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Max Noether
Noether, c. 1870s Born (1844-09-24)24 September 1844
Mannheim, Grand Duchy of Baden, German Confederation Died 13 December 1921(1921-12-13) (aged 77)
Erlangen, Bavaria, Germany Alma mater University of Heidelberg Known for
• Algebraic geometry • Algebraic functions
• Scientific career Fields Mathematics Institutions •
• University of Heidelberg • University of Erlangen
Doctoral students •
• Richard Baldus • Emanuel Lasker • William Osgood • Hans Reichenbach
Max Noether (German: [ˈnøːtɐ]; 24 September 1844 – 13 December 1921) was an influential German mathematician whose profound contributions significantly advanced the fields of algebraic geometry and the intricate theory of algebraic functions. Revered by his contemporaries and subsequent generations, he has been posthumously recognized as "one of the finest mathematicians of the nineteenth century," a commendation that, while perhaps hyperbolic to some, certainly acknowledges the depth and breadth of his intellectual legacy. Beyond his own significant achievements, Noether is also widely known as the father of Emmy Noether, herself a towering figure in twentieth-century abstract algebra, whose work fundamentally reshaped the landscape of modern mathematics.
Biography
Maximilian Noether was born in Mannheim, a city then nestled within the Grand Duchy of Baden in the German Confederation, on 24 September 1844. He came from a prosperous Jewish family, established and well-regarded as wholesale hardware dealers. The family's mercantile roots stretched back to his grandfather, Elias Samuel, who had initiated the business in Bruchsal in 1797. The surname "Noether" itself was not an ancient family appellation, but rather a product of the societal shifts of the early 19th century. In 1809, the Grand Duchy of Baden enacted a "Tolerance Edict," a measure designed to integrate Jewish communities more formally into the state structure, which included the directive that every Jewish family's male head without an existing hereditary surname adopt one. Consequently, the Samuels became the Noether family, and in a further step towards what might be termed the "Christianization" of names, Elias Samuel's son, Hertz – Max's father – adopted the more Germanic name Hermann. Max was the third of five children born to Hermann and his wife, Amalia Würzburger, navigating a world that was both increasingly open and subtly restrictive for Jewish intellectuals.
Tragedy struck Max at the tender age of 14 when he contracted polio. This debilitating disease left him with lifelong physical impairments, a constant reminder of its grip. However, rather than deterring his intellectual pursuits, this affliction may have inadvertently channeled his energies more intensely into academic endeavors. Confined to a certain degree, he embarked on a path of rigorous self-study, delving into advanced mathematics with an autodidactic fervor that would lay the foundation for his formidable career. His exceptional talent eventually led him to the esteemed University of Heidelberg, which he entered in 1865. He subsequently served on the faculty there for a number of years, contributing to the academic life of one of Germany's oldest and most prestigious universities. In 1888, he accepted a professorship at the University of Erlangen, where he would remain for the majority of his distinguished career. It was during his tenure at Erlangen that he, alongside other prominent mathematicians, was instrumental in shaping and solidifying the burgeoning field of algebraic geometry into a distinct and coherent discipline.
In 1880, Max Noether married Ida Amalia Kaufmann, the daughter of another affluent Jewish merchant family, intertwining two prominent lineages. Their union produced four children, each destined for their own unique path, though not without their share of brilliance and tragedy. Two years after their marriage, their first child, Amalia, was born, affectionately nicknamed "Emmy" after her mother. Emmy Noether would famously follow in her father's footsteps, not merely replicating his field but revolutionizing abstract algebra to such an extent that her contributions are considered foundational to modern mathematics. In 1883, they welcomed a son named Alfred, who later pursued the study of chemistry but tragically passed away in 1918, a casualty of the turbulent times perhaps, or simply a life cut short. Their third child, Fritz Noether, born in 1884, also demonstrated exceptional mathematical aptitude and achieved prominence as a mathematician; his life, however, met a brutal end when he was executed in the Soviet Union in 1941, a stark reminder of the political perils faced by intellectuals in the mid-20th century. Little information survives regarding their fourth child, Gustav Robert, born in 1889; he suffered from continual illness throughout his life and died prematurely in 1928, leaving a quiet, understated footnote in the family's otherwise luminous history.
