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132-711: He was a student at the University of Chicago and at New York University , where he completed his Ph.D. under Richard Courant in 1952. He spent much of his career at Princeton University , as a research scientist at the Plasma Physics Laboratory starting in 1951, and then as a professor of astronomy (1961), founder and chair of the Program in Applied and Computational Mathematics (1968), and professor of mathematics (1979). He retired from Princeton University in 1989 and joined

264-661: A 15-day sit-in at the college's administration building in a protest over the university's segregationist off-campus rental policies. After continued turmoil, a university committee in 1967 issued what became known as the Kalven Report. The report, a two-page statement of the university's policy in "social and political action," declared that "To perform its mission in the society, a university must sustain an extraordinary environment of freedom of inquiry and maintain an independence from political fashions, passions, and pressures." The report has since been used to justify decisions such as

396-496: A coordinate system for the Schwarzschild geometry for a black hole . These coordinates have the advantage that they cover the entire spacetime manifold of the maximally extended Schwarzschild solution and are well-behaved everywhere outside the physical singularity. There is no coordinate singularity at the horizon. The Kruskal–Szekeres coordinates also apply to space-time around a spherical object, but in that case do not give

528-443: A gravitational singularity . Technically, this means that a maximally extended spacetime is either "geodesically complete" (meaning any geodesic can be extended to arbitrarily large positive or negative values of its 'affine parameter', which in the case of a timelike geodesic could just be the proper time ), or if any geodesics are incomplete, it can only be because they end at a singularity. In order to satisfy this requirement, it

660-404: A light cone drawn in a Kruskal–Szekeres diagram will look just the same as a light cone in a Minkowski diagram in special relativity . The event horizons bounding the black hole and white hole interior regions are also a pair of straight lines at 45 degrees, reflecting the fact that a light ray emitted at the horizon in a radial direction (aimed outward in the case of the black hole, inward in

792-442: A 19th-century mystery dating back to work by John Scott Russell who, in 1834, observed what we now call a soliton, propagating in a canal, and chased it on horseback. In spite of his observations of solitons in wave tank experiments, Scott Russell never recognized them as such, because of his focus on the "great wave of translation," the largest amplitude solitary wave. His experimental observations, presented in his Report on Waves to

924-453: A Kruskal–Szekeres diagram (this can be derived from the metric equation given above, which guarantees that if d X = ± d T {\displaystyle dX=\pm dT\,} then the proper time d s = 0 {\displaystyle ds=0} ). All timelike world lines of slower-than-light objects will at every point have a slope closer to the vertical time axis (the T coordinate) than 45 degrees. So,

1056-518: A Schwarzschild t -coordinate of − ∞ {\displaystyle -\infty } , reflecting the fact that in Schwarzschild coordinates an infalling particle takes an infinite coordinate time to reach the horizon (i.e. the particle's distance from the horizon approaches zero as the Schwarzschild t -coordinate approaches infinity), and a particle traveling up away from the horizon must have crossed it an infinite coordinate time in

1188-479: A collision with other such waves. Because of the particle-like properties of such a wave, they named it a " soliton ", a term that caught on almost immediately. This work was partly motivated by the near- recurrence paradox that had been observed in a very early computer simulation of a certain nonlinear lattice by Enrico Fermi , John Pasta , Stanislaw Ulam and Mary Tsingou at Los Alamos in 1955. Those authors had observed long-time nearly recurrent behavior of

1320-423: A copy of every book and journal published by the University of Chicago Press at no cost. The agreement provided that either party could terminate the affiliation on proper notice. Several University of Chicago professors disliked the program, as it involved uncompensated additional labor on their part, and they believed it cheapened the academic reputation of the university. The program was ended by 1910. In 1929,

1452-480: A description of space-time inside the radius of the object. Space-time in a region where a star is collapsing into a black hole is approximated by the Kruskal–Szekeres coordinates (or by the Schwarzschild coordinates ). The surface of the star remains outside the event horizon in the Schwarzschild coordinates, but crosses it in the Kruskal–Szekeres coordinates. (In any "black hole" which we observe , we see it at

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1584-466: A grand reading room for the university library and prevents the need for an off-campus book depository. The site of Chicago Pile-1 is a National Historic Landmark and is marked by the Henry Moore sculpture Nuclear Energy . Robie House , a Frank Lloyd Wright building acquired by the university in 1963, is a UNESCO World Heritage Site as well as a National Historic Landmark, as is room 405 of

1716-518: A large study space and an automated book storage and retrieval system. The John Crerar Library contains more than 1.4 million volumes in the biological, medical and physical sciences and collections in general science and the philosophy and history of science, medicine, and technology. The university also operates a number of special libraries, including the D'Angelo Law Library, the Social Service Administration Library, and

1848-676: A location on the University of Chicago campus. Since 1983, the University of Chicago has maintained the University of Chicago School Mathematics Project , a mathematics program used in urban primary and secondary schools. The university runs a program called the Council on Advanced Studies in the Humanities and Social Sciences, which administers interdisciplinary workshops to provide a forum for graduate students, faculty, and visiting scholars to present scholarly work in progress. The university also operates

1980-507: A major role in the development of many academic disciplines, including economics, law, literary criticism, mathematics, physics, religion, sociology, and political science, establishing the Chicago schools of thought in various fields. Chicago's Metallurgical Laboratory produced the world's first human-made, self-sustaining nuclear reaction in Chicago Pile-1 beneath the viewing stands of

2112-622: A member of the Baptist clergy who believed that a great university should maintain the study of faith as a central focus. To fulfill this commitment, he brought the Baptist seminary that had begun as an independent school "alongside" the Old University of Chicago and separated from the old school decades earlier to Morgan Park. This became the Divinity School in 1891, the first professional school at

2244-590: A nearby particle physics laboratory, as well as a stake in the Apache Point Observatory in Sunspot, New Mexico . Faculty and students at the adjacent Toyota Technological Institute at Chicago collaborate with the university. In 2013, the university formed an affiliation with the formerly independent Marine Biological Laboratory in Woods Hole, Mass. The National Opinion Research Center maintains an office at

