1.
Fellow of the Royal Society of Edinburgh
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Fellowship of the Royal Society of Edinburgh is an award granted to individuals that the Royal Society of Edinburgh judges to be eminently distinguished in their subject. Around 50 new fellows are elected each year in March, as of 2016 there are around 1650 Fellows, including 71 Honorary Fellows and 76 Corresponding Fellows. Fellows are entitled to use the post-nominal letters FRSE, examples of fellows include Peter Higgs and Jocelyn Bell Burnell. Previous fellows have included Melvin Calvin and Benjamin Franklin, see the Category, Fellows of the Royal Society of Edinburgh for more examples

2.
St Andrews
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St Andrews is a town on the east coast of Fife in Scotland,10 miles southeast of Dundee and 30 miles northeast of Edinburgh. The town is home to the University of St Andrews, the third oldest university in the English-speaking world, according to some rankings, it is ranked as the third best university in the United Kingdom, behind Oxbridge. The University is an part of the burgh and during term time students make up approximately one third of the towns population. St Andrews has a population of 16,800, the town is named after Saint Andrew the Apostle. There has been an important church in St Andrews since at least the 8th century, the settlement grew to the west of St Andrews cathedral with the southern side of the Scores to the north and the Kinness burn to the south. The burgh soon became the capital of Scotland, a position which was held until the Scottish Reformation. The famous cathedral, the largest in Scotland, now lies in ruins, St Andrews is also known worldwide as the home of golf. Visitors travel to St Andrews in great numbers for several courses ranked amongst the finest in the world, as well as for the sandy beaches. The Martyrs Memorial, erected to the honour of Patrick Hamilton, George Wishart, the civil parish has a population of 18,421. The earliest recorded name the area is Muckross, after the founding of a religious settlement in Muckross in around 370 AD, the name changed to Cennrígmonaid. This is Old Gaelic and composed of the elements cenn, ríg and this became Cell Rígmonaid and was anglicised Kilrymont. The modern Gaelic spelling is Cill Rìmhinn, the name St Andrews derives from the towns claim to be the resting place of bones of the apostle Andrew. According to legend, St Regulus brought the relics to Kilrymont and this is the origin of a third name for the town Kilrule. The first inhabitants who settled on the fringes of the rivers Tay. This was followed by the people who settled around the modern town around 4,500 BC as farmers clearing the area of woodland. In AD877, king Causantín mac Cináeda built a new church for the Culdees at St Andrews and later the same year was captured and executed after defending against Viking raiders. In AD906, the became the seat of the bishop of Alba. In 940 Constantine III abdicated and took the position of abbot of the monastery of St Andrews, the establishment of the present town began around 1140 by Bishop Robert on an L-shaped vill, possibly on the site of the ruined St Andrews Castle

