Donald G. Saari | |
---|---|
Born | March 1940 (age 84) |
Nationality | American |
Alma mater | |
Awards |
|
Scientific career | |
Fields | |
Institutions | |
Thesis | Singularities of the n-Body Problem of Celestial Mechanics (1967) |
Doctoral advisor | Harry Pollard |
Doctoral students |
Donald Gene Saari (born March 1940) is an American mathematician, a Distinguished Professor of Mathematics and Economics and former director of the Institute for Mathematical Behavioral Sciences at the University of California, Irvine. His research interests include the n-body problem, the Borda count voting system, and application of mathematics to the social sciences.
Saari has been widely quoted as an expert in voting systems [1] and lottery odds. [2] He is opposed to the use of the Condorcet criterion in evaluating voting systems, [3] and among positional voting schemes he favors using the Borda count over plurality voting, because it reduces the frequency of paradoxical outcomes (which however cannot be avoided entirely in ranking systems because of Arrow's impossibility theorem). [4] For instance, as he has pointed out, plurality voting can lead to situations where the election outcome would remain unchanged if all voters' preferences were reversed; this cannot happen with the Borda count. [5] Saari has defined, as a measure of the inconsistency of a voting method, the number of different combinations of outcomes that would be possible for all subsets of a field of candidates. According to this measure, the Borda count is the least inconsistent possible positional voting scheme, while plurality voting is the most inconsistent. [3] However, other voting theorists such as Steven Brams, while agreeing with Saari that plurality voting is a bad system, disagree with his advocacy of the Borda count, because it is too easily manipulated by tactical voting. [4] [6] Saari also applies similar methods to a different problem in political science, the apportionment of seats to electoral districts in proportion to their populations. [3] He has written several books on the mathematics of voting. [S94] [S95a] [S01a] [S01b] [S08]
In economics, Saari has shown that natural price mechanisms that set the rate of change of the price of a commodity proportional to its excess demand can lead to chaotic behavior rather than converging to an economic equilibrium, and has exhibited alternative price mechanisms that can be guaranteed to converge. However, as he also showed, such mechanisms require that the change in price be determined as a function of the whole system of prices and demands, rather than being reducible to a computation over pairs of commodities. [SS] [S85] [S95b]
In celestial mechanics, Saari's work on the n-body problem "revived the singularity theory" of Henri Poincaré and Paul Painlevé, and proved Littlewood's conjecture that the initial conditions leading to collisions have measure zero. [7] He also formulated the "Saari conjecture", that when a solution to the Newtonian n-body problem has an unchanging moment of inertia relative to its center of mass, its bodies must be in relative equilibrium. [8] More controversially, Saari has taken the position that anomalies in the rotation speeds of galaxies, discovered by Vera Rubin, can be explained by considering more carefully the pairwise gravitational interactions of individual stars instead of approximating the gravitational effects of a galaxy on a star by treating the rest of the galaxy as a continuous mass distribution (or, as Saari calls it, "star soup"). In support of this hypothesis, Saari showed that simplified mathematical models of galaxies as systems of large numbers of bodies arranged symmetrically on circular shells could be made to form central configurations that rotate as a rigid body rather than with the outer bodies rotating at the speed predicted by the total mass interior to them. According to his theories, neither dark matter nor modifications to the laws of gravitational force are needed to explain galactic rotation speeds. However, his results do not rule out the existence of dark matter, as they do not address other evidence for dark matter based on gravitational lenses and irregularities in the cosmic microwave background. [9] His works in this area include two more books. [SX] [S05]
Overviewing his work in these diverse areas, Saari has argued that his contributions to them are strongly related. In his view, Arrow's impossibility theorem in voting theory, the failure of simple pricing mechanisms, and the failure of previous analysis to explain the speeds of galactic rotation stem from the same cause: a reductionist approach that divides a complex problem (a multi-candidate election, a market, or a rotating galaxy) into multiple simpler subproblems (two-candidate elections for the Condorcet criterion, two-commodity markets, or the interactions between individual stars and the aggregate mass of the rest of the galaxy) but, in the process, loses information about the initial problem making it impossible to combine the subproblem solutions into an accurate solution to the whole problem. [S15] Saari credits some of his research success to a strategy of mulling over research problems on long road trips, without access to pencil or paper. [10]
Saari is also known for having some discussion with Theodore J. Kaczynski in 1978, prior to the mail bombings that led to Kaczynski's 1996 arrest. [11]
Saari grew up in a Finnish American copper mining community in the Upper Peninsula of Michigan, the son of two labor organizers there. Frequently in trouble for talking in his classes, he spent his detention time in private mathematics lessons with a local algebra teacher, Bill Brotherton. He was accepted to an Ivy League university, but his family could only afford to send him to the local state university, Michigan Technological University, which gave him a full scholarship. He majored in mathematics there, his third choice after previously trying chemistry and electrical engineering. [12] While attending Michigan Tech, Saari joined the Beta Chapter of Theta Tau Professional Engineering Fraternity.
