Gerald Walter Schwarz (born February 15, 1946,

Portland, Oregon Portland (, ) is a port city in the Pacific Northwest and the largest city in the U.S. state of Oregon. Situated at the confluence of the Willamette and Columbia rivers, Portland is the county seat of Multnomah County, the most populou ...

, United States) is an American mathematician and Professor Emeritus at

Brandeis University Brandeis University is a Private university, private research university in Waltham, Massachusetts. Founded in 1948 as a nonsectarian, non-sectarian, coeducational institution sponsored by the Jews, Jewish community, Brandeis was established on t ...

. Schwarz specializes in

invariant theory Invariant theory is a branch of abstract algebra dealing with actions of groups on algebraic varieties, such as vector spaces, from the point of view of their effect on functions. Classically, the theory dealt with the question of explicit descri ...

, algebraic

group action In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphi ...

s and

invariant differential operator In mathematics and theoretical physics, an invariant differential operator is a kind of mathematical map from some objects to an object of similar type. These objects are typically functions on \mathbb^n, functions on a manifold, vector valued fu ...

Early life and education

Of German descent, Schwarz's father, Ernst, was one of the 30,000 Jews seized during

Kristallnacht () or the Night of Broken Glass, also called the November pogrom(s) (german: Novemberpogrome, ), was a pogrom against Jews carried out by the Nazi Party's Sturmabteilung, (SA) paramilitary and Schutzstaffel, (SS) paramilitary forces along ...

. He was imprisoned at the

Buchenwald Buchenwald (; literally 'beech forest') was a Nazi concentration camp established on hill near Weimar, Germany, in July 1937. It was one of the first and the largest of the concentration camps within Germany's 1937 borders. Many actual or su ...

concentration camp until his wife, Elaine, managed to secure a visa to travel abroad. Upon his release from the camp, the couple fled to England where Gerald's older brother, Maurice, was born. In November 1939, Ernst, Elaine and Maurice arrived in the United States, eventually settling in Portland, Oregon. Gerald was born seven years later. He spent his childhood in Portland, then moved to Cambridge, Massachusetts to attend school. Schwarz earned his B.S. and M.S. degrees from the

Massachusetts Institute of Technology The Massachusetts Institute of Technology (MIT) is a Private university, private Land-grant university, land-grant research university in Cambridge, Massachusetts. Established in 1861, MIT has played a key role in the development of modern t ...

(MIT) in 1969 and his Ph.D. in mathematics from MIT in 1972.

Career

Schwarz began his career at the

University of Pennsylvania The University of Pennsylvania (also known as Penn or UPenn) is a private research university in Philadelphia. It is the fourth-oldest institution of higher education in the United States and is ranked among the highest-regarded universit ...

(1972–74) as a

postdoctoral researcher A postdoctoral fellow, postdoctoral researcher, or simply postdoc, is a person professionally conducting research after the completion of their doctoral studies (typically a PhD). The ultimate goal of a postdoctoral research position is to pu ...

, then joined the faculty at Brandeis University in

Waltham, Massachusetts Waltham ( ) is a city in Middlesex County, Massachusetts, United States, and was an early center for the labor movement as well as a major contributor to the American Industrial Revolution. The original home of the Boston Manufacturing Company, ...

(1974). He spent the next academic year 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 ...

in Princeton, New Jersey (1975–76), where he recognized that the solution of the

homotopy In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a defor ...

/ isotopy lifting problem requires

algebraic group In mathematics, an algebraic group is an algebraic variety endowed with a group structure which is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory. ...

s. The resulting

theorem In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of ...

helps mathematicians classify smooth compact

lie group In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the addit ...

actions on

manifolds In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a ...

. The proof of the theorem appears in the paper ''Lifting smooth homotopies of orbit spaces'' and led to a

tenure Tenure is a category of academic appointment existing in some countries. A tenured post is an indefinite academic appointment that can be terminated only for cause or under extraordinary circumstances, such as financial exigency or program disco ...

d position at Brandeis in 1978. Four years later, Schwarz was promoted to full Professor. Schwarz has written or co-authored over 60 journal articles in the field of mathematics. In 1996, he was one of the founding editors of the journal ''Transformation Groups,'' and continued as one of its Managing Editors until February 2000. In 2012, he became a member of the inaugural class of fellows 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 ...

which recognizes mathematicians who have made significant contributions to the field.

Honors

*Poste Rouge, Centre National de Recherche Scientifiques (1996) *Invited Speaker,

International Congress of Mathematicians The International Congress of Mathematicians (ICM) is the largest conference for the topic of mathematics. It meets once every four years, hosted by the International Mathematical Union (IMU). The Fields Medals, the Nevanlinna Prize (to be rename ...

Zürich , neighboring_municipalities = Adliswil, Dübendorf, Fällanden, Kilchberg, Maur, Oberengstringen, Opfikon, Regensdorf, Rümlang, Schlieren, Stallikon, Uitikon, Urdorf, Wallisellen, Zollikon , twintowns = Kunming, San Francisco Zürich () i ...

(1994)Schwarz, Gerald W. "Invariant differential operators." In ''Proceedings of the International Congress of Mathematicians'' (Zürich, 1994), pp. 333–341. 1995. *Member, Institut des Hautes Études Scientifiques (1982) *Member, Institute for Advanced Study (1975)

Selected publications

*''Smooth functions invariant under the action of a compact Lie group''
Topology
14 (1975), 63–68. *''Representations of simple Lie groups with regular rings of invariants''
Inventiones mathematicae
49 (1978), 167–191. *''Lifting smooth homotopies of orbit spaces''
Publications Mathématiques de l'Institut des Hautes Études Scientifiques
51 (1980), 37–132. *(with C. Procesi) ''Inequalities defining orbit spaces'',
Inventiones mathematicae
81 (1985), 539–554. *''Invariant theory of G and Spin'',
Commentarii Mathematici Helvetici
(1988), 624–663. *(with H. Kraft) ''Reductive groups actions with one-dimensional quotient''
Publications Mathématiques de l'Institut des Hautes Études Scientifiques
76 (1993), 1-97. *(with D. Wehlau) ''Invariants of four subspaces'', Ann. Inst. Fourier, 48 No. 3 (1998) 667–697. *''Lifting differential operators from orbit spaces''
Annales scientifiques de l'École Normale Supérieure
Sup. 28 (1995), 253-305 *(with P. Heinzner) ''Cartan decomposition of the moment map''
Mathematische Annalen
337 (2007), 197–232. *(with L. Helminck) ''Real double coset spaces and their invariants''
Journal of Algebra
322 (2009), 219-236 *(with F. Kutzschebauch and F. Larusson) ''Sufficient Conditions for Holomorphic Linearisation'', (2015).

References

External links

The Mathematics Genealogy ProjectHome Page
{{DEFAULTSORT:Schwarz, Gerald Living people Scientists from Portland, Oregon Massachusetts Institute of Technology School of Science alumni 20th-century American mathematicians American people of German-Jewish descent Jewish American scientists Fellows of the American Mathematical Society 1946 births 21st-century American mathematicians Brandeis University faculty