Bernhard Riemann
| Bernhard Riemann | |
|---|---|
| |
| Bernhard Riemann, German mathematician | |
| Tradition | Mathematics, Analysis, Differential geometry, Number theory |
| Influenced by | Carl Friedrich Gauss, Joseph Fourier, Leonhard Euler |
| Lifespan | 1826–1866 |
| Notable ideas | Riemann hypothesis; Riemann integral; Riemann surfaces; Riemannian geometry |
| Occupation | Mathematician |
| Influenced | David Hilbert, Albert Einstein, Modern mathematics, General relativity |
| Wikidata | Q42299 |
Bernhard Riemann (1826–1866) was a German mathematician whose innovative ideas reshaped many fields of mathematics and laid the groundwork for modern geometry and analysis. His work bridged complex analysis, number theory and the geometry of space, influencing both pure mathematics and theoretical physics (especially Einstein’s relativity). Despite a life cut short by illness at age 39, Riemann introduced ideas and methods of lasting significance. He is remembered above all for the notion of a Riemann surface in complex analysis, the Riemann integral in real analysis, the formulation of curved multi-dimensional space in differential geometry, and his famous hypothesis about the zeros of the zeta function. Riemann was known to be shy and religious, and he preferred elegant, intuitive reasoning to lengthy symbolic calculation. His lectures and papers were often terse and challenging, but they inspired generations of mathematicians.
Early Life and Education
Riemann was born on September 17, 1826, in Breselenz (then in Hanover, now in Germany) to a modest Lutheran pastor and his wife. His father, Friedrich Riemann, valued education and taught his children at home. Riemann grew up studying classical subjects – including Latin, Greek, Hebrew and theology – at local schools. He demonstrated exceptional mathematical talent early on: according to later accounts, as a teenager he rapidly mastered complex subjects well beyond his years. A local teacher noticed young Riemann’s aptitude and lent him an advanced book on number theory, which Riemann reportedly read in six days and claimed to have memorized.
In 1846 Riemann entered the University of Göttingen intending to study theology. At Göttingen he attended lectures by the great mathematician Carl Friedrich Gauss. Gauss recognized Riemann’s promise and urged him to switch to mathematics, with his father’s permission. Riemann moved first to the University of Berlin (1847–49) and studied under leading figures such as Peter Gustav Lejeune Dirichlet, Jacobi and Steiner. Dirichlet in particular became an important mentor. While in Berlin, Riemann worked on complex analysis and elliptic functions. He returned to Göttingen in 1849 and completed his doctorate in 1851 under Gauss’s supervision. His doctoral thesis developed a “general theory of functions of a complex variable” and introduced the concept of what are now called Riemann surfaces.
After earning his Ph.D., Riemann remained at Göttingen as a postdoctoral lecturer. Over three years he prepared his Habilitation dissertation (completed 1854), focusing on trigonometric series and the conditions for integrability. At this time he also suffered personal tragedies: his mother died in 1847 and several of his siblings died young, which may have contributed to his lifelong poor health and anxious disposition. In 1854 Riemann completed the degree that allowed him to teach (the “Habilitation”) and gave a celebrated lecture “On the Hypotheses which Underlie Geometry.” This work, delivered at Göttingen, proposed a new way of understanding space and metrics in any dimension. It astonished his examiner Gauss for its originality, although it was not immediately appreciated by others. Nevertheless this lecture laid the foundation of what is now known as Riemannian geometry.
Riemann remained at Göttingen for the rest of his life. In 1857 he finally obtained a paid professorship after years of poorly paid academic employment. He married Elise Koch in 1862, shortly before falling seriously ill with tuberculosis. Seeking a warmer climate, he spent his final years traveling in Italy and died in Selasca on July 20, 1866, at age 39. Throughout his career Riemann published relatively few papers – each was dense and abstract – and he would not publish anything he regarded as incomplete. Some of his manuscripts remained unpublished at his death, and his collected works were later edited by his friends Richard Dedekind and Heinrich Weber.