He received his Bachelor of Science in Mathematics in 1962 from Michigan Tech, and his Master of Science and PhD in Mathematics from Purdue University in 1964 and 1967, respectively. [13] At Purdue, he began working with his doctoral advisor, Harry Pollard, because of a shared interest in pedagogy, but soon picked up Pollard's interests in celestial mechanics and wrote his doctoral dissertation on the n-body problem. [12]
After taking a temporary position at Yale University, he was hired at Northwestern University by Ralph P. Boas Jr., who had also been doing similar work in celestial mechanics. [12] From 1968 to 2000, he served as assistant, associate, and full professor of mathematics at Northwestern, and eventually became Pancoe Professor of Mathematics there. [14] He was led to mathematical economics by discovering the high caliber of the economics students enrolling in his courses in functional analysis, [12] and added a second position as Professor of Economics. [14] He then moved to the University of California, Irvine at the invitation of R. Duncan Luce, who had founded the Institute for Mathematical Behavioral Sciences (IMBS) in the UCI School of Social Sciences in 1989. [12] At UC Irvine, he took over the directorship of the IMBS in 2003, and stepped down as director in 2017. [15] He is a trustee of the Mathematical Sciences Research Institute. [16]
He was editor in chief of the Bulletin of the American Mathematical Society from 1998 to 2005, [17] and published a book on the early history of the journal. [S03]
S94. | Geometry of Voting, Studies in Economic Theory 3, Springer-Verlag, 1994.
|
S95a. |
S01a. | Chaotic Elections! A Mathematician Looks at Voting, American Mathematical Society, 2001.
|
S01b. | Decisions and Elections; Explaining the Unexpected, Cambridge University Press, 2001.
|
S05. | Collisions, Rings, and Other Newtonian N-Body Problems, American Mathematical Society, 2005.
|
S08. | Disposing Dictators, Demystifying Voting Paradoxes: Social Choice Analysis, Cambridge University Press, 2008.
|
SX. | Hamiltonian Dynamics and Celestial Mechanics (with Z. Xia), Contemporary Mathematics 198, American Mathematical Society, 1996. |
S03. | The Way it Was: Mathematics From the Early Years of the Bulletin, American Mathematical Society, 2003.
|
SS. | Saari, Donald G.; Simon, Carl P. (1978), "Effective price mechanisms" (PDF), Econometrica , 46 (5): 1097–1125, doi:10.2307/1911438, JSTOR 1911438 . |
SU. | Saari, Donald G.; Urenko, John B. (1984), "Newton's method, circle maps, and chaotic motion", American Mathematical Monthly , 91 (1): 3–17, doi:10.2307/2322163, JSTOR 2322163
|
S85. | Saari, Donald G. (1985), "Iterative price mechanisms", Econometrica , 53 (5): 1117–1131, doi:10.2307/1911014, JSTOR 1911014 . |
S90. | Saari, Donald G. (1990), "A Visit to the Newtonian N-body problem via elementary complex variables", American Mathematical Monthly , 97 (2): 105–119, doi:10.2307/2323910, JSTOR 2323910
|
S95b. | Saari, Donald (1995), "Mathematical complexity of simple economics", Notices of the American Mathematical Society , 42 (2): 222–230. |
SV. | Saari, Donald G.; Valognes, Fabrice (1998), "Geometry, voting, and paradoxes", Mathematics Magazine , 71 (4): 243–259, doi:10.2307/2690696, JSTOR 2690696
|
S15. | Saari, Donald G. (2015), "From Arrow's Theorem to 'Dark Matter'", British Journal of Political Science , 46 (1): 1–9, doi:10.1017/s000712341500023x, S2CID 154799988 |
The following outline is provided as an overview of and topical guide to physics:
In physics, gravity (from Latin gravitas 'weight') is a fundamental interaction primarily observed as mutual attraction between all things that have mass. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the strong interaction, 1036 times weaker than the electromagnetic force and 1029 times weaker than the weak interaction. As a result, it has no significant influence at the level of subatomic particles. However, gravity is the most significant interaction between objects at the macroscopic scale, and it determines the motion of planets, stars, galaxies, and even light.
Édouard Albert Roche was a French astronomer and mathematician, who is best known for his work in the field of celestial mechanics. His name was given to the concepts of the Roche sphere, Roche limit, and Roche lobe. He also was the author of works in meteorology.
