Emily was an influential German mathematician known for her contributions to abstract algebra and theoretical physics. She revolutionized the theories of rings, fields, and algebras. Noethers theorem explains the fundamental connection between symmetry and conservation laws in physics. In 1924, Dutch mathematician B. L. van der Waerden joined her circle and soon became the leading expositor of Noether’s ideas: her work was the foundation for the second volume of his influential 1931 textbook, Moderne Algebra. The following year, Germany’s Nazi government dismissed Jews from university positions, and Noether moved to the United States to take up a position at Bryn Mawr College in Pennsylvania. In 1935 she underwent surgery for an ovarian cyst and, despite signs of a recovery, died four days later at the age of 53.
Noether’s mathematical work has been divided into three “epochs”. In the first (1908-19), she made significant contributions to the theories of algebraic invariants and number fields. In the second epoch (1920-26), she began work that “changed the face of [abstract] algebra”. In her classic paper Idealtheorie in Ringbereichen (Theory of Ideals in Ring Domains, 1921) Noether developed the theory of ideals in commutative rings into a powerful tool with wide-ranging applications. She made elegant use of the ascending chain condition, and objects satisfying it are named Noetherian in her honor. In the third epoch (1927-35), she published major works on noncommutative algebras and hypercomplex numbers and united the representation theory of groups with the theory of modules and ideals. Emily was a very noble person and credited those who helped her with her discoveries.
From 1913 to 1916 Noether published several papers extending and applying Hilbert’s methods to mathematical objects such as fields of rational functions and the invariants of finite groups. This phase marks the beginning of her engagement with abstract algebra, the field of mathematics to which she would make groundbreaking contributions. In the spring of 1915, Noether was invited to return to the University of Göttingen by David Hilbert and Felix Klein. Their effort to recruit her, however, was blocked by the philologists and historians among the philosophical faculty because they did not believe that woman could do the job. In 1915 David Hilbert invited Noether to join the Göttingen mathematics department, challenging the views of some of his colleagues that a woman should not be allowed to teach at a University. Noether left for Göttingen in late April; two weeks later her mother died suddenly in Erlangen. At about the same time Noether’s father retired and her brother joined the German Army to serve in World War I. Emily then returned to Erlangen to care for her father. During her first years teaching at Göttingen she did not have an official position and was not paid; her family paid for her room and board and supported her academic work. Her lectures often were advertised under Hilbert’s name, and Noether would provide “assistance”. Soon after arriving at Göttingen she demonstrated her capabilities by proving the theorem now known as Noether’s theorem, which shows that a conservation law is associated with any differentiable symmetry of a physical system.
When World War I ended, the German Revolution brought a huge change for woman’s rights. In 1919 the University of Göttingen allowed Noether to proceed with her habilitation. Three years later she received a letter from the Prussian Minister for Science, Art, and Public Education, in which he conferred on her the title of an untenured professor with limited internal administrative rights and functions. This was an unpaid professorship, not the higher “ordinary” professorship, which was a civil-service position. Although it recognized the importance of her work, the position still didn’t pay. Noether was not paid for her lectures until she was appointed to the special position of Lehrbeauftragte für Algebra a year later. Although Noether’s theorem had a profound effect upon physics, among mathematicians she is best remembered for her seminal contributions to abstract algebra. The development of abstract algebra, which is one of the most distinctive innovations of twentieth century mathematics, is largely due to her. Noether’s groundbreaking work in algebra began in 1920. In collaboration with W. Schmeidler, she then published a paper about the theory of ideals in which they defined left and right ideals in a ring. The following year she published a landmark paper called Idealtheorie in Ringbereichen, in which she analyzed the ascending chain conditions with regard to mathematical ideals.
In 1931 Waerden published Moderne Algebra in which its second volume borrowed heavily from Noether’s work. Although Emily did not seek recognition, he included as a note in the seventh edition “based in part on lectures by E. Artin and E. Noether”. Emily wouldn’t mind if students received credit for her ideas in fact she would gladly allow them so that it could help with careers and get them started. Because of these and other noble acts Noether was very well respected for her consideration that she had for others even when some would act rude to her and argue with her, however she handled the situations very well and gained a reputation for always being helpful and being patient with her students. At first her lifestyle wasn’t all that good because she was being denied pay for her work but eventually the University began paying her a small salary in 1923, but Emily continued to live a simple and modest life. She then got paid more generously later in her life, but saved half of her salary for her nephew, Gottfried E. Noether.
