of Jean-Pierre Serre and Alexander Grothendieck. This led to the introduction of schemes and greater emphasis on topological methods, including various cohomology theories. One of seven Millennium Prize problems, the Hodge conjecture, is a question in algebraic geometry.

The study of low-dimensional algebraic varieties, algebraic curves, algebraic surfaces and algebraic varieties of dimension 3 ("algebraic threefolds"), has been far advanced. Gröbner basis theory and real algebraic geometry are among more applied subfields of modern algebraic geometry. Arithmetic geometry is an active field combining algebraic geometry and number theory. Other directions of research involve moduli spaces and complex geometry. Algebro-geometric methods are commonly applied in string and brane theory.