Bernhard Riemann

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Tiraspol State University

The Faculty of Physics, Mathematics and Information Technologies

Individual Work

REPORT

Bernhard Riemann

Performed by: Galusceac Daria, gr.11M

Verifield by: PhD, univ.lect., R. Burdujan

Chisinau, 2020

PLAN

Introduction

Family

Education

Homeschool

High School

Johanneum Gymnasium

Mathematical Genius to Serve God

Mathematics Chair, Marriage, and Ill Health

An Italian Finale

Mathematical discoveries

Used sources

List of terms associated with Riemann's name

Introduction

Georg Friedrich Bernhard Riemann was a German mathematician who made contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the first rigorous formulation of the integral, the Riemann integral, and his work on Fourier series. His contributions to complex analysis include most notably the introduction of Riemann surfaces, breaking new ground in a natural, geometric treatment of complex analysis. His famous 1859 paper on the prime-counting function, containing the original statement of the Riemann hypothesis, is regarded as one of the most influential papers in analytic number theory. Through his pioneering contributions to differential geometry, Riemann laid the foundations of the mathematics of general relativity. He is considered by many to be one of the greatest mathematicians of all time

Family

Georg Friedrich Bernhard Riemann was born on September 17, 1826 in the rural village of Breselenz, in northern Germany. He was the second of his parents' six children.

His mother was Charlotte Ebell, the daughter of a government employee in the city of Hanover. His father was Friedrich Bernhard Riemann, a Lutheran pastor. family home training johanneum

Life was harsh for the family - they lived in near poverty. In accordance with Lutheran practice, a solemn mood pervaded the home at all times. Although it was austere, Bernhard's home was also loving and caring.

The threat of death from tuberculosis was present in the very air the family breathed. Their diet and nutrition was poor and young Bernhard was plagued with persistent constipation. Although his brother and his four sisters reached adulthood, they all, except for his sister Ida, died at relatively young ages.

Education

Homeschool

Bernhard grew up 10 miles from Breselenz in the tiny village of Quickborn, where his father became the pastor when Bernhard was a toddler.

Bernhard's parents believed the most important thing they could give their children was a solid education. Bernhard attended some classes in the village school, but most of his education came from homeschooling with his father.

In his earliest years Bernhard was interested in history, particularly ancient history. But it was always clear that this stuttering, painfully shy boy had one exceptional talent: mathematics. His father enlisted a local teacher by the name of Schultz to teach 10-year-old Bernhard arithmetic and geometry, but soon Bernhard was teaching his teacher!

High School

In the spring of 1840, age 13, Bernhard was sent to live with his maternal grandmother in the city of Hanover. There he attended the Tertia des Lyceums Gymnasium, a school whose students were expected to go to college.

Bernhard was behind his classmates in most subjects, and he suffered terribly from homesickness. Hanover was about 90 miles from Quickborn - too far for him to travel home to see his cherished family. Despite his homesickness, he worked hard and made good progress academically.

Johanneum Gymnasium as it looked when Bernhard Riemann attended it.

Bernhard's grandmother died two years after Bernhard arrived in Hanover to live with her.

Bernhard moved again, this time to the Johanneum Gymnasium in the small city of Lьneburg.

Lьneburg was about 45 miles from Quickborn, close enough for him to walk home for vacations. These treks strained his frail body badly, but knowing he could get home to Quickborn took the edge off his appalling homesickness. His mother agonized over his safety and health when he was on the road.

Mathematical Genius to Serve God

More than half of Bernhard's school day was devoted to Latin, Greek, Hebrew, and German, but Mathematics formed a significant part of his curriculum: his ability in this subject was outstanding.

One of his teachers, Herr Schmalfuss, recognized Bernhard's flair and began lending him advanced college-level mathematics texts, including works by Leonhard Euler and Adrien-Marie Legendre. The first time he did this, Herr Schmalfuss was astonished when Bernhard, after just a few days, returned the book to him. He questioned Bernhard about the book's themes, and it became clear that his student truly had read and understood mathematical material that a typical advanced college student would have taken weeks or months to absorb.