Emily wasn’t one to be so concerned about appearance or manners; she would rather be more focused on her studies than romance and fashion. Noether had unique ideas and would not follow a lesson plan for her lectures, which frustrated some students. Instead her lectures were spontaneous and based on the discussion the students had for that day. This actually benefited Emily because some of her most important results were developed in these lectures, and the lecture notes of her students formed the basis for several important textbooks, such as those of van der Waerden and Deuring. Several of her colleagues attended her lectures, and Emily would allow them to use some of her ideas, such as the crossed product of associative algebras, to be published by others.
Noether was recorded as having given at least five semester-long courses at Göttingen:
Winter 1924/25: Gruppentheorie und hyperkomplexe Zahlen (Group Theory and Hypercomplex Numbers)
Winter 1927/28: Hyperkomplexe Grössen und Darstellungstheorie (Hypercomplex Quantities and Representation Theory)
Summer 1928: Nichtkommutative Algebra (Noncommutative Algebra)
Summer 1929: Nichtkommutative Arithmetik (Noncommutative Arithmetic)
Winter 1929/30: Algebra der hyperkomplexen Grössen (Algebra of Hypercomplex Quantities).
These courses often preceded major publications in these areas.
Noether spoke very quickly reflecting the speed of her thoughts and demanded great concentration from her students. She had a bond with those colleagues and students who thought along similar lines to her, and often exclude those who did not. “Outsiders” who occasionally visited Noether’s lectures usually spent only 30 minutes in the room before leaving in frustration or confusion.
Noether taught at the Moscow State University during the winter of 1928-29. In the winter of 1928-29 Noether accepted an invitation to Moscow State University, where she continued working with P. S. Alexandrov. In addition to carrying on with her research, she taught classes in abstract algebra and algebraic geometry. Although politics was not central to her life, Noether took a keen interest in political matters andshowed considerable support for the Russian Revolution (1917). She was especially happy to see Soviet advancements in the fields of science and mathematics, which she considered indicative of new opportunities made possible by the Bolshevik project, however this attitude caused her problems in Germany. During the 1930s, and in 1935 she made plans for a return to the Soviet Union. Meanwhile her brother, Fritz accepted a position at the Research Institute for Mathematics and Mechanics in Tomsk, in the Siberian Federal District of Russia, after losing his job in Germany.
In 1932 Emmy Noether and Emil Artin received the Ackermann-Teubner Memorial Award for their contributions to mathematics. The prize carried a monetary reward of 500 Reichsmarks and was seen as a long-overdue official recognition of her considerable work in the field. Nevertheless, her colleagues expressed frustration at the fact that she was not elected to the Göttingen Gesellschaft der Wissenschaften (academy of sciences) and was never promoted to the position of Ordentlicher Professor.
Noether’s colleagues celebrated her fiftieth birthday in 1932, in typical mathematicians’ style. Helmut Hasse dedicated an article to her in the Mathematische Annalen, wherein he confirmed her suspicion that some aspects of non-commutative algebra are simpler than those of commutative algebra, by proving a non-commutative reciprocity law. This pleased her immensely. He also sent her a mathematical riddle, which she solved immediately; the riddle has been lost.
In November of the same year, Noether delivered a plenary address on “Hyper-complex systems in their relations to commutative algebra and to number theory” at the International Congress of Mathematicians in Zürich. Apparently, Noether’s prominent speaking position was a recognition of the importance of her contributions to mathematics. The 1932 congress is sometimes described as the high point of her career.
When Adolf Hitler became the German Reichskanzler in January 1933, Nazi activity around the country increased dramatically. At the University of Göttingen the German Student Association led the attack on the “un-German spirit” attributed to Jews and was aided by a Werner Weber, a former student of Emmy Noether. Antisemitic attitudes created a climate hostile to Jewish professors. One young protester reportedly demanded: “Aryan students want Aryan mathematics and not Jewish mathematics.”
One of the first actions of Hitler’s administration was the Law for the Restoration of the Professional Civil Service which removed Jews and politically suspect government employees from their jobs unless they had “demonstrated their loyalty to Germany” by serving in World War I. In April 1933 Noether received a notice from the Prussian Ministry for Sciences, Art, and Public Education which read: “On the basis of paragraph 3 of the Civil Service Code of 7 April 1933, I hereby withdraw from you the right to teach at the University of Göttingen.” Noether accepted the decision calmly, providing support for others during this difficult time. Typically, Noether remained focused on mathematics, gathering students in her apartment to discuss class field theory. When one of her students appeared in the uniform of the Nazi paramilitary organization Sturmabteilung (SA), she showed no sign of agitation and, reportedly, even laughed about it later.