Major Works and Ideas
Riemann made fundamental contributions across several areas of mathematics. His major achievements include:
- Complex Analysis and Riemann Surfaces. In his 1851 doctoral dissertation, Riemann developed a geometric approach to complex functions. He showed that multi-valued complex functions (like the solutions of algebraic equations in two complex variables) can be understood on a new kind of space called a Riemann surface. A Riemann surface is, roughly speaking, a two-dimensional surface (perhaps with holes) on which a complex function becomes single-valued and analytic. Riemann classified these surfaces by an integer called the genus (the number of holes), showing that topological features determine properties of complex functions on them. He also introduced what became known as the Riemann sphere to compactify the complex plane by adding a point at infinity. These ideas blended analysis and topology and were some of the first instances of topology being used in function theory. Riemann also proved key theorems about conformal maps (angle-preserving transformations of surfaces) and the mapping between simply connected domains and the unit disk – results that matured into the Riemann mapping theorem in complex analysis.
- Real Analysis and the Riemann Integral. In his Habilitation thesis (around 1854), Riemann rigorously defined what it means for a function to have an integral, extending the notion of area under a curve to a wide class of functions. A Riemann sum approximates the area by summing the value of the function at sample points times small intervals; taking the limit of such sums yields the Riemann integral. Riemann gave the conditions under which a bounded function is integrable: roughly, a function on an interval is Riemann-integrable precisely if its set of discontinuities is small (in the sense of having zero length). This clarified which functions could be integrated in the classical sense. Riemann also studied Fourier series: he asked under what conditions a function can be represented by a trigonometric (Fourier) series and how that representation behaves. He explored functions with many discontinuities and how their Fourier series converge. In fact, he discovered examples of functions whose Fourier series exhibit surprising behavior, highlighting limitations of earlier intuitions.
- Differential Geometry and the Geometry of Space. Riemann’s 1854 lecture “On the Hypotheses which Underlie Geometry” proposed a radical generalization of Euclidean space. He introduced the notion of an n-dimensional manifold, a space that locally looks like n-dimensional Euclidean space, but possibly curved. On such a manifold one can have a smoothly varying method of measuring distances and angles, represented by a metric tensor. Riemann argued that the properties of space should be determined experimentally rather than assumed flat, so he allowed geometry to have arbitrary curvature. He defined geodesics (generalized “straight lines” that locally minimize distance) and derived how curvature can be measured. His ideas implied that space might not obey Euclid’s postulates on the largest scales. Although these ideas were far ahead of their time, they later became the mathematical framework for Einstein’s general theory of relativity. Riemannian geometry – the study of such manifolds with curvature – was born from this work, and his notion of the Riemann curvature tensor quantified exactly how space deviates from being flat. This contribution united analysis and geometry in a profound way.
- Topology and Algebraic Geometry. Building on his work with Riemann surfaces, Riemann effectively pioneered the topological study of algebraic curves. His work on Abelian and elliptic functions (mid-1850s) examined multi-valued integrals on these curved surfaces. He proved what is now the Riemann–Roch theorem, relating the number of independent meromorphic functions on a surface to the surface’s topology (genus) and a chosen divisor of points. This theorem is a cornerstone of modern algebraic geometry; it connects the solutions to polynomial equations with the underlying shape of the curve. In his “Theory of Abelian Functions” (1857), Riemann further developed these ideas: he solved inversion problems for integrals on Riemann surfaces, generalizing earlier results of Abel and Jacobi. These advances framed much of what would become algebraic topology and complex manifold theory.
