Paul Alexander Schweitzer SJ (born July 21, 1937) is an American mathematician specializing in

differential topology In mathematics, differential topology is the field dealing with the topological properties and smooth properties of smooth manifolds. In this sense differential topology is distinct from the closely related field of differential geometry, which ...

geometric topology In mathematics, geometric topology is the study of manifolds and maps between them, particularly embeddings of one manifold into another. History Geometric topology as an area distinct from algebraic topology may be said to have originated i ...

, and

algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify topological spaces up t ...

. Schweitzer has done research on

foliation In mathematics (differential geometry), a foliation is an equivalence relation on an ''n''-manifold, the equivalence classes being connected, injectively immersed submanifolds, all of the same dimension ''p'', modeled on the decomposition of ...

knot theory In the mathematical field of topology, knot theory is the study of knot (mathematics), mathematical knots. While inspired by knots which appear in daily life, such as those in shoelaces and rope, a mathematical knot differs in that the ends are ...

, and

3-manifold In mathematics, a 3-manifold is a space that locally looks like Euclidean 3-dimensional space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane to a small enough observer, all 3-manifolds lo ...

s. In 1974 he found a counterexample to the

Seifert conjecture In mathematics, the Seifert conjecture states that every nonsingular, continuous vector field on the 3-sphere has a closed orbit. It is named after Herbert Seifert. In a 1950 paper, Seifert asked if such a vector field exists, but did not phrase ...

that every non-vanishing vector field on the 3-sphere has a closed integral curve. In 1995 he demonstrated that Sergei Novikov's compact leaf theorem cannot be generalized to manifolds with dimension greater than 3. Specifically, Schweitzer proved that a smooth, compact, connected manifold with

Euler characteristic In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space ...

zero and dimension > 3 has a ''C''¹ codimension-one

that has no compact leaf.

Education and career

Schweitzer studied at the

College of the Holy Cross The College of the Holy Cross is a private, Jesuit liberal arts college in Worcester, Massachusetts, about 40 miles (64 km) west of Boston. Founded in 1843, Holy Cross is the oldest Catholic college in New England and one of the oldest ...

Worcester, Massachusetts Worcester ( , ) is a city and county seat of Worcester County, Massachusetts, United States. Named after Worcester, England, the city's population was 206,518 at the 2020 United States census, 2020 census, making it the second-List of cities i ...

, with a B.S. in 1958 and then received his Ph.D. in 1962 at

Princeton University Princeton University is a private university, private research university in Princeton, New Jersey. Founded in 1746 in Elizabeth, New Jersey, Elizabeth as the College of New Jersey, Princeton is the List of Colonial Colleges, fourth-oldest ins ...

under

Norman Steenrod Norman Earl Steenrod (April 22, 1910October 14, 1971) was an American mathematician most widely known for his contributions to the field of algebraic topology. Life He was born in Dayton, Ohio, and educated at Miami University and University of ...

with thesis ''Secondary cohomology operations induced by the diagonal mapping''. He received a degree in philosophy (Ph.L.) in 1966 from

Weston College Weston College of Further and Higher Education is a general college of further and higher education in Weston-super-Mare, Somerset, England. It provides education and vocational training from age 14 to adult. The college provided education to ...

Weston, Massachusetts Weston is a town in Middlesex County, Massachusetts, Middlesex County, Massachusetts, about 15 miles west of Boston. At the time of the 2020 United States Census, the population of Weston was 11,851. Weston was incorporated in 1713, and protectio ...

, and a bachelor's degree in 1970 in theology (B. Div.) from the

Weston Jesuit School of Theology The Boston College School of Theology and Ministry (STM) is a Jesuit school of graduate theology at Boston College. It is an ecclesiastical faculty of theology that trains men and women, both lay and religious, for scholarship and service, espe ...