As dozens of newly unemployed professors began searching for positions outside of Germany, their colleagues in the United States sought to provide assistance and job opportunities for them.. Noether was contacted by representatives of two educational institutions, Bryn Mawr College in the United States and Somerville College at the University of Oxford in England. After a series of negotiations with the Rockefeller Foundation, a grant to Bryn Mawr was approved for Noether and she took a position there, starting in late 1933.
In 1934, Noether began lecturing at the Institute for Advanced Study in Princeton upon the invitation of Abraham Flexner and Oswald Veblen. She also worked with and supervised Abraham Albert and Harry Vandiver. However, she remarked about Princeton University that she was not welcome at the “men’s university, where nothing female is admitted”.
Her time in the United States was pleasant, surrounded as she was by supportive colleagues and absorbed in her favorite subjects. In the summer of 1934 she briefly returned to Germany to see Emil Artin and her brother Fritz before he left for Tomsk.
Noether’s remains were placed under the walkway surrounding the cloisters of Bryn Mawr’s M. Carey Thomas Library
In April 1935 doctors discovered a tumor in Noether’s pelvis. Worried about complications from surgery, they ordered two days of bed rest first. During the operation they discovered an ovarian cyst “the size of a large cantaloupe”. Two smaller tumors in her uterus appeared to be benign and were not removed, to avoid prolonging surgery. For three days she appeared to convalesce normally, and she recovered quickly from a circulatory collapse on the fourth. On 14 April she fell unconscious, her temperature soared to 109 °F (42.8 °C), and she died. A few days after Noether’s death her friends and associates at Bryn Mawr held a small memorial service at College President Park’s house. Her body was cremated and the ashes interred under the walkway around the cloisters of the M. Carey Thomas Library at Bryn Mawr.
In the first epoch (1907-19), Noether dealt primarily with differential and algebraic invariants, beginning with her dissertation under Paul Gordan. Her mathematical horizons broadened, and her work became more general and abstract, as she became acquainted with the work of David Hilbert, through close interactions with a successor to Gordan, Ernst Sigismund Fischer. After moving to Göttingen in 1915, she produced her seminal work for physics, the two Noether’s theorems. In the second epoch (1920-26), Noether devoted herself to developing the theory of mathematical rings.
In the third epoch (1927-35), Noether focused on noncommutative algebra, linear transformations, and commutative number fields
A group consists of a set of elements and a single operation which combines a first and a second element and returns a third. The operation must satisfy certain constraints for it to determine a group: It must be closed (when applied to any pair of elements of the associated set, the generated element must also be a member of that set), it must be associative, there must be an identity element (an element which, when combined with another element using the operation, results in the original element, such as adding zero to a number or multiplying it by one), and for every element there must be an inverse element.
A ring likewise, has a set of elements, but now has two operations. The first operation must make the set a group, and the second operation is associative and distributive with respect to the first operation. It may or may not be commutative; this means that the result of applying the operation to a first and a second element is the same as to the second and first-the order of the elements does not matter. If every non-zero element has a multiplicative inverse (an element x such that ax = xa = 1), the ring is called a division ring. A field is defined as a commutative division ring.
Groups are frequently studied through group representations. In their most general form, these consist of a choice of group, a set, and an action of the group on the set, that is, an operation which takes an element of the group and an element of the set and returns an element of the set. Most often, the set is a vector space, and the group represents symmetries of the vector space. For example, there is a group which represents the rigid rotations of space. This is a type of symmetry of space, because space itself does not change when it is rotated even though the positions of objects in it do. Noether used these sorts of symmetries in her work on invariants in physics.
A powerful way of studying rings is through their modules. A module consists of a choice of ring, another set, usually distinct from the underlying set of the ring and called the underlying set of the module, an operation on pairs of elements of the underlying set of the module, and an operation which takes an element of the ring and an element of the module and returns an element of the module. The underlying set of the module and its operation must form a group. A module is a ring-theoretic version of a group representation: Ignoring the second ring operation and the operation on pairs of module elements determines a group representation. The real utility of modules is that the kinds of modules that exist and their interactions, reveal the structure of the ring in ways that are not apparent from the ring itself. An important special case of this is an algebra. An algebra consists of a choice of two rings and an operation which takes an element from each ring and returns an element of the second ring. This operation makes the second ring into a module over the first. Often the first ring is a field.
