John G. Thompson
John Griggs Thompson is a mathematician at the University of Florida noted for his work in the field of finite groups. He was awarded the Fields Medal in the Wolf Prize in 1992 and the 2008 Abel Prize, he received his B. A. from Yale University in 1955 and his doctorate from the University of Chicago in 1959 under the supervision of Saunders Mac Lane. After spending some time on the Mathematics faculty at the University of Chicago, he moved in 1970 to the Rouse Ball Professorship in Mathematics at the University of Cambridge and moved to the Mathematics Department of the University of Florida as a Graduate Research Professor, he is a Professor Emeritus of Pure Mathematics at the University of Cambridge, professor of mathematics at the University of Florida. He received the Abel Prize 2008 together with Jacques Tits. Thompson's doctoral thesis introduced new techniques, included the solution of a problem in finite group theory which had stood for around sixty years, the nilpotency of Frobenius kernels.
At the time, this achievement was noted in The New York Times. Thompson became a figure in the progress toward the classification of finite simple groups. In 1963, he and Walter Feit proved that all nonabelian finite simple groups are of order; this work was recognised by the award of the 1965 Cole Prize in Algebra of the American Mathematical Society. His N-group papers classified all finite simple groups for which the normalizer of every non-identity solvable subgroup is solvable; this included, as a by-product, the classification of all minimal finite simple groups. This work had some influence on developments in the classification of finite simple groups, was quoted in the citation by Richard Brauer for the award of Thompson's Fields Medal in 1970; the Thompson group Th is one of the 26 sporadic finite simple groups. Thompson made major contributions to the inverse Galois problem, he found a criterion for a finite group to be a Galois group, that in particular implies that the monster simple group is a Galois group.
In 1971, Thompson was elected to the United States National Academy of Sciences. In 1982, he was awarded the Senior Berwick Prize of the London Mathematical Society, in 1988, he received the honorary degree of Doctor of Science from the University of Oxford. Thompson was awarded the United States National Medal of Science in 2000, he is a Fellow of the Royal Society, a recipient of its Sylvester Medal in 1985. He is a member of the Norwegian Academy of Letters. Feit–Thompson theorem McKay–Thompson series Quadratic pair Thompson factorization Thompson order formula Thompson subgroup Thompson transitivity theorem Thompson uniqueness theorem O'Connor, John J.. John G. Thompson at the Mathematics Genealogy Project List of mathematical articles by John G. Thompson Biography from the Abel Prize center
Lester S. Hill
Lester S. Hill was an American mathematician and educator, interested in applications of mathematics to communications, he received a Bachelor's degree from Columbia College and a Ph. D. from Yale University. He taught at the University of Montana, Princeton University, the University of Maine, Yale University, Hunter College. Among his notable contributions was the Hill cipher, he developed methods for detecting errors in telegraphed code numbers and wrote two books. Rosen, Kenneth. Elementary Number Theory and its Applications, fifth edition, Addison-Wesley, p. 292
Columbia University is a private Ivy League research university in Upper Manhattan, New York City. Established in 1754, Columbia is the oldest institution of higher education in New York and the fifth-oldest institution of higher learning in the United States, it is one of nine colonial colleges founded prior to the Declaration of Independence, seven of which belong to the Ivy League. It has been ranked by numerous major education publications as among the top ten universities in the world. Columbia was established as King's College by royal charter of George II of Great Britain in reaction to the founding of Princeton University in New Jersey, it was renamed Columbia College in 1784 following the Revolutionary War and in 1787 was placed under a private board of trustees headed by former students Alexander Hamilton and John Jay. In 1896, the campus was moved from Madison Avenue to its current location in Morningside Heights and renamed Columbia University. Columbia scientists and scholars have played an important role in the development of notable scientific fields and breakthroughs including: brain-computer interface.
The Columbia University Physics Department has been affiliated with 33 Nobel Prize winners as alumni, faculty or research staff, the third most of any American institution behind MIT and Harvard. In addition, 22 Nobel Prize winners in Physiology and Medicine have been affiliated with Columbia, the third most of any American institution; the university's research efforts include the Lamont-Doherty Earth Observatory, Goddard Institute for Space Studies and accelerator laboratories with major technology firms such as IBM. Columbia is one of the fourteen founding members of the Association of American Universities and was the first school in the United States to grant the M. D. degree. The university administers the Pulitzer Prize annually. Columbia is organized into twenty schools, including three undergraduate schools and numerous graduate schools, it maintains research centers outside of the United States known as Columbia Global Centers. In 2018, Columbia's undergraduate acceptance rate was 5.1%, making it one of the most selective colleges in the United States, the second most selective in the Ivy League after Harvard.