2376-615: A new school could go by the name of the city. In 1890, the American Baptist Education Society incorporated a new University of Chicago as a coeducational institution, using $ 400,000 donated to the ABES to supplement a $ 600,000 donation from Standard Oil co-founder John D. Rockefeller , and land donated by Marshall Field . While the Rockefeller donation provided money for academic operations and long-term endowment, it

2508-412: A new timelike coordinate T and a new spacelike coordinate X {\displaystyle X} : for the exterior region r > 2 G M {\displaystyle r>2GM} outside the event horizon and: for the interior region 0 < r < 2 G M {\displaystyle 0<r<2GM} . Here G M {\displaystyle GM}

2640-460: A new tool to be exploited." Kruskal received the National Medal of Science in 1993 "for his influence as a leader in nonlinear science for more than two decades as the principal architect of the theory of soliton solutions of nonlinear equations of evolution". In an article surveying the state of mathematics at the turn of the millennium, the eminent mathematician Philip A. Griffiths wrote that

2772-446: A one-dimensional chain of anharmonic oscillators, in contrast to the rapid thermalization that had been expected. Kruskal and Zabusky simulated the KdV equation, which Kruskal had obtained as a continuum limit of that one-dimensional chain, and found solitonic behavior, which is the opposite of thermalization. That turned out to be the heart of the phenomenon. Solitary wave phenomena had been

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2904-454: A pair of event horizons at 45 degrees, while lines of constant t -coordinate in Schwarzschild coordinates always look like straight lines at various angles passing through the center of the diagram. The black hole event horizon bordering exterior region I would coincide with a Schwarzschild t -coordinate of + ∞ {\displaystyle +\infty } while the white hole event horizon bordering this region would coincide with

3036-497: A pioneering computer simulation by Kruskal and Norman Zabusky (with some assistance from Harry Dym ) of a nonlinear equation known as the Korteweg–de Vries equation (KdV). The KdV equation is an asymptotic model of the propagation of nonlinear dispersive waves. But Kruskal and Zabusky made the startling discovery of a "solitary wave" solution of the KdV equation that propagates non-dispersively and even regains its shape after

3168-553: A related equation known as the Modified Korteweg–de Vries (MKdV) equation. With these conservation laws, Miura showed a connection (called the Miura transformation) between solutions of the KdV and MKdV equations. This was a clue that enabled Kruskal, with Clifford S. Gardner , John M. Greene , and Miura (GGKM), to discover a general technique for exact solution of the KdV equation and understanding of its conservation laws. This

3300-464: A single exterior region and a single interior region, such as regions I and II in the Kruskal–Szekeres diagram. The Kruskal–Szekeres coordinate system, on the other hand, can cover a "maximally extended" spacetime which includes the region covered by Schwarzschild coordinates. Here, "maximally extended" refers to the idea that the spacetime should not have any "edges": any geodesic path can be extended arbitrarily far in either direction unless it runs into

3432-472: A sphere, the point ( T , X ) = ( 0 , 0 ) {\displaystyle (T,X)=(0,0)} (corresponding to the event horizon r = 2 G M {\displaystyle r=2GM} in the Schwarzschild picture) corresponds not to a sphere but to the projective plane R P 2 = S 2 / ± {\displaystyle \mathbf {RP} ^{2}=S^{2}/\pm } instead, and

3564-505: A time when its matter has not yet finished collapsing, so it is not really a black hole yet.) Similarly, objects falling into a black hole remain outside the event horizon in Schwarzschild coordinates, but cross it in Kruskal–Szekeres coordinates. Kruskal–Szekeres coordinates on a black hole geometry are defined, from the Schwarzschild coordinates ( t , r , θ , ϕ ) {\displaystyle (t,r,\theta ,\phi )} , by replacing t and r by

3696-415: Is Katherine Baicker since March 2023. The current president of the University of Chicago is chemist Paul Alivisatos , who assumed the role on September 1, 2021. Robert Zimmer , the previous president, transitioned into the new role of chancellor of the university. The university's endowment was the 12th largest among American educational institutions and state university systems in 2013 and as of 2020

3828-590: Is a private research university in Chicago, Illinois , United States. Its main campus is in the Hyde Park neighborhood on Chicago's South Side , near the shore of Lake Michigan about 7 miles (11 km) from the Loop . The university is composed of the undergraduate College of the University of Chicago and four graduate research divisions: Biological Science, Humanities, Physical Science, and Social Science, which also include various organized institutes. In addition,

3960-423: Is a free action by the group Z / 2 Z {\displaystyle \mathbb {Z} /2\mathbb {Z} } preserving the metric, this gives a well-defined Lorentzian manifold (everywhere except at the singularity). It identifies the limit t (II) = − ∞ {\displaystyle t^{\text{(II)}}=-\infty } of the interior region II corresponding to

4092-416: Is a relativistic wave equation in 1+1 dimensions that also exhibits the soliton phenomenon and which became an important model of solvable relativistic field theory. In seminal work preceding AKNS, Zakharov and Shabat discovered an inverse scattering method for the nonlinear Schrödinger equation. Solitons are now known to be ubiquitous in nature, from physics to biology. In 1986, Kruskal and Zabusky shared

Martin David Kruskal - Misplaced Pages Continue

4224-457: Is determined in terms of Kruskal–Szekeres coordinates as the (unique) solution of the equation: Using the Lambert W function the solution is written as: Moreover one sees immediately that in the region external to the black hole T 2 − X 2 < 0 ,   X > 0 {\displaystyle T^{2}-X^{2}<0,\ X>0} whereas in

4356-497: Is not so widely used as the term soliton . Asymptotic methods of various types have been successfully used since almost the birth of science itself. Nevertheless, Kruskal tried to show that asymptotology is a special branch of knowledge, intermediate, in some sense, between science and art. His proposal has been found to be very fruitful. Kruskal's honors and awards included: University of Chicago The University of Chicago ( UChicago , Chicago , U of C , or UChi )

4488-499: Is the gravitational constant multiplied by the Schwarzschild mass parameter, and this article is using units where c {\displaystyle c} = 1. It follows that on the union of the exterior region, the event horizon and the interior region the Schwarzschild radial coordinate r {\displaystyle r} (not to be confused with the Schwarzschild radius r s = 2 G M {\displaystyle r_{\text{s}}=2GM} ),