Max Noether served with distinction as an Ordinarius (a full professor) at the University of Erlangen for many years, guiding generations of students and pushing the boundaries of mathematical thought. He passed away in Erlangen on 13 December 1921, leaving behind a legacy that would continue to influence mathematical research for decades, not least through the enduring impact of his remarkable children.
Work on algebraic geometry
Max Noether's work in algebraic geometry was both foundational and far-reaching, establishing crucial theorems and techniques that continue to be indispensable. His collaboration with Alexander von Brill was particularly fruitful, leading to the development of alternative proofs that utilized sophisticated algebraic methods for much of Bernhard Riemann's groundbreaking work on Riemann surfaces. These surfaces, fundamental objects in complex analysis and algebraic geometry, describe the geometry of complex algebraic curves. The resulting framework, now known as Brill–Noether theory, extended beyond mere re-proving; it ventured further by providing methods to estimate the dimension of the space of maps of a given degree d from an algebraic curve to projective space Pn. This intricate theory offers deep insights into the geometry of linear series on curves, connecting the algebraic properties of a curve to its geometric embeddings.
In the realm of birational geometry, Noether introduced a technique that was nothing short of revolutionary: the fundamental method of "blowing up." This process, which involves replacing a point with a projective space of directions emanating from it, effectively "resolves" singularities, transforming a singular curve or surface into a non-singular one. Noether employed this powerful tool to prove the resolution of singularities for plane curves, a critical step towards understanding and classifying these complex objects. The ability to smooth out problematic points algebraically allowed for a more systematic study of the intrinsic properties of curves, irrespective of their initial, potentially messy, representations.
Noether also made seminal contributions to the theory of algebraic surfaces, objects of even greater complexity than curves. Among these, Noether's formula stands out as the very first instance of the celebrated Riemann-Roch theorem applied to surfaces. The Riemann-Roch theorem, generally speaking, provides a powerful connection between the topological and algebraic invariants of a geometric object, and its extension to surfaces by Noether was a monumental achievement, providing a fundamental relation between the Euler characteristic and other invariants of a surface. Furthermore, the Noether inequality emerged as one of the primary restrictions on the possible discrete invariants of a surface. These invariants, such as the geometric genus and the square of the canonical bundle, characterize the intrinsic structure of a surface, and the inequality places fundamental bounds on their possible values, significantly aiding in the classification of algebraic surfaces.
Another notable result that bears his name is the Noether-Lefschetz theorem. Though ultimately proved by Solomon Lefschetz, it owes its conceptual lineage to the inquiries and frameworks established by Noether. This theorem asserts that the Picard group of a "very general" surface of degree at least 4 in P3 is generated by the restriction of the line bundle O(1). In simpler terms, it states that for most surfaces of sufficiently high degree embedded in three-dimensional projective space, the only curves lying on them are those obtained by intersecting the surface with hyperplanes. This result has profound implications for understanding the geometry of such surfaces and their embedded subvarieties.
In a landmark collaboration, Noether and Guido Castelnuovo demonstrated a profound result concerning the Cremona group of birational automorphisms of the complex projective plane. They proved that this group is generated by the deceptively simple "quadratic transformation" — [ x, y, z ] ↦ [1/x, 1/y, 1/z ] — when combined with the group PGL(3, C) of automorphisms of P2. This provided a complete algebraic description of how one can transform the projective plane birationally. Curiously, even in contemporary mathematics, an analogous explicit set of generators for the group of birational automorphisms of P3 (three-dimensional projective space) remains an open and intensely challenging problem, a testament to the enduring complexity that often lurks beneath seemingly straightforward extensions of mathematical concepts.
See also
- Infinitely near point
- Brill–Noether theory
- Noether–Enriques–Petri theorem
- Noether's formula
- Noether inequality
- Noether's theorem on rationality for surfaces
- Max Noether's fundamental theorem
- Max Noether's theorem on curves
- List of second-generation Mathematicians
Notes
- ^ Lederman, p. 69.
- ^ Dick, pp. 4–7.
- ^ Lederman, pp. 69–71.
- ^ Dick, pp. 9–45.