The Genius of Bernhard Riemann: Contributions to Maths & Geometry - Research Paper

Introduction

George Friedrich Bernhard Riemann takes credit for his significant contribution to differential geometry, number theory, and analysis (Baker 1). Born on 17th September, 1826, Riemann made significant contributions in Fourier series, rigorous formulation, and Riemann integral. Most of Riemann's explorations involved complex analysis and notably Riemann surfaces, contributing to the development in natural geometry treatment involving multifaceted breakdowns. One of Riemann's famous works, the prime-counting function contained in his 1859 paper, entailed the first analysis and statement of Riemann's theory (Baker 2). Riemann takes the most credit in laying a strong foundation in general relativity and takes recognition as one of the most influential mathematicians of the 19thcentury (Baker 2). This paper provides a historical background of Friedrich Bernhard Riemann, a man considered as one of the greatest mathematicians.

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Early Years

Riemann grew up in the Kingdom of Hanover, a remote town close to Dannenberg, to Friedrich Bernhard Riemann, a German Lutheran pastor, and Charlotte Ebell (Baker 3). The couple had six children. At a young age, Riemann suffered from a nervous breakdown and lost his mother long before he reached adulthood (Baker 3). Riemann started showing great interest in mathematics at an early age. In class, Riemannoften suggesting new mathematics solutions and helping teachers solves complex mathematical problems. Riemann explained all mathematical problems teachers gave him, and in one of the class lessons, Riemann tried to validate a particular book using mathematical models and formulas (Papadopoulos 1). Riemann school environment played a critical role in improving his mathematical skills. Riemann took part in solving many mathematics lessons with professors who marveled at his outstanding skills and abilities.

Early Education

Riemann enrolled in middle school in 1840 while living with his grandmother (Baker 3). According to Baker, after the death of Riemann's grandmother, he attended high school in Johanneum Luneburg (5). Riemann studied biblical teachings exhaustively and found himself sidetracked by mathematics in high school. Riemann outperformed in nearly all operations and took great interest in mathematical problems. In 1846 Papadopoulos stated that at only aged nineteen years, Riemann focused his attention on Christian theology and philology with the passion for serving in his local church while helping with domestic finances (2). Riemann formed and formulated his Christian beliefs during this period.

Riemann's father played a critical role in his interest and involvement in mathematical problems. Riemann's father encouraged him to partake in mathematical challenges and later enrolled him at Gottingen to study theology and philology (Papadopoulos 3). Riemann's father had plans to enroll Riemann at theological course to enable him take up the role of a clergy in the local church. Riemann, enthusiastic to follow his father's counsel enrolled at Gottingen for a course in theology and philology (Baker 10). Riemann continued with his interest in mathematics at Gottingen. Riemannhad exceptional skills and abilities in calculation and mathematics. However, he suffered from fear and nervousness in social places.

Academics

Riemann focused his attention on Carl Friedrich Gauss at the University of Gottingen. His attention zeroed on Gauss's method of least squares. Riemann let go of Christian theology and focused more on mathematics Under Gauss's advice (Baker 12). Riemann's father approved of his academic desires in 1847 and moved him to the University of Berlin the same year. Some of his most exceptional tutors included Gotthold Eisenstein, Jakob Steiner, and Peter Gustav, among others. He studied for slightly over two years and returned home in 1849. He committed to finishing his doctorate while in Gottingen. Papadopoulos stated that Riemann received his Ph. D. in 1851 with his thesis dealing in Gaussian mathematics and the theory of complex functions (4). The Ph.D. served as a significant pointer to his great achievements in mathematical methods.

Greatest Contributions

According to Elizalde, Riemann's most significant contributions included Riemannian geometry that formed the basis of topology currently used in mathematical physics (1). His Riemannian geometry gave forth differential geometry and fundamental Riemann curvature tensor. Other significant contributions included a complex analysis that he derived from his dissertation. Riemann founded the Riemann surfaces, one-to-one functions, the Riemann mapping theorem, the Dirichlet principle, and his famed "uniformization theorem" (Elizalde 3). The theta function, Jacobian variety, and the Riemann-Roch thesis constitute some of his major works and contributions in the field of mathematics (Papadopoulos 4). In Real Analysis, he came up with the Riemann integral. Riemann also indicated continuous functions and also the Riemann-Stieltje integral (Elizalde 3). For example, Dirichlet contributed to the piecewise-differentiable tasks in which he exemplified the Fourier series that represented the differentiation functionality.

Persona

According to Baker, Riemann remained a staunch dedicated Christian all his life. In his mathematical discoveries, he believed it all entailed serving God (2). Riemann closely held onto his faith, an aspect he found remained the most critical aspect of his life. Riemann had a shy personality and took a great deal of time in preparing even to address a gathering (Baker 4). As a perfectionist, Riemann never revealed much of his work and only published the best. In most biographies, descriptions pointed to Riemann as "one of the most profound and imaginative mathematicians of all time" (Baker 7). Baker states that in 1866, Riemannfled Gottingen after an invasion by the Prussia and Hanover armies (7).Riemann continued with his theological work and mathematical discoveries in Gottingen.

Death

Riemann contracted tuberculosis while working as a professor at Gottingen. Riemannspent much of his later life touring Italy and Gottingen, reportedly searching for a favorable climate. He died in the presence of his wife while in the middle of prayer in Selasca on 16thJune 1866 at the prime age of 39 (Baker 12). As a staunch Christian who dedicated much of his time in serving God, Riemann's tomb has the inscription referring to Romans 8:28 (Baker 12). Riemann works continue to open up great frontiers in the field of geometry and analysis, and his works form the basis of topology used in mathematical physics (Papadopoulos 9). Some of Riemannmost exceptional applications widely used today includes the complex manifold, algebraic geometry, and the Riemann geometry, and the Riemann surfaces.

Conclusion

George Friedrich Bernhard Riemann discoveries, theories, and mathematical analysis defined much of his lifetime. George Friedrich Bernhard Riemann remains one of the greatest mathematicians to live with many of his findings and approaches used and applied in various mathematical and physics settings. He takes credit for many innovations and theorems that continue to shape and define the mathematics and physics field.

Works Cited

Baker, Roger. An introduction to Riemann's life, his mathematics, and his work on the zeta function. Exploring the Riemann zeta function. Springer, vol. 2, no. 1, 2017, 1-12.

Elizalde, Emilio. Bernhard Riemann, archetypical mathematical-physicist? Frontiers in Physics, vol.2, no.1, 2013, 1-5.

Papadopoulos, Athanase. Mathematics, physics and philosophy in Riemann's work and beyond. Springer, vol. 1, no. 2, 2017, 1-18.