Columbia is ranked as the 3rd best university in the United States by U. S. News & World Report behind Princeton and Harvard. In athletics, the Lions field varsity teams in 29 sports as a member of the NCAA Division I Ivy League conference; the university's endowment stood at $10.9 billion in 2018, among the largest of any academic institution. As of 2018, Columbia's alumni and affiliates include: five Founding Fathers of the United States — among them an author of the United States Constitution and co-author of the Declaration of Independence. S. presidents. Discussions regarding the founding of a college in the Province of New York began as early as 1704, at which time Colonel Lewis Morris wrote to the Society for the Propagation of the Gospel in Foreign Parts, the missionary arm of the Church of England, persuading the society that New York City was an ideal community in which to establish a college. However, it was not until the founding of the College of New Jersey across the Hudson River in New Jersey that the City of New York considered founding a college.
In 1746, an act was passed by the general assembly of New York to raise funds for the foundation of a new college. In 1751, the assembly appointed a commission of ten New York residents, seven of whom were members of the Church of England, to direct the funds accrued by the state lottery towards the foundation of a college. Classes were held in July 1754 and were presided over by the college's first president, Dr. Samuel Johnson. Dr. Johnson was the only instructor of the college's first class, which consisted of a mere eight students. Instruction was held in a new schoolhouse adjoining Trinity Church, located on what is now lower Broadway in Manhattan; the college was founded on October 31, 1754, as King's College by royal charter of King George II, making it the oldest institution of higher learning in the state of New York and the fifth oldest in the United States. In 1763, Dr. Johnson was succeeded in the presidency by Myles Cooper, a graduate of The Queen's College, an ardent Tory. In the charged political climate of the American Revolution, his chief opponent in discussions at the college was an undergraduate of the class of 1777, Alexander Hamilton.
The American Revolutionary War broke out in 1776, was catastrophic for the operation of King's College, which suspended instruction for eight years beginning in 1776 with the arrival of the Continental Army. The suspension continued through the military occupation of New York City by British troops until their departure in 1783; the college's library was looted and its sole building requisitioned for use as a military hospital first by American and British forces. Loyalists were forced to abandon their King's College in New York, seized by the rebels and renamed Columbia College; the Loyalists, led by Bishop Charles Inglis fled to Windsor, Nova Scotia, where the
Taftville is a small village in eastern Connecticut. It has its own post office, it was established in 1866 as site for the large Taftville Mill Ponemah Mill. The village is listed on the National Register of Historic Places as Taftville and as alternative name Taftville/Ponemah Mill National Register Historic District. Redevelopment of the large mill is being conducted by The O'Neill Group in conjunction with OneKey LLC; the National Park Service will oversee the historic preservation of the structure, to ensure the historic elements are sustained. The 430,000-square-foot Ponemah Mill is being converted into commercial space; the Ponemah Mills, a cotton textile factory, was built on the Shetucket River where a large dam could be built to provide power. The large mill building was purported to be the largest weave-shed under one roof at that time; the original workers were predominantly Irish immigrants, they were hard hit by the depression of the 1870s that began with the Panic of 1873. Unemployment rose and wages dropped appreciably from 1873 to 1875, causing bitter relations between workers and management in many places.
In April 1875, the 1,200 workers went on strike. The mill owners had raised rents in company-owned housing as well as prices at the company-owned store. Wages at the time were under $10 for a 67-hour work week. In one often-cited anecdote, a workingman said he and his daughter had worked full-time for more than three months but only had four dollars between them to show for it; the immediate cause of the strike was a pay cut of 12 percent in an attempt to stop unionization. Workers were told half of the pay cut would be restored to anyone who had not participated in trying to form a union at the company; the company replaced the workers with French Canadians, who would come to number more than 70 percent of the population. Workers were evicted from company-owned housing, the Connecticut General Assembly passed a strict "tramp law" aimed at workers who became drifters after their strikes were broken. Ponemah Mills operated for about 100 years, it sat abandoned for over 40 years until being redeveloped into residential apartments.