- Number Theory and the Zeta Function. Riemann made a single but enormous contribution to analytic number theory in an 1859 paper on the distribution of prime numbers. He considered the zeta function, an analytic continuation of the series ζ(s) = ∑ n^(-s) first studied by Euler for real arguments. Riemann treated ζ(s) as a complex function and studied its properties. He proved the functional equation that relates ζ(s) to ζ(1–s), and he carefully analyzed its zeros. Riemann showed that all nontrivial zeros of ζ(s) lie in the so-called critical strip (complex s with real part between 0 and 1). Crucially, he conjectured that every nontrivial zero has real part exactly 1/2. This statement became the famous Riemann Hypothesis. Riemann also used his complex-analytic insights to derive a formula (now involving an integral and the zeros of ζ) that gives the number of primes less than a given number. His approach laid the foundation of analytic number theory. Later mathematicians (Hadamard and de la Vallée Poussin) proved Riemann’s main result on the prime counting function, but the hypothesis about the zeros remains one of mathematics’ great unsolved problems.
- Partial Differential Equations and Physics. Riemann introduced new analytic methods in the theory of partial differential equations. For example, he developed techniques for handling wave equations and fluid dynamics. One notable contribution is the study of what are now called Riemann problems and Riemann invariants in gas dynamics: he analyzed how shock waves and discontinuities propagate in compressible fluids. These ideas were later seen as pioneering work in solving hyperbolic PDEs and applied in physics and engineering. Riemann also explored topics such as minimal surfaces (surfaces of least area) using complex function theory, connecting analysis, geometry, and physics.
Throughout his work, Riemann often emphasized unifying concepts over computational details. He believed that concrete examples and existence arguments (even non-constructive ones) could reveal the essence of mathematical phenomena. For instance, he frequently used physical intuition – such as analogies with heat flow or electricity – in formulating theorems in function theory. His notion of using a “variational principle” (the Dirichlet principle) to guarantee the existence of certain solutions was influential but also controversial among his contemporaries.
Method and Philosophy
Bernhard Riemann’s mathematical approach combined deep intuition with broad generality. He is remembered for relying on geometric and physical ideas to suggest results, often before rigorous proofs were available. Riemann valued existence proofs: rather than constructing explicit solutions, he would argue that an object (such as a function or a metric) exists because it would minimize some energy or satisfy some principle. For example, in solving problems about complex functions, Riemann often invoked the Dirichlet principle: an idea from potential theory that a certain energy integral must attain a minimum. He used this to show solutions exist to particular boundary-value problems, even though a rigorous foundation for that principle was not yet established. (Later, it was discovered that the principle required additional conditions to be valid; Weierstrass and others pointed out these issues, and Hilbert eventually fixed the principle at the end of the 19th century.)
Riemann also had a preference for a global and conceptual viewpoint. He would consider mathematics in as general a setting as possible. An example is his definition of a manifold: he did not restrict to two or three dimensions but allowed n-dimensional spaces. This abstraction allowed him to pose very general questions about space, like why our physical space has three dimensions or how to measure distance in higher dimensions. He was even willing to consider infinite-dimensional spaces for abstract problems. His work showed that many results could be discovered by thinking about the overall structure, rather than by heavy symbolic manipulation. In this sense, he was more interested in the “big picture” and the connections between areas (analysis, geometry, topology) than in detailed calculations.
At the same time, Riemann’s methods sometimes blurred the line between mathematics and physics. He often used insights from physics as motivation. For instance, his road to non-Euclidean geometry was partly inspired by the notion that forces like gravity should not act at a distance without mediation. He suggested that geometry might be tied to physical phenomena that propagate locally. This physical intuition helped him introduce the metric properties of space in his geometry lecture. Although Riemann was devoutly religious and saw his work as a service to God, he firmly believed that mathematics ultimately must accord with observable reality. In practical terms, this meant he allowed hypothesis testing of geometry by experiment – a revolutionary idea before Einstein.