In addition to his love of mathematics, Bernhard was also passionate about his religion. A devout Lutheran, he decided to study Theology and Philology at the University of Gцttingen, hoping to follow in his father's footsteps and become a pastor.

This was a curious decision, because Bernhard was introverted, terrified of public speaking, and generally uncomfortable around people. On the other hand, his father was his hero, and trying to emulate his hero was probably instinctive.

Interestingly, Leonhard Euler was also the son of a Protestant pastor and also seemed destined to join the clergy.

Mathematics Chair, Marriage, and Ill Health

In 1859, Gцttingen's chair of mathematics, Lejeune Dirichlet, died. He was replaced by Riemann, who was 32 years old.

In 1862, Riemann found the courage to propose marriage to 27-year-old Elise Koch, one of his sister's friends. Elise accepted, and they had one daughter, Ida, born in 1863.

A month after his wedding, Riemann suffered an attack of pleurisy, a painful inflammation of the lungs that, among other things, can be caused by tuberculosis.

Riemann was a dedicated Christian, the son of a Protestant minister, and saw his life as a mathematician as another way to serve God. During his life, he held closely to his Christian faith and considered it to be the most important aspect of his life. At the time of his death, he was reciting the Lord's Prayer with his wife and died before they finished saying the prayer.^[9] Meanwhile, in Gцttingen his housekeeper discarded some of the papers in his office, including much unpublished work. Riemann refused to publish incomplete work, and some deep insights may have been lost forever.

Riemann's tombstone in Biganzolo (Italy) refers to Romans 8:28

Here rests in God

Georg Friedrich Bernhard Riemann

Professor in Gцttingen

born in Breselenz, 17 September 1826

died in Selasca, 20 July 1866

For those who love God, all things must work together for the best

An Italian Finale

In his final years, Riemann made several trips to Italy, where the milder climate eased his tuberculosis. Indeed, his daughter Ida was born in Pisa. Riemann enjoyed life in Italy; he loved the artworks he saw there, and he felt more carefree and relaxed than he did in Gцttingen.

On his final day of life, he knew the end was near. He sat under a fig tree, working on mathematics, and enjoying the view. His Christian faith remained strong to the end. He died just after saying, “forgive us our debts,” while he and his wife Elise recited the Lord's Prayer together.

Bernhard Riemann died, age 39, of tuberculosis on July 20, 1866 in Selasca, Italy. He was buried in Selasca.

His friend Richard Dedekind collected Riemann's surviving papers and published them in 1868, bringing most of Riemann's work to a wider audience for the first time. Sadly, however, much of Riemann's research never came to light, because a cleaner, unaware of its importance, burned it shortly after his death.

Mathematical discoveries

Riemannian geometry

Riemann's published works opened up research areas combining analysis with geometry. These would subsequently become major parts of the theories of Riemannian geometry, algebraic geometry, and complex manifold theory. The theory of Riemann surfaces was elaborated by Felix Klein and particularly Adolf Hurwitz. This area of mathematics is part of the foundation of topology and is still being applied in novel ways to mathematical physics.

In 1853, Gauss asked Riemann, his student, to prepare a Habilitationsschrift on the foundations of geometry. Over many months, Riemann developed his theory of higher dimensions and delivered his lecture at Gцttingen in 1854 entitled "Ueber die Hypothesenwelche der GeometriezuGrundeliegen" ("On the hypotheses which underlie geometry"). It was only published twelve years later in 1868 by Dedekind, two years after his death. Its early reception appears to have been slow but it is now recognized as one of the most important works in geometry.

The subject founded by this work is Riemannian geometry. Riemann found the correct way to extend into n dimensions the differential geometry of surfaces, which Gauss himself proved in his theoremaegregium. The fundamental object is called the Riemann curvature tensor. For the surface case, this can be reduced to a number (scalar), positive, negative, or zero; the non-zero and constant cases being models of the known non-Euclidean geometries.

Riemann's idea was to introduce a collection of numbers at every point in space (i.e., a tensor) which would describe how much it was bent or curved. Riemann found that in four spatial dimensions, one needs a collection of ten numbers at each point to describe the properties of a manifold, no matter how distorted it is. This is the famous construction central to his geometry, known now as a Riemannian metric.