A similar mill and village community is Wauregan Historic District, NRHP-listed. Taftville has a public elementary school operated by the Norwich Public School System called Wequonnoc Magnet Elementary School and a private elementary school called Sacred Heart School. Students attend Kelly Middle School. After graduating from there, they move on to either Norwich Free Academy, Norwich Technical High School, or other surrounding high schools. There are two churches: the Taftville Congregational Church. Among the more accomplished Taftville residents was Ned Hanlon who managed the Pittsburgh Alleghenys, Pittsburgh Burghers, Baltimore Orioles, Brooklyn Superbas, the Cincinnati Reds, he was inducted into the Baseball Hall of Fame in 1996. Another ballplayer was the Quebec-born right fielder, Augustine "Lefty" Dugas, whose family settled in Taftville, he played for the Pirates, Philadelphia Phillies and Washington Senators between 1930 and 1934. In the academic arena Saunders Mac Lane, was the son of the Minister of the Taftville Congregational Church and a mathematician of world note, who spent his career at the University of Chicago and Harvard.
He was the co-author of A Survey of Modern Algebra, a book, the standard work in that field for many years. Neighborhoods of Norwich, Connecticut National Register of Historic Places listings in New London County, Connecticut Ponemah Mills 1894 electric locomotive No. 1386 at Connecticut Trolley Museum
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though most classify up to homotopy equivalence. Although algebraic topology uses algebra to study topological problems, using topology to solve algebraic problems is sometimes possible. Algebraic topology, for example, allows for a convenient proof that any subgroup of a free group is again a free group. Below are some of the main areas studied in algebraic topology: In mathematics, homotopy groups are used in algebraic topology to classify topological spaces; the first and simplest homotopy group is the fundamental group, which records information about loops in a space. Intuitively, homotopy groups record information about the basic shape, or holes, of a topological space. In algebraic topology and abstract algebra, homology is a certain general procedure to associate a sequence of abelian groups or modules with a given mathematical object such as a topological space or a group.
In homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a co-chain complex. That is, cohomology is defined as the abstract study of cochains and coboundaries. Cohomology can be viewed as a method of assigning algebraic invariants to a topological space that has a more refined algebraic structure than does homology. Cohomology arises from the algebraic dualization of the construction of homology. In less abstract language, cochains in the fundamental sense should assign'quantities' to the chains of homology theory. A manifold is a topological space. Examples include the plane, the sphere, the torus, which can all be realized in three dimensions, but the Klein bottle and real projective plane which cannot be realized in three dimensions, but can be realized in four dimensions. Results in algebraic topology focus on global, non-differentiable aspects of manifolds. Knot theory is the study of mathematical knots. While inspired by knots that appear in daily life in shoelaces and rope, a mathematician's knot differs in that the ends are joined together so that it cannot be undone.
In precise mathematical language, a knot is an embedding of a circle in 3-dimensional Euclidean space, R 3. Two mathematical knots are equivalent if one can be transformed into the other via a deformation of R 3 upon itself. A simplicial complex is a topological space of a certain kind, constructed by "gluing together" points, line segments and their n-dimensional counterparts. Simplicial complexes should not be confused with the more abstract notion of a simplicial set appearing in modern simplicial homotopy theory; the purely combinatorial counterpart to a simplicial complex is an abstract simplicial complex. A CW complex is a type of topological space introduced by J. H. C. Whitehead to meet the needs of homotopy theory; this class of spaces is broader and has some better categorical properties than simplicial complexes, but still retains a combinatorial nature that allows for computation. An older name for the subject was combinatorial topology, implying an emphasis on how a space X was constructed from simpler ones.
In the 1920s and 1930s, there was growing emphasis on investigating topological spaces by finding correspondences from them to algebraic groups, which led to the change of name to algebraic topology. The combinatorial topology name is still sometimes used to emphasize an algorithmic approach based on decomposition of spaces. In the algebraic approach, one finds a correspondence between spaces and groups that respects the relation of homeomorphism of spaces; this allows one to recast statements about topological spaces into statements about groups, which have a great deal of manageable structure making these statement easier to prove. Two major ways in which this can be done are through fundamental groups, or more homotopy theory, through homology and cohomology groups; the fundamental groups give us basic information about the structure of a topological space, but they are nonabelian and can be difficult to work with. The fundamental group of a simplicial complex does have a finite presentation.