Cambridge, Massachusetts Cambridge ( ) is a city in Middlesex County, Massachusetts, United States. As part of the Boston metropolitan area, the cities population of the 2020 U.S. census was 118,403, making it the fourth most populous city in the state, behind Boston, ...

and was ordained in 1970 as a Catholic priest. In 1963 he became a member of the

Jesuits The Society of Jesus ( la, Societas Iesu; abbreviation: SJ), also known as the Jesuits (; la, Iesuitæ), is a religious order (Catholic), religious order of clerics regular of pontifical right for men in the Catholic Church headquartered in Rom ...

. He became in 1971 a professor extraordinarius and in 1980 a professor ordinarius at the

Pontifical Catholic University of Rio de Janeiro The Pontifical Catholic University of Rio de Janeiro ( pt, Pontifícia Universidade Católica do Rio de Janeiro, PUC-Rio) is a Jesuit, Catholic, pontifical university in Rio de Janeiro, Brazil. It is the joint responsibility of the Catholic Ar ...

. Schweitzer has been a visiting professor at the

University of Notre Dame The University of Notre Dame du Lac, known simply as Notre Dame ( ) or ND, is a private Catholic research university in Notre Dame, Indiana, outside the city of South Bend. French priest Edward Sorin founded the school in 1842. The main campu ...

, the

Fairfield University Fairfield University is a private Jesuit university in Fairfield, Connecticut. It was founded by the Jesuits in 1942. In 2017, the university had about 4,100 full-time undergraduate students and 1,100 graduate students, including full-time ...

Northwestern University Northwestern University is a private research university in Evanston, Illinois. Founded in 1851, Northwestern is the oldest chartered university in Illinois and is ranked among the most prestigious academic institutions in the world. Charte ...

Boston College Boston College (BC) is a private Jesuit research university in Chestnut Hill, Massachusetts. Founded in 1863, the university has more than 9,300 full-time undergraduates and nearly 5,000 graduate students. Although Boston College is classifie ...

Harvard University Harvard University is a private Ivy League research university in Cambridge, Massachusetts. Founded in 1636 as Harvard College and named for its first benefactor, the Puritan clergyman John Harvard, it is the oldest institution of higher le ...

, and the

University of Strasbourg The University of Strasbourg (french: Université de Strasbourg, Unistra) is a public research university located in Strasbourg, Alsace, France, with over 52,000 students and 3,300 researchers. The French university traces its history to the ea ...

. For the academic years 1970–1971 and 1981–1982 he was at the

Institute for Advanced Study The Institute for Advanced Study (IAS), located in Princeton, New Jersey, in the United States, is an independent center for theoretical research and intellectual inquiry. It has served as the academic home of internationally preeminent scholar ...

. Since 1978 he has been on the board of the

Brazilian Mathematical Society The Brazilian Mathematical Society ( pt, Sociedade Brasileira de Matemática, SBM) is a professional association founded in 1969 at Instituto de Matemática Pura e Aplicada to promote mathematics education in Brazil. Presidents *1969–1971 ...

. He was elected a Fellow of the

American Mathematical Society The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, ...

in 2012. He was an Invited Speaker at the ICM in 1974 in

Vancouver Vancouver ( ) is a major city in western Canada, located in the Lower Mainland region of British Columbia. As the List of cities in British Columbia, most populous city in the province, the 2021 Canadian census recorded 662,248 people in the ...

References

External links

Paul A. Schweitzer, S.J., Full Professor, Dept. of Mathematics, PUC-Rio
{{DEFAULTSORT:Schweitzer, Paul A 1937 births 20th-century American mathematicians 21st-century American mathematicians 20th-century American Jesuits 21st-century American Jesuits Topologists College of the Holy Cross alumni Princeton University alumni Boston College School of Theology and Ministry alumni Academic staff of the Pontifical Catholic University of Rio de Janeiro Fellows of the American Mathematical Society Living people People from Yonkers, New York Mathematicians from New York (state)