3.
Scotland
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Scotland is a country that is part of the United Kingdom and covers the northern third of the island of Great Britain. It shares a border with England to the south, and is surrounded by the Atlantic Ocean, with the North Sea to the east. In addition to the mainland, the country is made up of more than 790 islands, including the Northern Isles, the Kingdom of Scotland emerged as an independent sovereign state in the Early Middle Ages and continued to exist until 1707. By inheritance in 1603, James VI, King of Scots, became King of England and King of Ireland, Scotland subsequently entered into a political union with the Kingdom of England on 1 May 1707 to create the new Kingdom of Great Britain. The union also created a new Parliament of Great Britain, which succeeded both the Parliament of Scotland and the Parliament of England. Within Scotland, the monarchy of the United Kingdom has continued to use a variety of styles, titles, the legal system within Scotland has also remained separate from those of England and Wales and Northern Ireland, Scotland constitutes a distinct jurisdiction in both public and private law. Glasgow, Scotlands largest city, was one of the worlds leading industrial cities. Other major urban areas are Aberdeen and Dundee, Scottish waters consist of a large sector of the North Atlantic and the North Sea, containing the largest oil reserves in the European Union. This has given Aberdeen, the third-largest city in Scotland, the title of Europes oil capital, following a referendum in 1997, a Scottish Parliament was re-established, in the form of a devolved unicameral legislature comprising 129 members, having authority over many areas of domestic policy. Scotland is represented in the UK Parliament by 59 MPs and in the European Parliament by 6 MEPs, Scotland is also a member nation of the British–Irish Council, and the British–Irish Parliamentary Assembly. Scotland comes from Scoti, the Latin name for the Gaels, the Late Latin word Scotia was initially used to refer to Ireland. By the 11th century at the latest, Scotia was being used to refer to Scotland north of the River Forth, alongside Albania or Albany, the use of the words Scots and Scotland to encompass all of what is now Scotland became common in the Late Middle Ages. Repeated glaciations, which covered the land mass of modern Scotland. It is believed the first post-glacial groups of hunter-gatherers arrived in Scotland around 12,800 years ago, the groups of settlers began building the first known permanent houses on Scottish soil around 9,500 years ago, and the first villages around 6,000 years ago. The well-preserved village of Skara Brae on the mainland of Orkney dates from this period and it contains the remains of an early Bronze Age ruler laid out on white quartz pebbles and birch bark. It was also discovered for the first time that early Bronze Age people placed flowers in their graves, in the winter of 1850, a severe storm hit Scotland, causing widespread damage and over 200 deaths. In the Bay of Skaill, the storm stripped the earth from a large irregular knoll, when the storm cleared, local villagers found the outline of a village, consisting of a number of small houses without roofs. William Watt of Skaill, the laird, began an amateur excavation of the site, but after uncovering four houses

4.
Pure mathematics
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Broadly speaking, pure mathematics is mathematics that studies entirely abstract concepts. Even though the pure and applied viewpoints are distinct philosophical positions, in there is much overlap in the activity of pure. To develop accurate models for describing the world, many applied mathematicians draw on tools. On the other hand, many pure mathematicians draw on natural and social phenomena as inspiration for their abstract research, ancient Greek mathematicians were among the earliest to make a distinction between pure and applied mathematics. Plato helped to create the gap between arithmetic, now called number theory, and logistic, now called arithmetic. Euclid of Alexandria, when asked by one of his students of what use was the study of geometry, the term itself is enshrined in the full title of the Sadleirian Chair, founded in the mid-nineteenth century. The idea of a discipline of pure mathematics may have emerged at that time. The generation of Gauss made no sweeping distinction of the kind, in the following years, specialisation and professionalisation started to make a rift more apparent. At the start of the twentieth century mathematicians took up the axiomatic method, in fact in an axiomatic setting rigorous adds nothing to the idea of proof. Pure mathematics, according to a view that can be ascribed to the Bourbaki group, is what is proved, Pure mathematician became a recognized vocation, achievable through training. One central concept in mathematics is the idea of generality. One can use generality to avoid duplication of effort, proving a general instead of having to prove separate cases independently. Generality can facilitate connections between different branches of mathematics, category theory is one area of mathematics dedicated to exploring this commonality of structure as it plays out in some areas of math. Generalitys impact on intuition is both dependent on the subject and a matter of preference or learning style. Often generality is seen as a hindrance to intuition, although it can function as an aid to it. Each of these branches of abstract mathematics have many sub-specialties. A steep rise in abstraction was seen mid 20th century, in practice, however, these developments led to a sharp divergence from physics, particularly from 1950 to 1983. Later this was criticised, for example by Vladimir Arnold, as too much Hilbert, the point does not yet seem to be settled, in that string theory pulls one way, while discrete mathematics pulls back towards proof as central