Homology and cohomology groups, on the other hand, are abelian and in many important cases finitely generated. Finitely generated abelian groups are classified and are easy to work with. In general, all constructions of algebraic topology are functorial. Fundamental groups and homology and cohomology groups are not only invariants of the underlying topological space, in the sense that two topological spaces which are homeomorphic have the same associated groups, but their associated morphisms correspond — a continuous mapping of spaces induces a group homomorphism on the associated groups, these homomorphisms can be used to show non-existence of mappings. One of the first mathematicians to work with different types of cohomology was Georges de Rham. One can use the differential structure of smooth manifolds via de Rham cohomology, or Čech or sheaf co
Presbyterianism is a part of the reformed tradition within Protestantism, which traces its origins to Britain Scotland. Presbyterian churches derive their name from the presbyterian form of church government, governed by representative assemblies of elders. A great number of Reformed churches are organized this way, but the word Presbyterian, when capitalized, is applied uniquely to churches that trace their roots to the Church of Scotland, as well as several English dissenter groups that formed during the English Civil War. Presbyterian theology emphasizes the sovereignty of God, the authority of the Scriptures, the necessity of grace through faith in Christ. Presbyterian church government was ensured in Scotland by the Acts of Union in 1707, which created the Kingdom of Great Britain. In fact, most Presbyterians found in England can trace a Scottish connection, the Presbyterian denomination was taken to North America by Scots and Scots-Irish immigrants; the Presbyterian denominations in Scotland hold to the Reformed theology of John Calvin and his immediate successors, although there is a range of theological views within contemporary Presbyterianism.
Local congregations of churches which use presbyterian polity are governed by sessions made up of representatives of the congregation. The roots of Presbyterianism lie in the Reformation of the 16th century, the example of John Calvin's Republic of Geneva being influential. Most Reformed churches that trace their history back to Scotland are either presbyterian or congregationalist in government. In the twentieth century, some Presbyterians played an important role in the ecumenical movement, including the World Council of Churches. Many Presbyterian denominations have found ways of working together with other Reformed denominations and Christians of other traditions in the World Communion of Reformed Churches; some Presbyterian churches have entered into unions with other churches, such as Congregationalists, Lutherans and Methodists. Presbyterians in the United States came from Scottish immigrants, Scotch-Irish immigrants, from New England Yankee communities, Congregational but changed because of an agreed-upon Plan of Union of 1801 for frontier areas.
Along with Episcopalians, Presbyterians tend to be wealthier and better educated than most other religious groups in United States, are disproportionately represented in the upper reaches of American business and politics. Presbyterian tradition that of the Church of Scotland, traces its early roots to the Church founded by Saint Columba, through the 6th century Hiberno-Scottish mission. Tracing their apostolic origin to Saint John, the Culdees practiced Christian monasticism, a key feature of Celtic Christianity in the region, with a presbyter exercising "authority within the institution, while the different monastic institutions were independent of one another." The Church in Scotland kept the Christian feast of Easter at a date different from the See of Rome and its monks used a unique style of tonsure. The Synod of Whitby in 664, ended these distinctives as it ruled "that Easter would be celebrated according to the Roman date, not the Celtic date." Although Roman influence came to dominate the Church in Scotland, certain Celtic influences remained in the Scottish Church, such as "the singing of metrical psalms, many of them set to old Celtic Christianity Scottish traditional and folk tunes", which became a "distinctive part of Scottish Presbyterian worship".
Presbyterian history is part of the history of Christianity, but the beginning of Presbyterianism as a distinct movement occurred during the 16th-century Protestant Reformation. As the Catholic Church resisted the reformers, several different theological movements splintered from the Church and bore different denominations. Presbyterianism was influenced by the French theologian John Calvin, credited with the development of Reformed theology, the work of John Knox, a Scotsman and a Roman Catholic Priest, who studied with Calvin in Geneva, Switzerland, he brought back Reformed teachings to Scotland. The Presbyterian church traces its ancestry back to England and Scotland. In August 1560 the Parliament of Scotland adopted the Scots Confession as the creed of the Scottish Kingdom. In December 1560, the First Book of Discipline was published, outlining important doctrinal issues but establishing regulations for church government, including the creation of ten ecclesiastical districts with appointed superintendents which became known as presbyteries.