4620-463: Is the largest gift in the university's history and the largest gift ever to any business school. In 2009, planning or construction on several new buildings, half of which cost $ 100 million or more, was underway. Since 2011, major construction projects have included the Jules and Gwen Knapp Center for Biomedical Discovery, a ten-story medical research center, and further additions to the medical campus of

4752-561: Is the time-reversed black hole, sometimes dubbed a " white hole ". Particles can escape from a white hole but they can never return. The maximally extended Schwarzschild geometry can be divided into 4 regions each of which can be covered by a suitable set of Schwarzschild coordinates. The Kruskal–Szekeres coordinates, on the other hand, cover the entire spacetime manifold. The four regions are separated by event horizons. The transformation given above between Schwarzschild and Kruskal–Szekeres coordinates applies only in regions I and II (if we take

4884-732: The George Herbert Jones Laboratory , where Glenn T. Seaborg and his team were the first to isolate plutonium. Hitchcock Hall , an undergraduate dormitory, is on the National Register of Historic Places . The campus is soon to be the home of the Obama Presidential Center , the Presidential Library for the 44th president of the United States with expected completion in 2026. The Obamas settled in

5016-674: The Great Depression . Nonetheless, in 1933, Hutchins proposed an unsuccessful plan to merge the University of Chicago and Northwestern University . During World War II, the university's Metallurgical Laboratory made ground-breaking contributions to the Manhattan Project . The university was the site of the first isolation of plutonium and of the creation of the first artificial, self-sustained nuclear reaction by Enrico Fermi in 1942. The university did not provide standard oversight of Bruno Bettelheim and his tenure as director of

5148-585: The Howard N. Potts Gold Medal from the Franklin Institute "for contributions to mathematical physics and early creative combinations of analysis and computation, but most especially for seminal work in the properties of solitons". In awarding the 2006 Steele Prize to Gardner, Greene, Kruskal, and Miura, the American Mathematical Society stated that before their work "there was no general theory for

5280-743: The John U. Nef Committee on Social Thought . The university contains eight professional schools: the University of Chicago Law School , the Pritzker School of Medicine , the Booth School of Business , the University of Chicago Divinity School , the University of Chicago Harris School of Public Policy , the University of Chicago School of Social Service Administration , the Graham School of Continuing Liberal and Professional Studies (which offers non-degree courses and certificates as well as degree programs) and

5412-638: The Kruskal–Shafranov instability and the Bernstein–Greene–Kruskal (BGK) modes . With I. B. Bernstein, E. A. Frieman, and R. M. Kulsrud, he developed the MHD (or magnetohydrodynamic ) Energy Principle. His interests extended to plasma astrophysics as well as laboratory plasmas. In 1960, Kruskal discovered the full classical spacetime structure of the simplest type of black hole in general relativity . A spherically symmetric spacetime can be described by

Martin David Kruskal - Misplaced Pages Continue

5544-798: The Miller–Urey experiment , was conducted at the university. REM sleep was discovered at the university in 1953 by Nathaniel Kleitman and Eugene Aserinsky . The University of Chicago (Department of Astronomy and Astrophysics) operated the Yerkes Observatory in Williams Bay, Wisconsin from 1897 until 2018, where the largest operating refracting telescope in the world and other telescopes are located. Kruskal%E2%80%93Szekeres coordinates In general relativity , Kruskal–Szekeres coordinates , named after Martin Kruskal and George Szekeres , are

5676-515: The Orthogenic School for Disturbed Children from 1944 to 1973. In the early 1950s, student applications declined as a result of increasing crime and poverty in the Hyde Park neighborhood. In response, the university became a major sponsor of a controversial urban renewal project for Hyde Park , which profoundly affected both the neighborhood's architecture and street plan. During this period

5808-761: The Pritzker School of Molecular Engineering . The Law School is accredited by the American Bar Association , the Divinity School is accredited by the Commission on Accrediting of the Association of Theological Schools in the United States and Canada , and Pritzker is accredited by the Liaison Committee on Medical Education . The university runs a number of academic institutions and programs apart from its undergraduate and postgraduate schools. It operates

5940-444: The Schwarzschild solution , which was discovered in the early days of general relativity. However, in its original form, this solution only describes the region exterior to the event horizon of the black hole. Kruskal (in parallel with George Szekeres ) discovered the maximal analytic continuation of the Schwarzschild solution, which he exhibited elegantly using what are now called Kruskal–Szekeres coordinates . This led Kruskal to

6072-665: The University of Chicago Laboratory Schools (a private day school for K-12 students and day care ), and a public charter school with four campuses on the South Side of Chicago administered by the university's Urban Education Institute. In addition, the Hyde Park Day School, a school for students with learning disabilities, and the Sonia Shankman Orthogenic School , a residential treatment program for those with behavioral and emotional problems, maintains

6204-506: The University of Chicago Medical Center . In 2014 the university launched the public phase of a $ 4.5 billion fundraising campaign. In September 2015, the university received $ 100 million from The Pearson Family Foundation to establish The Pearson Institute for the Study and Resolution of Global Conflicts and The Pearson Global Forum at the Harris School of Public Policy . In 2019,

6336-831: The University of Chicago Press , and the University of Chicago Medical Center , and oversees several laboratories, including Fermi National Accelerator Laboratory (Fermilab) , Argonne National Laboratory , and the Marine Biological Laboratory . The university is accredited by The Higher Learning Commission . The university runs on a quarter system in which the academic year is divided into four terms: Summer (June–August), Autumn (September–December), Winter (January–March), and Spring (April–June). Full-time undergraduate students take three to four courses every quarter for approximately nine weeks before their quarterly academic breaks. The school year typically begins in late September and ends in late May. After its foundation in

6468-529: The University of Chicago Press , the largest university press in the United States. The University of Chicago Library system encompasses six libraries that contain a total of 11 million volumes, the 9th most among library systems in the United States. The university's main library is the Regenstein Library , which contains one of the largest collections of print volumes in the United States. The Joe and Rika Mansueto Library , built in 2011, houses