Riemann did not shy away from novelty. He introduced terminology and concepts that did not exist before (such as “manifold,” “tensor,” or “Abelian function”) and he frequently cast aside previous limitations. Yet because his style was not focused on formal rigor by modern standards, some contemporaries found his proofs sketchy. Riemann himself was aware when criticisms arose: after Weierstrass showed a flaw in Riemann’s use of the Dirichlet principle, Riemann admitted the logical gap but argued that his results could still be correct (and indeed were later proven valid by other means). In sum, Riemann’s philosophy was to explore sweeping generalizations and to trust broadly conceived existence arguments, believing that more technical details could be filled in later.
Influence and Reception
Riemann’s ideas had profound and lasting influence, though that influence took time to unfold. In the mid-19th century, his groundbreaking lectures and papers were read by relatively few mathematicians. His low salary and shyness meant he did not travel widely, and many of his insights went unnoticed at first. Notably, Gauss – his mentor – immediately recognized the depth of Riemann’s 1854 geometry lecture and praised it enthusiastically. Others in his audience initially did not grasp the significance. Riemann’s writing style—abstract and succinct—also hindered immediate comprehension. For several decades after his death, his theorems were accepted by the mathematical community only after other mathematicians provided fully rigorous proofs. For example, Riemann’s colleagues Karl Weierstrass and Hermann Schwarz took up the task of reworking Riemann’s results in complex analysis to remove any logical gaps.
Nevertheless, key figures did appreciate Riemann’s originality. Richard Dedekind, who attended Riemann’s lectures as a student, became one of his staunchest advocates and helped publish Riemann’s unfinished notes. Other contemporaries like Enrico Betti and Eugenio Beltrami were inspired by Riemann’s geometric ideas to advance topology and non-Euclidean geometry. Perhaps the ultimate tribute came in the early 20th century when Albert Einstein used Riemannian geometry precisely as described by Riemann to formulate general relativity. Einstein’s theory relies on a curved four-dimensional spacetime, with gravity as a manifestation of curvature – a direct application of Riemann’s 1854 framework. In physics, it is often said that “without Riemann, Einstein could not have happened.”
In mathematics, Riemann’s influence grew significantly through the work of later geometers and analysts. The field of Riemann surfaces became centrally important in both complex analysis and algebraic geometry. Felix Klein and Adolf Hurwitz, among others, developed Riemann’s ideas into the classical theory of symmetry and moduli of surfaces. Riemannian geometry became a cornerstone of differential geometry, built upon by mathematicians like Ricci, Levi-Civita, and later Élie Cartan. The systematic study of smooth manifolds and curvature traces directly to Riemann. In number theory, Riemann’s insight that ζ(s) encodes prime numbers guided 20th-century developments in analytic number theory; his name appears everywhere in that field (for example, the Riemann–Siegel formula is used to compute zeros of ζ(s)).
By the late 19th century, the importance of Riemann’s work was widely recognized. David Hilbert, in his famous address of 1900 that set out 23 unsolved problems for the new century, explicitly mentioned the Riemann Hypothesis as one of the main problems to tackle. Riemann’s approach of using broad existence theorems presaged Hilbert’s own style of abstract axiomatic thinking. In fact, Hilbert tackled Riemann’s use of the Dirichlet principle and formulated a correct variational method to justify Riemann’s earlier results about harmonic functions. Felix Klein helped establish Göttingen as a world center of mathematics (alongside Gauss and Riemann as its iconic figures), and German mathematicians in the late 19th and early 20th centuries often traced their lineage back to Riemann at Göttingen. Riemann’s collected works (published after his death) and lectures continued to be studied and celebrated as embodying deep conceptual leaps in mathematics.
Even beyond mathematics, Riemann’s legacy touched science broadly. His notion that space could have more than three dimensions opened theoretical doors. Space-time in physics is now modeled as a Riemannian manifold of four dimensions (three of space and one of time). In topology and string theory, it is common to consider spaces of even higher dimension. Riemann’s recognition that the dimension of a space could be an independent question was an idea far ahead of its time.