In time, the Scots Confession would be supplanted by the Westminster Confession of Faith, the Larger and Shorter Catechisms, which were formulated by the Westminster Assembly between 1643 and 1649. Presbyterians distinguish themselves from other denominations by doctrine, institutional organization and worship; the origins of the Presbyterian churches are in Calvinism. Many branches of Presbyterianism are remnants of previous splits from larger groups; some of the splits have been due to doctrinal controversy, while some have been caused by disagreement concerning the degree to which those ordained to church office should be required to agree with the Westminster Confession of Faith, which serves as an important confessional document – second only to the Bible, yet directing particularities in the standardization and translation of the Bible – in Presbyterian churches. Presbyteria
Montclair, New Jersey
Montclair is a township in Essex County, New Jersey, United States. As of the 2010 United States Census, the township's population was 37,669, reflecting a decline of 1,308 from the 38,977 counted in the 2000 Census, which had in turn increased by 1,248 from the 37,729 counted in the 1990 Census; as of 2010, it was the 60th-most-populous municipality in New Jersey. Montclair was first formed as a township on April 15, 1868, from portions of Bloomfield Township, so that a second railroad could be built to Montclair. After a referendum held on February 21, 1894, Montclair was reincorporated as a town, effective February 24, 1894, it derives its name from the French mont clair, meaning "clear mountain" or "bright mountain."In 1980, after multiple protests filed by Montclair officials regarding the inequities built into the federal revenue sharing system, Montclair passed a referendum changing its name to the "Township of Montclair," becoming the third of more than a dozen Essex County municipalities to reclassify themselves as townships to take advantage of federal revenue sharing policies that allocated townships a greater share of government aid to municipalities on a per capita basis.
Montclair, which opened the state's first dispensary in December 2012, joins Bellmawr, Egg Harbor Township and Woodbridge Township as one of the five municipalities that have authorized dispensaries for the sale of medical marijuana. According to the United States Census Bureau, the township has a total area of 6.315 square miles, including 6.308 square miles of land and 0.007 square mile of water. Montclair is on the east side of the First Mountain of the Watchung Mountains; some higher locations in the township provide excellent views of the surrounding area and of the New York City skyline about 12 miles away. Named localities in the township include Church Street, Frog Hollow, Montclair Heights, South End, Upper Montclair and Watchung Plaza. Montclair citizens use two main ZIP codes; the central and southern parts of the township are designated 07042. Upper Montclair lies north of Watchung Avenue and has a separate ZIP code, 07043; because the ZIP codes do not match municipal boundaries, a few homes near the borders with neighboring towns fall into the ZIP codes for those communities.
A few homes in some adjoining municipalities use one of the two ZIP codes assigned to Montclair, as does HackensackUMC Mountainside, whose campus straddles the border with Glen Ridge. Small areas in the southeast of the township fall into the Glen Ridge ZIP code 07028. Several streams flow eastward through Montclair: Toney's Brook in the center, Nishuane Brook in the southeast, the Wigwam Brook in the southwest, Pearl Brook in the northwest, Yantacaw Brook in the northeast – all in the Passaic River watershed. Yantacaw and Toney's brooks are dammed in parks to create ponds. Wigwam and Toney's brooks flow into the Second River, the others flow into the Third River. Montclair lies just north of the northernmost extent of the Rahway River watershed; the southern border of Montclair is a straight line between Eagle Rock, on the ridge of the First Watchung Mountain, the point where Orange Road begins at the foot of Ridgewood Avenue. The eastern border is a straight line between that point and a point just southwest of where Broad Street crosses the Third River.
The western border runs along the ridge of the First Watchung Mountain between Eagle Rock and the Essex County/Passaic County border. The northern border is the border between those two counties. Montclair has a temperate climate, with warm-to-hot, humid summers and cool to cold winters, as is characteristic of the Köppen climate classification humid continental climate. January tends to be the coldest month, with average high temperatures in the upper 30s Fahrenheit and lows averaging 21. July, the warmest month, features high temperatures in the mid-80s and lows in the 70s, with an average high of 86 degrees. From April to June and from September to early November, Montclair experiences temperatures from the lower 60s to the lower 70s. Montclair gets 50 inches of rain per year, above the United States average of 39 inches. Snowfall is common from December to early March, totals about 30 inches annually; the number of days each year in Montclair with any measurable precipitation is 90. Montclair is one or two degrees warmer than the neighboring towns of Verona and Cedar Grove because of the mountain between them, which sometimes blocks winds and clouds, including warmer air from the ocean to the east.
The township has long celebrated a feature that has attracted many to the community. The African American population has been stable at around 30% for decades, although it fell from 32% in 2000 to 27% in 2010. Montclair has attracted many who work for major media organizations in New York City, including The New York Times and Newsweek. A March 11, 2007, posting in the blog Gawker.com listed some of those who work in the media and live in Montclair. Many residents are commuters to the Metro Area; as of the 2010 United States Census, there were 37,669 people, 15,089 households, 9,445.714 families residing in the township. The population density was 5,971.2 per square mile. There were 15,911 housing units at an average density of 2,522.2 per square mile. The racial makeup of the township was 62.16% White, 27.16% Black or African American, 0.16% Native American, 3.81% Asian, 0.02% Pacific Islander, 2.19% from other races, 4.50% f