Structure is a fundamental, tangible or intangible notion referring to the recognition, observation, nature, and permanence of patterns and relationships of entities. This notion may itself be an object, such as a built structure, or an attribute, such as the structure of society. From a child's verbal description of a snowflake, to the detailed scientific analysis of the properties of magnetic fields, the concept of structure is now often an essential foundation of nearly every mode of inquiry and discovery in science, philosophy, and art.^[1] In early 20th-century and earlier thought, form^{[disambiguation needed ]} often plays a role comparable to that of structure in contemporary thought. The neo-Kantianism of Ernst Cassirer (cf. his Philosophy of Symbolic Forms, completed in 1929 and published in English translation in the 1950s) is sometimes regarded as a precursor of the later shift to structuralism and poststructuralism.^[2]

The description of structure implicitly offers an account of what a system is made of: a configuration of items, a collection of inter-related components or services. A structure may be a hierarchy (a cascade of one-to-many relationships), a network featuring many-to-many links, or a lattice featuring connections between components that are neighbors in space.

Types of structure[link]

Physical structure[link]

In engineering and architecture, a structure is a body or assemblage of bodies in space to form a system capable of supporting loads. Physical structures include man-made and natural arrangements. Buildings, aircraft, soap films, skeletons, anthills, beaver dams and salt domes are all examples of physical structures. The effects of loads on physical structures are determined through structural analysis. Structural engineering refers to engineering of physical structures.

Built structures are a subset of physical structures resulting from construction. These are divided into buildings and nonbuilding structures, and make up the infrastructure of a human society. Built structures are composed of structural elements such as columns, beams and trusses. Built structures are broadly divided by their varying design approaches and standards, into categories including Building structures, Architectural structures, Civil engineering structures and Mechanical structures.

Biological structure[link]

In biology, structures exist at all levels of organization, ranging hierarchically from the atomic and molecular to the cellular, tissue, organ, organismic, population and ecosystem level. Usually, a higher-level structure is composed of multiple copies of a lower-level structure.

Chemical structure[link]

Chemistry is the science treating matter at the atomic to macromolecular scale, the reactions, transformations and aggregations of matter, as well as accompanying energy and entropy changes during these processes. The chemical structure refers to both molecular geometry and to electronic structure. The structural formula of a chemical compound is a graphical representation of the molecular structure showing how the atoms are arranged. A protein structure is the three dimensional coordinates of the atoms within (macro) molecules made of protein.

Musical composition[link]

Music is an art form consisting of sound and silence expressed through time. The term musical form, a type of structure, refers to two related concepts:

• the type of composition (for example, a musical work can have the form of a symphony, a concerto, or other generic type)
• the structure of a particular piece (for example, a piece can be written in binary form, sonata form, as a fugue, etc.)

Social structure[link]

A social structure is a pattern of relations. They are social organizations of individuals in various life situations. Structures are applicable to people in how a society is as a system organized by a characteristic pattern of relationships. This is known as the social organization of the group. Sociologists have studied the changing structure of these groups. Structure and agency are two confronted theories about human behaviour. The debate surrounding the influence of structure and agency on human thought is one of the central issues in sociology. In this context "agency" refers to the capacity of individual humans to act independently and to make their own free choices. "Structure" here refers to those factors such as social class, religion, gender, ethnicity, customs etc. which seem to limit or influence the opportunities that individuals have.

Data structure[link]

In computer science, a data structure is a way of storing data in a computer so that it can be used efficiently. Often a carefully chosen data structure will allow the most efficient algorithm to be used. The choice of the data structure often begins from the choice of an abstract data type. A well-designed data structure allows a variety of critical operations to be performed, using as few resources, both execution time and memory space, as possible. Data structures are implemented in a programming language as data types and the references (e.g. relationships, links and pointers) and operations that are possible with them. For structure tables and structure functions, see data structure.