Critiques and Debates
While Riemann’s results were brilliant, his methods sparked debate. The most famous critique concerned his use of the Dirichlet principle in complex analysis. When Riemann originally used this variational argument to prove the existence of certain harmonic functions (in effect, solving boundary-value problems), he assumed that a minimum of the appropriate integral existed. After Riemann’s death, his contemporary Karl Weierstrass exhibited examples showing that a minimizer might not exist in the class of functions considered, invalidating Riemann’s argument. This led to a period when some mathematicians questioned Riemann’s reliance on such arguments. Weierstrass himself stayed convinced of Riemann’s conclusions but insisted on finding alternative proofs. Eventually, around 1870, Hermann Schwarz and others provided more rigorous demonstrations of the theorems Riemann had claimed. Finally, David Hilbert (around 1899) fully justified the Dirichlet principle using more advanced methods from the calculus of variations. In retrospect, the controversy over rigor had a productive outcome: it spurred significant progress in analysis and led to new algebraic insights (by Clebsch, Gordan, Brill, etc.) while mathemeticians reworked Riemann’s classical results.
More broadly, Riemann’s abstract style was sometimes criticized by the stricter analysts of his era. For example, Weierstrass and others, scrutinizing convergence of series and functions, generally preferred explicit constructions and “epsilon-delta” arguments. They felt Riemann’s intuitive arguments needed more rigor. Some later historians note that Riemann’s intuition and geometric vision complemented Weierstrass’s rigor and symbolic proof style. Even today, Riemann’s work can be challenging to read in original form, because he often omitted routine details and many calculations. Mathematicians sometimes debate how to balance conceptual insight versus formal proofs; Riemann’s career is often cited in such discussions.
Another debate revolves around the Riemann Hypothesis itself. Riemann offered rather few details or evidence for the truth of his hypothesis beyond numerical observations and analogy with simpler functions. Over the years, many mathematicians attempted to prove it, and some claimed proofs (none of which held up under scrutiny). The status of the hypothesis – unproven but supported by extensive computational evidence – has at times been controversial within the mathematical community. However, this debate is more about modern speculative work and is not a criticism of Riemann himself (he posed the problem, but did not claim to have proven it). It does highlight Riemann’s legacy of presenting open, deep questions that continue to engage mathematicians.
It is worth noting that, aside from issues of mathematical rigor, Riemann faced some friction in his career for non-mathematical reasons. Finances at Göttingen were tight, and he sometimes complained about lack of resources and travel restrictions. His non-confrontational personality meant he rarely campaigned for promotions or recognition. In the mid-1850s, there was an unsuccessful effort to advance him to a higher professorship, and he often felt frustrated with academic politics. These personal and institutional challenges limited his output but do not reflect on the mathematical quality of his work.
Legacy
Riemann’s legacy lies in the vast areas of mathematics and physics that he helped to create. Today, many fundamental concepts bear his name, reflecting his lasting influence. Some terms named after him include Riemann surfaces (central in complex analysis and algebraic geometry), Riemannian manifolds (the setting of modern differential geometry), the Riemann curvature tensor (key to curvature in geometry and physics), the Riemann integral, and the Riemann zeta function. The Riemann Hypothesis remains a touchstone problem in number theory and has even influenced fields such as random matrix theory and quantum chaos. The Riemann–Roch theorem underpins modern algebraic geometry, connecting geometry and algebra in sophisticated ways. In short, anyone working in geometry, analysis or number theory is following paths that Riemann blazed.
In physics, Riemann’s concepts of curved space underpin the entire framework of general relativity and cosmology. Gravitational theory treats spacetime as a four-dimensional Riemannian manifold. Without Riemann’s generalization of metric geometry, the precise mathematical formulation of gravity (through Einstein’s field equations) could not have been written. In this sense, Riemann’s ideas extend far beyond pure mathematics into how we describe the universe.