6600-519: The 1890s, the university, fearful that its vast resources would injure smaller schools by drawing away good students, affiliated with several regional colleges and universities: Des Moines College , Kalamazoo College , Butler University , and Stetson University . In 1896, the university affiliated with Shimer College in Mount Carroll, Illinois. Under the terms of the affiliation, the schools were required to have courses of study comparable to those at

6732-491: The 1970s, when the thermal nature of black hole physics was discovered, the wormhole property of the Schwarzschild solution turned out to be an important ingredient. Nowadays, it is considered a fundamental clue in attempts to understand quantum gravity . Kruskal's most widely known work was the discovery in the 1960s of the integrability of certain nonlinear partial differential equations involving functions of one spatial variable as well as time. These developments began with

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6864-503: The 1980s, Kruskal developed an acute interest in the Painlevé equations. They frequently arise as symmetry reductions of soliton equations, and Kruskal was intrigued by the intimate relationship that appeared to exist between the properties characterizing these equations and completely integrable systems. Much of his subsequent research was driven by a desire to understand this relationship and to develop new direct and simple methods for studying

6996-571: The 2-sphere. If we think of regions III and IV as having spherical coordinates but with a negative choice for the square root to compute r {\displaystyle r} , then we just correspondingly use opposite points on the sphere to denote the same point in space, so e.g. This means that r (I) Ω (I) = r (III) Ω (III) = r Ω {\displaystyle r^{\text{(I)}}\Omega ^{\text{(I)}}=r^{\text{(III)}}\Omega ^{\text{(III)}}=r\Omega } . Since this

7128-535: The 2018 Illinois Bicentennial, the University of Chicago Quadrangles were selected as one of the Illinois 200 Great Places by the American Institute of Architects Illinois component (AIA Illinois). After the 1940s, the campus's Gothic style began to give way to modern styles. In 1955, Eero Saarinen was contracted to develop a second master plan, which led to the construction of buildings both north and south of

7260-562: The British Association for the Advancement of Science in 1844, were viewed with skepticism by George Airy and George Stokes because their linear water wave theories were unable to explain them. Joseph Boussinesq (1871) and Lord Rayleigh (1876) published mathematical theories justifying Scott Russell's observations. In 1895, Diederik Korteweg and Gustav de Vries formulated the KdV equation to describe shallow water waves (such as

7392-654: The Chicago firm Skidmore, Owings & Merrill . Another master plan, designed in 1999 and updated in 2004, produced the Gerald Ratner Athletics Center (2003), the Max Palevsky Residential Commons (2001), South Campus Residence Hall and dining commons (2009), a new children's hospital, and other construction, expansions, and restorations. In 2011, the university completed the glass dome-shaped Joe and Rika Mansueto Library , which provides

7524-569: The Eckhart Library for mathematics and computer science. Harper Memorial Library is now a reading and study room. According to the National Science Foundation , University of Chicago spent $ 423.9 million on research and development in 2018, ranking it 60th in the nation. It is classified among "R1: Doctoral Universities – Very high research activity" and is a founding member of the Association of American Universities and

7656-469: The Hyde Park campus and is affiliated with multiple academic centers and institutes. The University of Chicago has been the site of some important experiments and academic movements. In economics, the university has played an important role in shaping ideas about the free market and is the namesake of the Chicago school of economics , the school of economic thought supported by Milton Friedman and other economists. The university's sociology department

7788-608: The Law . The university has an extensive record of producing successful business leaders and billionaires. The College of the University of Chicago grants Bachelor of Arts and Bachelor of Science degrees in 51 academic majors and 33 minors. The college's academics are divided into five divisions: the Biological Sciences Collegiate Division, the Physical Sciences Collegiate Division,

7920-547: The Main Quadrangles were designed by Cobb, Shepley, Rutan and Coolidge , Holabird & Roche , and other architectural firms in a mixture of the Victorian Gothic and Collegiate Gothic styles, patterned on the colleges of the University of Oxford. (Mitchell Tower, for example, is modeled after Oxford's Magdalen Tower , and the university Commons, Hutchinson Hall , replicates Christ Church Hall. ) In celebration of

8052-568: The Midway, including the Laird Bell Law Quadrangle (a complex designed by Saarinen); a series of arts buildings; a building designed by Ludwig Mies van der Rohe for the university's School of Social Service Administration, a building which is to become the home of the Harris School of Public Policy by Edward Durrell Stone , and the Regenstein Library , the largest building on campus, a brutalist structure designed by Walter Netsch of

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8184-422: The Old University of Chicago are recognized as alumni of the University of Chicago. William Rainey Harper became the university's president on July 1, 1891, and the Hyde Park campus opened for classes on October 1, 1892. Harper worked on building up the faculty and in two years he had a faculty of 120, including eight former university or college presidents. Harper was an accomplished scholar ( Semiticist ) and

8316-536: The Painlevé equations. Kruskal was rarely satisfied with the standard approaches to differential equations. The six Painlevé equations have a characteristic property called the Painlevé property: their solutions are single-valued around all singularities whose locations depend on the initial conditions. In Kruskal's opinion, since this property defines the Painlevé equations, one should be able to start with this, without any additional unnecessary structures, to work out all

8448-493: The Schwarzschild coordinates, the Schwarzschild radius r s = 2 G M {\displaystyle r_{\text{s}}=2GM} is the radial coordinate of the event horizon r = r s = 2 G M {\displaystyle r=r_{\text{s}}=2GM} . In the Kruskal–Szekeres coordinates the event horizon is given by T 2 − X 2 = 0 {\displaystyle T^{2}-X^{2}=0} . Note that

8580-622: The Social Sciences Collegiate Division, the Humanities Collegiate Division, and the New Collegiate Division. The first four are sections within their corresponding graduate divisions, while the New Collegiate Division administers interdisciplinary majors and studies which do not fit in one of the other four divisions. Undergraduate students are required to take a distribution of courses to satisfy

8712-449: The United States. The university's students, faculty, and staff has included 101 Nobel laureates . The university's faculty members and alumni also include 10 Fields Medalists, 4 Turing Award winners, 52 MacArthur Fellows , 26 Marshall Scholars , 54 Rhodes Scholars , 27 Pulitzer Prize winners, 20 National Humanities Medalists , 29 living billionaire graduates, and 8 Olympic medalists . The first University of Chicago