5.
University of St Andrews
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The University of St Andrews is a British public research university in St Andrews, Fife, Scotland. It is the oldest of the four ancient universities of Scotland, St Andrews was founded between 1410 and 1413, when the Avignon Antipope Benedict XIII issued a papal bull to a small founding group of Augustinian clergy. St Andrews is made up from a variety of institutions, including three constituent colleges and 18 academic schools organised into four faculties, the university occupies historic and modern buildings located throughout the town. The academic year is divided into two terms, Martinmas and Candlemas, in term time, over one-third of the towns population is either a staff member or student of the university. It is ranked as the third best university in the United Kingdom in national league tables, the Times Higher Education World Universities Ranking names St Andrews among the worlds Top 50 universities for Social Sciences, Arts and Humanities. St Andrews has the highest student satisfaction amongst all multi-faculty universities in the United Kingdom, St Andrews has many notable alumni and affiliated faculty, including eminent mathematicians, scientists, theologians, philosophers, and politicians. Six Nobel Laureates are among St Andrews alumni and former staff, a charter of privilege was bestowed upon the society of masters and scholars by the Bishop of St Andrews, Henry Wardlaw, on 28 February 1411. Wardlaw then successfully petitioned the Avignon Pope Benedict XIII to grant the university status by issuing a series of papal bulls. King James I of Scotland confirmed the charter of the university in 1432, subsequent kings supported the university with King James V confirming privileges of the university in 1532. A college of theology and arts called St Johns College was founded in 1418 by Robert of Montrose, St Salvators College was established in 1450, by Bishop James Kennedy. St Leonards College was founded in 1511 by Archbishop Alexander Stewart, St Johns College was refounded by Cardinal James Beaton under the name St Marys College in 1538 for the study of divinity and law. Some university buildings that date from this period are still in use today, such as St Salvators Chapel, St Leonards College Chapel, at this time, the majority of the teaching was of a religious nature and was conducted by clerics associated with the cathedral. During the 17th and 18th centuries, the university had mixed fortunes and was beset by civil. He described it as pining in decay and struggling for life, in the second half of the 19th century, pressure was building upon universities to open up higher education to women. In 1876, the University Senate decided to allow women to receive an education at St Andrews at a roughly equal to the Master of Arts degree that men were able to take at the time. The scheme came to be known as the L. L. A and it required women to pass five subjects at an ordinary level and one at honours level and entitled them to hold a degree from the university. In 1889 the Universities Act made it possible to admit women to St Andrews. Agnes Forbes Blackadder became the first woman to graduate from St Andrews on the level as men in October 1894

6.
Group (mathematics)
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In mathematics, a group is an algebraic structure consisting of a set of elements equipped with an operation that combines any two elements to form a third element. The operation satisfies four conditions called the group axioms, namely closure and it allows entities with highly diverse mathematical origins in abstract algebra and beyond to be handled in a flexible way while retaining their essential structural aspects. The ubiquity of groups in areas within and outside mathematics makes them a central organizing principle of contemporary mathematics. Groups share a kinship with the notion of symmetry. The concept of a group arose from the study of polynomial equations, after contributions from other fields such as number theory and geometry, the group notion was generalized and firmly established around 1870. Modern group theory—an active mathematical discipline—studies groups in their own right, to explore groups, mathematicians have devised various notions to break groups into smaller, better-understandable pieces, such as subgroups, quotient groups and simple groups. A theory has developed for finite groups, which culminated with the classification of finite simple groups. Since the mid-1980s, geometric group theory, which studies finitely generated groups as objects, has become a particularly active area in group theory. One of the most familiar groups is the set of integers Z which consists of the numbers, −4, −3, −2, −1,0,1,2,3,4. The following properties of integer addition serve as a model for the group axioms given in the definition below. For any two integers a and b, the sum a + b is also an integer and that is, addition of integers always yields an integer. This property is known as closure under addition, for all integers a, b and c, + c = a +. Expressed in words, adding a to b first, and then adding the result to c gives the final result as adding a to the sum of b and c. If a is any integer, then 0 + a = a +0 = a, zero is called the identity element of addition because adding it to any integer returns the same integer. For every integer a, there is a b such that a + b = b + a =0. The integer b is called the element of the integer a and is denoted −a. The integers, together with the operation +, form a mathematical object belonging to a class sharing similar structural aspects. To appropriately understand these structures as a collective, the abstract definition is developed