Words such as “element” and “combining operation” are very general, and can be applied to many real-world and abstract situations. Any set of things that obeys all the rules for one (or two) operation(s) is, by definition, a group (or ring), and obeys all theorems about groups (or rings). Integer numbers, and the operations of addition and multiplication, are just one example. For example, the elements might be computer data words, where the first combining operation is exclusive or and the second is logical conjunction. Theorems of abstract algebra are powerful because they are general; they govern many systems. It might be imagined that little could be concluded about objects defined with so few properties, but precisely therein lay Noether’s gift: to discover the maximum that could be concluded from a given set of properties, or conversely, to identify the minimum set, the essential properties responsible for a particular observation. Unlike most mathematicians, she did not make abstractions by generalizing from known examples; rather, she worked directly with the abstractions.
The maxim by which Emmy Noether was guided throughout her work might be formulated as follows: “Any relationships between numbers, functions, and operations become transparent, generally applicable, and fully productive only after they have been isolated from their particular objects and been formulated as universally valid concepts.”
For illustration, if a physical system behaves the same, regardless of how it is oriented in space, the physical laws that govern it are rotationally symmetric; from this symmetry, Noether’s theorem shows the angular momentum of the system must be conserved. The physical system itself need not be symmetric; a jagged asteroid tumbling in space conserves angular momentum despite its asymmetry. Rather, the symmetry of the physical laws governing the system is responsible for the conservation law. As another example, if a physical experiment has the same outcome at any place and at any time, then its laws are symmetric under continuous translations in space and time; by Noether’s theorem, these symmetries account for the conservation laws of linear momentum and energy within this system, respectively.
Noether’s theorem has become a fundamental tool of modern theoretical physics, both because of the insight it gives into conservation laws, and also, as a practical calculation tool. Her theorem allows researchers to determine the conserved quantities from the observed symmetries of a physical system. Conversely, it facilitates the description of a physical system based on classes of hypothetical physical laws. For illustration, suppose that a new physical phenomenon is discovered. Noether’s theorem provides a test for theoretical models of the phenomenon: if the theory has a continuous symmetry, then Noether’s theorem guarantees that the theory has a conserved quantity, and for the theory to be correct, this conservation must be observable in experiment. Noether became famous for her use of ascending or descending chain conditions. A sequence of non-empty subsets A1, A2, A3, etc. of a set S is usually said to be ascending, if each is a subset of the next
Conversely, a sequence of subsets of S is called descending if each contains the next subset:
A chain becomes constant after a finite number of steps if there is an n such that for all m â‰¥ n. A collection of subsets of a given set satisfies the ascending chain condition if any ascending sequence becomes constant after a finite number of steps. It satisfies the descending chain condition if any descending sequence becomes constant after a finite number of steps.
Ascending and descending chain conditions are general, meaning that they can be applied to many types of mathematical objects-and, on the surface, they might not seem very powerful. Noether showed how to exploit such conditions, however, to maximum advantage: for example, how to use them to show that every set of sub-objects has a maximal/minimal element or that a complex object can be generated by a smaller number of elements. These conclusions often are crucial steps in a proof. Noether’s work continues to be relevant for the development of theoretical physics and mathematics and she is consistently ranked as one of the greatest mathematicians of the twentieth century.
At an exhibition at the 1964 World’s Fair devoted to Modern Mathematicians, Noether was the only woman represented among the notable mathematicians of the modern world. Noether has been honored in several memorials, The Association for Women in Mathematics holds a Noether Lecture to honor women in mathematics every year; in its 2005 pamphlet for the event, the Association characterizes Noether as “one of the great mathematicians of her time, someone who worked and struggled for what she loved and believed in. Her life and work remain a tremendous inspiration”. Because of Emily’s great dedication to her students the University of Siegen houses its mathematics and physics departments in buildings on the Emmy Noether Campus.
Thanks to Emily’s many contributions several things have been done to honor her such as; The German Research Foundation operates the Emmy Noether Programme, a scholarship providing funding to promising young post-doctorate scholars in their further research and teaching activities, a street in her hometown, Erlangen, has been named after Emmy Noether and her father, Max Noether. The successor to the secondary school she attended in Erlangen has been renamed as the Emmy Noether School. In fiction, Emmy Nutter, the physics professor in “The God Patent” by Ransom Stephens, is based on Emmy Noether. The crater Nöther on the far side of the Moon is named after her. The 7001 Noether asteroid also is named for Emmy Noether
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