8844-481: The University of Chicago Hospitals (now called the University of Chicago Medical Center ) finished construction and enrolled their first medical students. Also, the philosophy oriented Committee on Social Thought , an institution distinctive of the university, was created. Money that had been raised during the 1920s and financial backing from the Rockefeller Foundation helped the school to survive through

8976-553: The University of Chicago announced plans to establish the Milton Friedman Institute , which attracted both support and controversy from faculty members and students. The institute would cost around $ 200 million and occupy the buildings of the Chicago Theological Seminary . During the same year, investor David G. Booth donated $ 300 million to the university's Booth School of Business , which

9108-601: The University of Chicago are known for their demanding standards, heavy workload and academic difficulty; according to Uni in the USA , "Among the academic cream of American universities – Harvard, Yale, Princeton, MIT, and the University of Chicago – it is UChicago that can most convincingly claim to provide the most rigorous, intense learning experience." The university graduate schools and committees are divided into four divisions: Biological Sciences, Humanities, Physical Sciences, and Social Sciences, and eight professional schools. In

9240-468: The University of Chicago consists of 217 acres (87.8 ha) in the Chicago neighborhoods of Hyde Park and Woodlawn , approximately eight miles (13 km) south of downtown Chicago . The northern and southern portions of campus are separated by the Midway Plaisance , a large, linear park created for the 1893 World's Columbian Exposition . In 2011, Travel+Leisure listed the university as one of

9372-462: The University of Chicago. Harper recruited acclaimed Yale baseball and football player Amos Alonzo Stagg from the Young Men's Christian Association training school at Springfield to coach the school's football program. Stagg was given a position on the faculty, the first such athletic position in the United States. While coaching at the university, Stagg invented the numbered football jersey and

9504-461: The University of Chicago. A student or faculty member of an affiliated school was entitled to free tuition at the University of Chicago, and Chicago students were eligible to attend an affiliated school on the same terms and receive credit for their work. The University of Chicago also agreed to provide affiliated schools with books and scientific apparatus and supplies at cost; special instructors and lecturers without cost except for travel expenses; and

9636-468: The ashes, a reference to the fire and foreclosure of the Old University of Chicago. A single stone from the rubble of the original Douglas Hall on 34th Place was set into the wall of the Classics Building. The dean of the college and University of Chicago and professor of history John Boyer has argued that the University of Chicago has "a plausible genealogy as a pre–Civil War institution". Alumni from

9768-420: The astonishing discovery that the interior of the black hole looks like a " wormhole " connecting two identical, asymptotically flat universes. This was the first real example of a wormhole solution in general relativity. The wormhole collapses to a singularity before any observer or signal can travel from one universe to the other. This is now believed to be the general fate of wormholes in general relativity. In

9900-741: The autumn quarter of 2022, the university enrolled 10,546 graduate students on degree-seeking courses: 569 in the Biological Sciences Division, 612 in the Humanities Division, 2,103 in the Physical Sciences Division, 972 in the Social Sciences Division, and 6,290 in the professional schools (including the Graham School). The university is home to several committees for interdisciplinary scholarship, including

10032-549: The board of trustees and donor of the Ryerson Physical Laboratory) Adolphus Clay Bartlett and Leon Mandel, who funded the construction of the gymnasium and assembly hall, and George C. Walker of the Walker Museum, a relative of Cobb who encouraged his inaugural donation for facilities. The new university acknowledged its predecessor. The university's coat of arms has a phoenix rising from

10164-453: The case of the white hole) would remain on the horizon forever. Thus the two black hole horizons coincide with the boundaries of the future light cone of an event at the center of the diagram (at T = X =0), while the two white hole horizons coincide with the boundaries of the past light cone of this same event. Any event inside the black hole interior region will have a future light cone that remains in this region (such that any world line within

10296-748: The coordinate line segment T = − X ,   T > 0 , X < 0 {\displaystyle T=-X,\ T>0,X<0} with the limit t (I) = − ∞ {\displaystyle t^{\text{(I)}}=-\infty } of the exterior region I corresponding to T = − X ,   T < 0 , X > 0 {\displaystyle T=-X,\ T<0,X>0} . The identification does mean that whereas each pair ( T , X ) ∼ ( − T , − X ) ≠ ( 0 , 0 ) {\displaystyle (T,X)\sim (-T,-X)\neq (0,0)} corresponds to

10428-527: The deed. Their three children are Karen (an attorney), Kerry (an author of children's books), and Clyde , a computer scientist. Martin Kruskal's scientific interests covered a wide range of topics in pure mathematics and applications of mathematics to the sciences. He had lifelong interests in many topics in partial differential equations and nonlinear analysis and developed fundamental ideas about asymptotic expansions , adiabatic invariants , and numerous related topics. His Ph.D. dissertation, written under

10560-437: The direction of Richard Courant and Bernard Friedman at New York University , was on the topic "The Bridge Theorem For Minimal Surfaces ". He received his Ph.D. in 1952. In the 1950s and early 1960s, he worked largely on plasma physics, developing many ideas that are now fundamental in the field. His theory of adiabatic invariants was important in fusion research. Important concepts of plasma physics that bear his name include

10692-617: The discovery of integrability of the KdV equation "exhibited in the most beautiful way the unity of mathematics. It involved developments in computation, and in mathematical analysis, which is the traditional way to study differential equations. It turns out that one can understand the solutions to these differential equations through certain very elegant constructions in algebraic geometry . The solutions are also intimately related to representation theory , in that these equations turn out to have an infinite number of hidden symmetries. Finally, they relate back to problems in elementary geometry." In

10824-756: The downtown Streeterville neighborhood of Chicago. The Center in Paris, a campus located on the left bank of the Seine in Paris, hosts various undergraduate and graduate study programs. In fall 2010, the university opened a center in Beijing, near Renmin University 's campus in Haidian District . The most recent additions are a center in New Delhi , India, which opened in 2014, and a center in Hong Kong which opened in 2018. In 2024,