See also[link]

• Crystal structure
• Data structure
• Mathematical structure
• Building
• Nonbuilding structure
• Structural analysis
• Structural engineering
• Structural geology
• Protein structure
• Biomolecular structure
  • Primary structure
  • Secondary structure
  • Tertiary structure
  • Quaternary structure

References[link]

Terence Chi-Shen Tao - Хуяо
Tao at the March 2006 at the Erdős Memorial Conference in Memphis, Tennessee
Born	(1975-07-17) 17 July 1975 (age 36) Adelaide, Australia
Residence	Los Angeles, California
Nationality	Australian[1] United States[1]
Fields	Mathematics
Institutions	University of California, Los Angeles
Alma mater	Flinders University Princeton University
Doctoral advisor	Elias M. Stein
Known for	Green–Tao theorem Tao's inequality Kakeya conjecture Horn conjecture
Notable awards	Salem Prize (2000) Bôcher Memorial Prize (2002) Clay Research Award (2003) Australian Mathematical Society Medal (2005) Ostrowski Prize (2005) SASTRA Ramanujan Prize (2006) Levi L.Conant Prize (2005) Fields Medal (2006) MacArthur Award (2007) Fellow of the Royal Society (2007) Alan T. Waterman Award (2008) Onsager Medal (2008) King Faisal International Prize (2010) Nemmers Prize in Mathematics (2010) Polya Prize (2010) Crafoord Prize (2012)

Terence "Terry" Chi-Shen Tao FRS (simplified Chinese: 陶哲轩; traditional Chinese: 陶哲軒) (born 17 July 1975, Adelaide), is an Australian born American mathematician working in harmonic analysis, partial differential equations, combinatorics, and analytic number theory. He currently holds the James and Carol Collins chair in mathematics at the University of California, Los Angeles. He was one of the recipients of the 2006 Fields Medal.

Personal life[link]

Tao was a child prodigy,^[2] one of the subjects in the longitudinal research on exceptionally gifted children by education researcher Miraca Gross.^[3] His father told the press that at the age of two, during a family gathering, Tao attempted to teach a 5-year-old child Arithmetic and English. According to Smithsonian Online Magazine, Tao could carry out basic arithmetic by the age of two. When asked by his father how he knew numbers and letters, he said he learned them from Sesame Street.^[4] Aside from English, Tao speaks Cantonese, but does not write Chinese.

Tao exhibited extraordinary mathematical abilities from an early age, attending university level mathematics courses at the age of nine. He is one of only two children (besides Lenhard Ng) in the history of the Johns Hopkins' Study of Exceptional Talent program to have achieved a score of 700 or greater on the SAT math section while just 8 years old (he scored a 760).^[5] In 1986, 1987, and 1988, Tao was the youngest participant to date in the International Mathematical Olympiad, first competing at the age of ten, winning a bronze, silver, and gold medal respectively. He remains the youngest winner of each of the three medals in the olympiad's history winning the gold medal when he barely turned thirteen. At age 14, Tao attended the Research Science Institute. When he was 15 he published his first assistant paper. He received his bachelor's and master's degrees (at the age of 16) from Flinders University under Garth Gaudry. In 1992 he won a Fulbright Scholarship to undertake postgraduate study in the United States. From 1992 to 1996, Tao was a graduate student at Princeton University under the direction of Elias Stein, receiving his Ph.D. at the age of 20.^[6] He joined the faculty of the University of California, Los Angeles in 1996.

When he was 24, he was promoted to full professor at UCLA and remains the youngest person ever appointed to that rank by the institution. Tao's father was born and grew up in Shanghai, and Tao's mother is Cantonese by ethnicity.^[7] His parents are first generation immigrants from Hong Kong to Australia.^[8] His father, Billy Tao (Chinese: 陶象國; pinyin: Táo Xiàngguó; Cantonese Yale: tòuh jeuhng gwok) is a pediatrician, and his mother is a physics and mathematics graduate from the University of Hong Kong, formerly a secondary school teacher of mathematics in Hong Kong.^[9]

Tao has two brothers living in Australia, both of whom represented Australia at the International Mathematical Olympiad.

• Nigel Tao is part of the team at Google Australia that created Google Wave.^[10]
• Trevor Tao has a double degree in maths and music and will soon be featured in a book on autistic savants.^[10]

Tao, his wife Laura (an engineer at NASA's Jet Propulsion Laboratory),^[11] their son and daughter live in Los Angeles, California.