Mathematicians and historians often rank Riemann among the greatest mathematicians ever. Carl Friedrich Gauss is sometimes called the “prince of mathematicians,” but Riemann is seen as one of Gauss’s great successors, carrying forward and expanding the mathematical program. By influencing Felix Klein, Hermann Weyl, David Hilbert and many others, Riemann helped shape the future of mathematics. The research agenda in mathematics changed significantly due to his vision: after Riemann, rigorous computation was not the only standard for important results – conceptual and structural insights gained equal status.
Selected compilations and collected works of Riemann were published after his death. In 1876, Dedekind and Weber edited Gesammelte mathematische Werke (Collected Mathematical Works), so Riemann’s lectures and unfinished notes became available. His name is now celebrated in commemorations (for example, the Riemann Hypothesis is one of the Millennium Prize Problems). Educational curricula in geometry and analysis often introduce “Riemannian geometry” and “Riemann integrals” because these are foundational ideas he created.
Riemann’s life story also adds to his legacy. He exemplifies the “brilliant but short-lived genius.” He shuffled personal hardship, shyness and sickness yet produced transformative mathematics. Near the end of his life, it was reported that he remained intellectually active under a fig tree on Lake Maggiore, working on geometry just before dying. His tombstone in Italy bears a reference to a Biblical verse (Romans 8:28) and calls him a Professor in Göttingen, testifying to his faith. The housekeeper who discarded some of his papers after his death unknowingly obscured parts of his gift to posterity. One wonders whether more lost insights might have changed mathematics even further.
'''Riemann’s influence endures in the many fields he opened. Every year, new books and papers build on Riemann’s concepts. The problems he posed – such as the Riemann Hypothesis – continue to challenge mathematicians. His style of geometric and global thinking is now standard in modern mathematics. In physics, experiments continue to probe the geometry of space (for instance, testing general relativity’s predictions); those too trace back to the seeds he planted. In summary, Riemann’s legacy is a vision of mathematics as a unified, geometric exploration of patterns, and a set of problems and theories that still drive discovery today.
Selected Works and Timeline
- 1851 – PhD Dissertation: Foundations for a general theory of functions of a complex variable (Göttingen). Introduced Riemann surfaces and function theory.
- 1854 – Lecture (Habilitation): “On the Hypotheses which Underlie Geometry.” Established the concept of an n-dimensional manifold with metric (foreshadowing Riemannian geometry).
- 1857 – “Theory of Abelian Functions.” Developed the theory of multi-valued integrals on Riemann surfaces and proved the Riemann–Roch theorem. This work was partly from lectures Riemann gave in Göttingen.
- 1859 – “On the Number of Primes Less Than a Given Magnitude.” First application of complex analysis to number theory; introduced the Riemann zeta function approach and stated the Riemann Hypothesis on the distribution of its zeros.
- 1861 – “On the Motion of Waves.” (Memoir on wave propagation and shock formation in fluids; published posthumously.)
- 1876 – Gesammelte mathematische Werke (Collected Mathematical Works) published by Dedekind and Weber, gathering Riemann’s published and unpublished papers and notes.
- 1892 – Another edition of Collected Works (edited by Heinrich Weber); the standard reference for Riemann’s writings for many years.
This timeline highlights Riemann’s key contributions during his short career. (Riemann’s actual output was modest in volume but immense in impact: each of the above works is celebrated as a major advance.)
Conclusion
Bernhard Riemann transformed mathematics by using deep insight and bold generalization. He bridged analysis, geometry and physics in ways never done before. His name lives on in the fundamental concepts of modern mathematics: Riemannian geometry, Riemann surfaces, Riemann sums, Riemann curvature, Riemann–Roch, and many more. His famous conjecture about prime numbers has become a guiding question for generations. Riemann showed that space could be curved and that analysis could be understood geometrically. Even now, mathematicians continue to unravel the ideas he set in motion. Though he died very young, Riemann’s mind bequeathed treasures that are still shaping science and mathematics today.