10956-436: The event's future light cone will eventually hit the black hole singularity, which appears as a hyperbola bounded by the two black hole horizons), and any event inside the white hole interior region will have a past light cone that remains in this region (such that any world line within this past light cone must have originated in the white hole singularity, a hyperbola bounded by the two white hole horizons). Note that although

11088-413: The exact solution of any important class of nonlinear differential equations". The AMS added, "In applications of mathematics, solitons and their descendants (kinks, anti-kinks, instantons, and breathers) have entered and changed such diverse fields as nonlinear optics, plasma physics, and ocean, atmospheric, and planetary sciences. Nonlinearity has undergone a revolution: from a nuisance to be eliminated, to

11220-497: The first singularity which occurs at T 2 − X 2 = 1 {\displaystyle T^{2}-X^{2}=1} . Thus the above metric is a solution of Einstein's equations throughout this region. The allowed values are Note that this extension assumes that the solution is analytic everywhere. In the maximally extended solution there are actually two singularities at r = 0, one for positive T and one for negative T . The negative T singularity

11352-454: The global top 10 for eleven consecutive years (from 2012 to 2022). The university's law and business schools rank among the top three professional schools in the United States. The business school is currently ranked first in the US by US News & World Report and first in the world by The Economist , while the law school is ranked third by US News & World Report and first by Above

11484-542: The horizon looks as though it is an outward expanding cone, the area of this surface, given by r is just 16 π M 2 {\displaystyle 16\pi M^{2}} , a constant. I.e., these coordinates can be deceptive if care is not exercised. It may be instructive to consider what curves of constant Schwarzschild coordinate would look like when plotted on a Kruskal–Szekeres diagram. It turns out that curves of constant r -coordinate in Schwarzschild coordinates always look like hyperbolas bounded by

11616-527: The huddle. Stagg is the namesake of the university's Stagg Field . The business school was founded in 1898, and the law school was founded in 1902. Harper died in 1906 and was replaced by a succession of three presidents whose tenures lasted until 1929. During this period, the Oriental Institute was founded to support and interpret archeological work in what was then called the Near East. In

11748-549: The inverse scattering method in terms of isospectral deformations and Lax pairs . The inverse scattering method has had an astonishing variety of generalizations and applications in different areas of mathematics and physics. Kruskal himself pioneered some of the generalizations, such as the existence of infinitely many conserved quantities for the sine-Gordon equation . This led to the discovery of an inverse scattering method for that equation by M. J. Ablowitz , D. J. Kaup, A. C. Newell, and H. Segur (AKNS). The sine-Gordon equation

11880-502: The late 1970s, and investigated by Kruskal with great tenacity, is whether sufficiently well behaved surreal functions possess definite integrals. This question was answered negatively in the full generality, for which Conway et al. had hoped, by Costin, Friedman and Ehrlich in 2015. However, the analysis of Costin et al. shows that definite integrals do exist for a sufficiently broad class of surreal functions for which Kruskal's vision of asymptotic analysis, broadly conceived, goes through. At

12012-655: The late 19th century, the University of Chicago quickly became established as one of the wealthiest and, according to Henry S. Webber, one of the most prestigious universities in America. To elevate higher education standards and practices, the university was a founder of the Association of American Universities in 1900. According to Jonathan R. Cole , universities such as Chicago leveraged endowments to fund research, attracting accomplished faculty and producing academic advancements, leading to sustained growth and maintenance of their institutional profile such that Chicago has been among

12144-526: The literature, the Kruskal–Szekeres coordinates sometimes also appear in their lightcone variant: in which the metric is given by and r is defined implicitly by the equation These lightcone coordinates have the useful feature that radially outgoing null geodesics are given by U = constant {\displaystyle U={\text{constant}}} , while radially ingoing null geodesics are given by V = constant {\displaystyle V={\text{constant}}} . Furthermore,

12276-571: The mathematics department of Rutgers University , holding the David Hilbert Chair of Mathematics. Apart from serious mathematical work, Kruskal was known for mathematical diversions. For example, he invented the Kruskal count , a magical effect that has been known to perplex professional magicians because it was based not on sleight of hand but on a mathematical phenomenon. Martin David Kruskal

12408-459: The metric (the Riemannian metric of the 2-sphere) is: Expressing the metric in this form shows clearly that radial null geodesics i.e. with constant Ω = Ω ( θ , ϕ ) {\displaystyle \Omega =\Omega (\theta ,\phi )} are parallel to one of the lines T = ± X {\displaystyle T=\pm X} . In

12540-535: The metric is perfectly well defined and non-singular at the event horizon. The curvature singularity is located at T 2 − X 2 = 1 {\displaystyle T^{2}-X^{2}=1} . The transformation between Schwarzschild coordinates and Kruskal–Szekeres coordinates defined for r  > 2 GM and − ∞ < t < ∞ {\displaystyle -\infty <t<\infty } can be extended, as an analytic function, at least to

12672-467: The most beautiful college campuses in the United States. The first buildings of the campus, which make up what is now known as the Main Quadrangles, were part of a master plan conceived by two University of Chicago trustees and plotted by Chicago architect Henry Ives Cobb . The Main Quadrangles consist of six quadrangles , each surrounded by buildings, bordering one larger quadrangle. The buildings of

12804-521: The most distinguished research universities in the US for more than a century. The university is described by the Encyclopedia Britannica as "one of the United States' most outstanding universities". ARWU has consistently placed the University of Chicago among the top 10 universities in the world, and the 2021 QS World University Rankings placed the university in 9th place worldwide. THE World University Rankings has ranked it among

12936-498: The number of required courses from 21 to 15. When The New York Times , The Economist , and other major news outlets picked up this story, the university became the focal point of a national debate on education. The changes were ultimately implemented, but the controversy played a role in Sonnenschein's decision to resign in 2000. From the mid-2000s, the university began a number of multimillion-dollar expansion projects. In 2008,

13068-412: The opposite Kruskal–Szekeres coordinates cannot be reduced to a null loop. Kruskal–Szekeres coordinates have a number of useful features which make them helpful for building intuitions about the Schwarzschild spacetime. Chief among these is the fact that all radial light-like geodesics (the world lines of light rays moving in a radial direction) look like straight lines at a 45-degree angle when drawn in