Research and awards[link]

Tao has won numerous honors and awards.^[12] He received the Salem Prize in 2000, the Bôcher Memorial Prize in 2002, and the Clay Research Award in 2003, for his contributions to analysis including work on the Kakeya conjecture and wave maps. In 2005 he received the American Mathematical Society's Levi L. Conant Prize with Allen Knutson, and in 2006 he was awarded the SASTRA Ramanujan Prize.

In 2004, Ben Green and Tao released a preprint proving what is now known as the Green–Tao theorem. This theorem states that there are arbitrarily long arithmetic progressions of prime numbers. The New York Times described it this way:^[13][14]

For this and other work Tao was awarded the Australian Mathematical Society Medal of 2004.

In August 2006, at the 25th International Congress of Mathematicians in Madrid, he became one of the youngest persons, the first Australian, and the first UCLA faculty member ever to be awarded a Fields Medal.^[15][16] An article by New Scientist^[17] writes of his ability:

Tao was a finalist to become Australian of the Year in 2007.^[18] He is a corresponding member of the Australian Academy of Science, and in 2007 was elected as a Fellow of the Royal Society.^[19][20] In the same year Tao also published Tao's inequality,^[21] an extension to the Szemerédi regularity lemma in the field of information theory.

In April 2008, Tao received the Alan T. Waterman Award, which recognizes an early career scientist for outstanding contributions in their field. In addition to a medal, Waterman awardees also receive a $500,000 grant for advanced research.^[22]

In December 2008, he was named The Lars Onsager lecturer^[23] of 2008, for "his combination of mathematical depth, width and volume in a manner unprecedented in contemporary mathematics". He was presented the Onsager Medal, and held his Lars Onsager lecture entitled "Structure and randomness in the prime numbers"^[24] at NTNU, Norway.

Tao was also elected a Fellow of the American Academy of Arts and Sciences in 2009.^{[citation needed]}

In 2010, he received the King Faisal International Prize jointly with Enrico Bombieri.^[25] Also in 2010, he was awarded the Nemmers Prize in Mathematics^[26] along with the Polya Prize(SIAM).^[27] Tao and Van H. Vu solved the circular law conjecture.

Tao also made contributions to the study of the Erdős-Straus conjecture in 2011 by showing that the number of solutions to the Erdős-Straus equation increases polylogarithmically as n tends to infinity.

In 2012 he and Jean Bourgain received the Crafoord Prize in Mathematics from the Royal Swedish Academy of Sciences.^[28][29]

He is a Fellow of the Royal Society, the Australian Academy of Sciences (Corresponding Member), the National Academy of Sciences (Foreign member), and the American Academy of Arts and Sciences. In 2006, he received the Fields Medal "for his contributions to partial differential equations, combinatorics, harmonic analysis and additive number theory", and in 2007, he was awarded the MacArthur Fellowship. He has been featured in The New York Times, CNN, USA Today, Popular Science, and many other media outlets. ^[30]

As of 2011 Tao has published over 230 research papers.^[31]

Publications[link]

• Solving Mathematical Problems: A Personal Perspective,^[32] Oxford University Press, 2006
• Analysis, Vols I and II, Hindustan Book Agency, 2006
• Additive Combinatorics, with Van H. Vu, Cambridge University Press, 2006
• Structure and Randomness: pages from year one of a mathematical blog, American Mathematical Society. 2008
• Poincaré's legacies: pages from year two of a mathematical blog, Vols. I and II, American Mathematical Society, 2009
• An Epsilon of Room, I: Real Analysis: pages from year three of a mathematical blog, American Mathematical Society, 2011 here
• An Epsilon of Room, II: pages from year three of a mathematical blog, American Mathematical Society, 2011 here

See also[link]

• Green-Tao theorem
• Tao's inequality
• Circular law
• Influence of non-standard analysis

References[link]

External links[link]

• Terence Tao's home page
• Tao's research blog
• Beautiful minds THE AUSTRALIAN 11 August 2007
• Terence Tao at the Mathematics Genealogy Project
• Terence Tao's results at the International Mathematical Olympiad