Jules Henri Poincaré was a French mathematician, theoretical physicist, engineer, and philosopher of science. He is often described as a polymath, and in mathematics as "The Last Universalist", since he excelled in all fields of the discipline as it existed during his lifetime. Due to his scientific success, influence and his discoveries, he has been deemed "the philosopher par excellence of modern science."
Celestial mechanics is the branch of astronomy that deals with the motions of objects in outer space. Historically, celestial mechanics applies principles of physics to astronomical objects, such as stars and planets, to produce ephemeris data.
George William Hill was an American astronomer and mathematician. Working independently and largely in isolation from the wider scientific community, he made major contributions to celestial mechanics and to the theory of ordinary differential equations. The importance of his work was explicitly acknowledged by Henri Poincaré in 1905. In 1909 Hill was awarded the Royal Society's Copley Medal, "on the ground of his researches in mathematical astronomy". Hill is remembered for the Hill differential equation, along with the Hill sphere.
Forest Ray Moulton was an American astronomer. He was the brother of Harold G. Moulton, a noted economist.
In physics, specifically classical mechanics, the three-body problem involves taking the initial positions and velocities of three point masses that orbit each other in space and calculating their subsequent trajectories using Newton's laws of motion and Newton's law of universal gravitation.
Positional voting is a ranked voting electoral system in which the options or candidates receive points based on their rank position on each ballot and the one with the most points overall wins. The lower-ranked preference in any adjacent pair is generally of less value than the higher-ranked one. Although it may sometimes be weighted the same, it is never worth more. A valid progression of points or weightings may be chosen at will or it may form a mathematical sequence such as an arithmetic progression, a geometric one or a harmonic one. The set of weightings employed in an election heavily influences the rank ordering of the candidates. The steeper the initial decline in preference values with descending rank, the more polarised and less consensual the positional voting system becomes.
Eric Stark Maskin is an American economist and mathematician. He was jointly awarded the 2007 Nobel Memorial Prize in Economic Sciences with Leonid Hurwicz and Roger Myerson "for having laid the foundations of mechanism design theory". He is the Adams University Professor and Professor of Economics and Mathematics at Harvard University.
The School of Social Sciences is an academic unit of the University of California, Irvine (UCI) that studies the social sciences. The School is the largest academic unit in the university with an enrollment of over 5,300 students. More than a third of the bachelor's degrees conferred at UCI are from the School of Social Sciences. It is home to the departments of Anthropology, Chicano-Latino Studies, Cognitive Science, Economics, Logic and Philosophy of Science, Political Science, International Studies, and Sociology.
The Borda method or order of merit is a positional voting rule which gives each candidate a number of points equal to the number of candidates ranked below them: the lowest-ranked candidate gets 0 points, the second-lowest gets 1 point, and so on. Once all votes have been counted, the option or candidate with the most points is the winner.
Florin Nicolae Diacu was a Romanian Canadian mathematician and author.
In physics, the n-body problem is the problem of predicting the individual motions of a group of celestial objects interacting with each other gravitationally. Solving this problem has been motivated by the desire to understand the motions of the Sun, Moon, planets, and visible stars. In the 20th century, understanding the dynamics of globular cluster star systems became an important n-body problem. The n-body problem in general relativity is considerably more difficult to solve due to additional factors like time and space distortions.
Harry Pollard was an American mathematician. He received his Ph.D from Harvard University in 1942 under the supervision of David Widder. He then taught at Cornell University, and was Professor of Mathematics at Purdue University from 1961 until his death in 1985. He is known for his work on celestial mechanics, orthogonal polynomials and the n-body problem as well as for the several textbooks he authored or co-authored. In the theory of Orthogonal polynomials, Pollard solved a conjecture of Antoni Zygmund, establishing mean convergence of the partial sums in norms for the Legendre polynomials and Jacobi polynomials in a series of three papers in the Transactions of the American Mathematical Society. The first of these papers deals with the fundamental case of Legendre polynomials. The end point cases in Pollard's theorem was established by Sagun Chanillo.
Richard Paul McGehee is an American mathematician, who works on dynamical systems with special emphasis on celestial mechanics.
Alain Chenciner is a French mathematician, specializing in dynamical systems with applications to celestial mechanics.
Zhihong "Jeff" Xia is a Chinese-American mathematician.
In celestial mechanics, a central configuration is a system of point masses with the property that each mass is pulled by the combined gravitational force of the system directly towards the center of mass, with acceleration proportional to its distance from the center. Central configurations are studied in n-body problems formulated in Euclidean spaces of any dimension, although only dimensions one, two, and three are directly relevant for celestial mechanics in physical space.
Frederic Yui-Ming Wan is a Chinese-American applied mathematician, academic, author and consultant. He is a Professor Emeritus of Mathematics at the University of California, Irvine (UCI), and an Affiliate Professor of Applied Mathematics at the University of Washington (UW).