13200-446: The past. This is just an artifact of how Schwarzschild coordinates are defined; a free-falling particle will only take a finite proper time (time as measured by its own clock) to pass between an outside observer and an event horizon, and if the particle's world line is drawn in the Kruskal–Szekeres diagram this will also only take a finite coordinate time in Kruskal–Szekeres coordinates. The Schwarzschild coordinate system can only cover

13332-416: The president are the provost, fourteen vice presidents (including the chief financial officer, chief investment officer , and vice president for campus life and student services), the directors of Argonne National Laboratory and Fermilab , the secretary of the university, and the student ombudsperson . As of May 2022, the current chairman of the board of trustees is David Rubenstein . The current provost

13464-440: The quantum process of Hawking radiation is unitary , 't Hooft proposed that the regions I and III, and II and IV are just mathematical artefacts coming from choosing branches for roots rather than parallel universes and that the equivalence relation should be imposed, where − Ω {\displaystyle -\Omega } is the antipode of Ω {\displaystyle \Omega } on

13596-401: The region internal to the black hole 0 < T 2 − X 2 < 1 ,   T > 0 {\displaystyle 0<T^{2}-X^{2}<1,\ T>0} In these new coordinates the metric of the Schwarzschild black hole manifold is given by written using the (− + + +) metric signature convention and where the angular component of

13728-457: The required information about their solutions. The first result was an asymptotic study of the Painlevé equations with Nalini Joshi , unusual at the time in that it did not require the use of associated linear problems. His persistent questioning of classical results led to a direct and simple method, also developed with Joshi, to prove the Painlevé property of the Painlevé equations. In the later part of his career, one of Kruskal's chief interests

13860-401: The square root as positive). A similar transformation can be written down in the other two regions. The Schwarzschild time coordinate t is given by In each region it runs from − ∞ {\displaystyle -\infty } to + ∞ {\displaystyle +\infty } with the infinities at the event horizons. Based on the requirements that

13992-426: The student movement. In 1978, history scholar Hanna Holborn Gray , then the provost and acting president of Yale University , became president of the University of Chicago, a position she held for 15 years. She was the first woman in the United States to hold the presidency of a major university. In 1999, then-President Hugo Sonnenschein announced plans to relax the university's famed core curriculum , reducing

14124-558: The time of his death, Kruskal was in the process of writing a book on surreal analysis with O. Costin. Kruskal coined the term asymptotology to describe the "art of dealing with applied mathematical systems in limiting cases". He formulated seven Principles of Asymptotology: 1. The Principle of Simplification; 2. The Principle of Recursion; 3. The Principle of Interpretation; 4. The Principle of Wild Behaviour; 5. The Principle of Annihilation; 6. The Principle of Maximal Balance; 7. The Principle of Mathematical Nonsense. The term asymptotology

14256-421: The topology of the underlying manifold is no longer R 4 − l i n e = R 2 × S 2 {\displaystyle \mathbb {R} ^{4}-\mathrm {line} =\mathbb {R} ^{2}\times S^{2}} . The manifold is no longer simply connected , because a loop (involving superluminal portions) going from a point in space-time back to itself but at

14388-411: The two interior regions. There are actually multiple possible ways to extend the exterior Schwarzschild solution into a maximally extended spacetime, but the Kruskal–Szekeres extension is unique in that it is a maximal, analytic , simply connected vacuum solution in which all maximally extended geodesics are either complete or else the curvature scalar diverges along them in finite affine time. In

14520-451: The university agreed to pay $ 13.5 million to settle a lawsuit that it and other universities conspired to limit financial aid to students. The academic bodies of the University of Chicago consist of the College , four divisions of graduate research, seven professional schools, and the Graham School of Continuing Liberal and Professional Studies . The university also contains a library system,

14652-444: The university created its first school in three decades, the Pritzker School of Molecular Engineering . On April 29, 2024, students at the University of Chicago set up an encampment on the university's main quad as a part of the nationwide movement in support of Palestine at institutions of higher learning across the country. The encampment was cleared by the University of Chicago Police Department on May 7. The main campus of

14784-619: The university includes eight professional schools, which also house academic research: the Booth School of Business ; Crown Family School of Social Work, Policy, and Practice ; Divinity School ; Graham School of Continuing Liberal and Professional Studies ; Harris School of Public Policy ; Law School ; Pritzker School of Medicine ; and Pritzker School of Molecular Engineering . The university has additional campuses and centers in London, Paris, Beijing, Delhi, and Hong Kong, as well as in downtown Chicago. University of Chicago scholars have played

14916-529: The university opened the John W. Boyer Center in Paris, designed by architectural firm Studio Gang and nearly tripling the size of the Center in Paris which had opened in 2003. The university is governed by a board of trustees. The board of trustees oversees the long-term development and plans of the university and manages fundraising efforts, and is composed of 55 members including the university president. Directly beneath

15048-426: The university's Stagg Field . Advances in chemistry led to the "radiocarbon revolution" in the carbon-14 dating of ancient life and objects. The university research efforts include administration of Fermi National Accelerator Laboratory and Argonne National Laboratory , as well as the Marine Biological Laboratory . The university is also home to the University of Chicago Press , the largest university press in

15180-813: The university's Hyde Park neighborhood where they raised their children and where Barack Obama began his political career. Michelle Obama served as an administrator at the university and founded the university's Community Service Center. In November 2021, a university graduate was robbed and fatally shot on a sidewalk in a residential area in Hyde Park near campus; a total of three University of Chicago students were killed by gunfire incidents in 2021. These incidents prompted student protests and an open letter to university leadership signed by more than 300 faculty members. The university also maintains facilities apart from its main campus. The university's Booth School of Business maintains campuses in Hong Kong , London , and

15312-512: The university's fifth president, 30-year-old legal philosophy scholar Robert Maynard Hutchins , took office. The university underwent many changes during his 24-year tenure. Hutchins reformed the undergraduate college's liberal-arts curriculum known as the Common Core, organized the university's graduate work into four divisions, and eliminated varsity football from the university in an attempt to emphasize academics over athletics. During his term,

15444-511: The university's general education requirements, commonly known as the Core Curriculum. In 2012–2013, the Core classes at Chicago were limited to 17 courses, and are generally led by a full-time professor (as opposed to a teaching assistant ). As of the 2013–2014 school year, 15 courses and demonstrated proficiency in a foreign language are required under the Core. Undergraduate courses at

15576-521: The university's refusal to divest from South Africa in the 1980s and Darfur in the late 2000s. In 1969, more than 400 students, angry about the dismissal of a popular professor, Marlene Dixon , occupied the Administration Building for two weeks. After the sit-in ended, when Dixon turned down a one-year reappointment, 42 students were expelled and 81 were suspended, the most severe response to student occupations of any American university during

15708-467: The university, like Shimer College and 10 others, adopted an early entrant program that allowed very young students to attend college; also, students enrolled at Shimer were enabled to transfer automatically to the University of Chicago after their second year, having taken comparable or identical examinations and courses. The university experienced its share of student unrest during the 1960s, beginning in 1962 when then-freshman Bernie Sanders helped lead

15840-433: The university, to notify the university early of any contemplated faculty appointments or dismissals, to make no faculty appointment without the university's approval, and to send copies of examinations for suggestions. The University of Chicago agreed to confer a degree on any graduating senior from an affiliated school who made a grade of A for all four years, and on any other graduate who took twelve weeks additional study at

15972-532: The university—and a number of National Resource Centers , including the Center for Middle Eastern Studies . Chicago also operates or is affiliated with several research institutions apart from the university proper. The university manages Argonne National Laboratory , part of the United States Department of Energy 's national laboratory system, and co-manages Fermi National Accelerator Laboratory (Fermilab) ,

16104-551: The waves in the canal observed by Russell), but the essential properties of this equation were not understood until the work of Kruskal and his collaborators in the 1960s. Solitonic behavior suggested that the KdV equation must have conservation laws beyond the obvious conservation laws of mass, energy, and momentum. A fourth conservation law was discovered by Gerald Whitham and a fifth one by Kruskal and Zabusky. Several new conservation laws were discovered by hand by Robert Miura , who also showed that many conservation laws existed for

16236-702: Was a member of the Committee on Institutional Cooperation from 1946 through June 29, 2016, when the group's name was changed to the Big Ten Academic Alliance . The University of Chicago is not a member of the rebranded consortium, but will continue to be a collaborator. The university operates more than 140 research centers and institutes on campus. Among these are the Institute for the Study of Ancient Cultures, West Asia & North Africa —a museum and research center for Near Eastern studies owned and operated by

16368-673: Was born to a Jewish family in New York City and grew up in New Rochelle . He was generally known as Martin to the world and David to his family. His father, Joseph B. Kruskal Sr., was a successful fur wholesaler. His mother, Lillian Rose Vorhaus Kruskal Oppenheimer , became a noted promoter of the art of origami during the early era of television and founded the Origami Center of America in New York City, which later became OrigamiUSA. He

16500-475: Was found that in addition to the black hole interior region (region II) which particles enter when they fall through the event horizon from the exterior (region I), there has to be a separate white hole interior region (region IV) which allows us to extend the trajectories of particles which an outside observer sees rising up away from the event horizon, along with a separate exterior region (region III) which allows us to extend some possible particle trajectories in

16632-415: Was founded by a small group of Baptist educators in 1856 through a land endowment from Senator Stephen A. Douglas . It closed in 1886 after years of financial struggle and a final annus horribilis in which the campus was badly damaged by fire and the school was foreclosed on by its creditors. Several years later, its trustees elected to change the school's name to the " Old University of Chicago " so that

16764-676: Was one of five children. His two brothers, both eminent mathematicians, were Joseph Kruskal (1928–2010; discoverer of multidimensional scaling , the Kruskal tree theorem , and Kruskal's algorithm ) and William Kruskal (1919–2005; discoverer of the Kruskal–Wallis test). Martin Kruskal's wife, Laura Kruskal, was a lecturer and writer about origami and originator of many new models. They were married for 56 years. Martin Kruskal also invented several origami models including an envelope for sending secret messages. The envelope could be easily unfolded, but it could not then be easily refolded to conceal

16896-446: Was stipulated that such money could not be used for buildings. The Hyde Park campus was financed by donations from wealthy Chicagoans such as Silas B. Cobb , who provided the funds for the campus's first building, Cobb Lecture Hall , and matched Marshall Field's pledge of $ 100,000. Other early benefactors included businessmen Charles L. Hutchinson (trustee, treasurer and donor of Hutchinson Commons ), Martin A. Ryerson (president of

17028-463: Was the inverse scattering method , a surprising and elegant method that demonstrates that the KdV equation admits an infinite number of Poisson-commuting conserved quantities and is completely integrable. This discovery gave the modern basis for understanding of the soliton phenomenon: the solitary wave is recreated in the outgoing state because this is the only way to satisfy all of the conservation laws. Soon after GGKM, Peter Lax famously interpreted

17160-533: Was the first independent sociology department in the United States and gave birth to the Chicago school of sociology . In physics, the university was the site of the Chicago Pile-1 (the first controlled, self-sustaining human-made nuclear chain reaction, part of the Manhattan Project ), of Robert Millikan 's oil-drop experiment that calculated the charge of the electron, and of the development of radiocarbon dating by Willard F. Libby in 1947. The chemical experiment that tested how life originated on early Earth,

17292-518: Was the theory of surreal numbers . Surreal numbers, which are defined constructively, have all the basic properties and operations of the real numbers. They include the real numbers alongside many types of infinities and infinitesimals. Kruskal contributed to the foundation of the theory, to defining surreal functions, and to analyzing their structure. He discovered a remarkable link between surreal numbers, asymptotics, and exponential asymptotics. A major open question, raised by Conway, Kruskal and Norton in

17424-418: Was valued at $ 10 billion. Since 2016, the university's board of trustees has resisted pressure from students and faculty to divest its investments from fossil fuel companies. Part of former university President Zimmer's financial plan for the university was an increase in accumulation of debt to finance large building projects. This drew both support and criticism from many in the university community. In 2023

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