Complex analysis

In his dissertation, he established a geometric foundation for complex analysis through Riemann surfaces, through which multi-valued functions like the logarithm (with infinitely many sheets) or the square root (with two sheets) could become one-to-one functions. Complex functions are harmonic functions (that is, they satisfy Laplace's equation and thus the Cauchy-Riemann equations) on these surfaces and are described by the location of their singularities and the topology of the surfaces. The topological "genus" of the Riemann surfaces is given {\displaystyle g=w/2-n+1}, where the surface has {\displaystyle n} leaves coming together at {\displaystyle w} branch points. For {\displaystyle g>1}the Riemann surface has {\displaystyle (3g-3)}parameters (the "Размещено на http://www.allbest.ru/

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moduli").His contributions to this area are numerous. The famous Riemann mapping theorem says that a simply connected domain in the complex plane is "biholomorphically equivalent" (i.e. there is a bijection between them that is holomorphic with holomorphic inverse) to either {\displaystyle \mathbb {C} } Размещено на http://www.allbest.ru/

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or to the interior of the unit circle. The generalization of the theorem to Riemann surfaces is the famous uniformization theorem, which was proved in the 19th century by Henri Poincarй and Felix Klein. Here, too, rigorous proofs were first given after the development of richer mathematical tools (in this case, topology). For the proof of the existence of functions on Riemann surfaces he used a minimality condition, which he called the Dirichlet principle. Karl Weierstrass found a gap in the proof: Riemann had not noticed that his working assumption (that the minimum existed) might not work; the function space might not be complete, and therefore the existence of a minimum was not guaranteed. Through the work of David Hilbert in the Calculus of Variations, the Dirichlet principle was finally established. Otherwise, Weierstrass was very impressed with Riemann, especially with his theory of abelian functions. When Riemann's work appeared, Weierstrass withdrew his paper from Crelle's Journal and did not publish it. They had a good understanding when Riemann visited him in Berlin in 1859.

Weierstrass encouraged his student Hermann Amandus Schwarz to find alternatives to the Dirichlet principle in complex analysis, in which he was successful.

An anecdote from Arnold Sommerfeld shows the difficulties which contemporary mathematicians had with Riemann's new ideas. In 1870, Weierstrass had taken Riemann's dissertation with him on a holiday to Rigi and complained that it was hard to understand. The physicist Hermann von Helmholtz assisted him in the work over night and returned with the comment that it was "natural" and "very understandable".

Other highlights include his work on abelian functions and theta functions on Riemann surfaces. Riemann had been in a competition with Weierstrass since 1857 to solve the Jacobian inverse problems for abelian integrals, a generalization of elliptic integrals. Riemann used theta functions in several variables and reduced the problem to the determination of the zeros of these theta functions. Riemann also investigated period matrices and characterized them through the "Riemannian period relations" (symmetric, real part negative). By Ferdinand Georg Frobenius and Solomon Lefschetz the validity of this relation is equivalent with the embedding of {\displaystyle \mathbb {C} ^{n}/\Omega } (where {\displaystyle \Omega } is the lattice of the period matrix) in a projective space by means of theta functions. For certain values of {\displaystyle n}, this is the Jacobian variety of the Riemann surface, an example of an abelian manifold.

Many mathematicians such as Alfred Clebsch furthered Riemann's work on algebraic curves. These theories depended on the properties of a function defined on Riemann surfaces. For example, the Riemann-Roch theorem (Roch was a student of Riemann) says something about the number of linearly independent differentials (with known conditions on the zeros and poles) of a Riemann surface.

According to DetlefLaugwitz,^[12] automorphic functions appeared for the first time in an essay about the Laplace equation on electrically charged cylinders. Riemann however used such functions for conformal maps (such as mapping topological triangles to the circle) in his 1859 lecture on hypergeometric functions or in his treatise on minimal surfaces.

Real analysis

In the field of real analysis, he discovered the Riemann integral in his habilitation. Among other things, he showed that every piecewise continuous function is integrable. Similarly, the Stieltjes integral goes back to the Gцttinger mathematician, and so they are named together the Riemann-Stieltjes integral.In his habilitation work on Fourier series, where he followed the work of his teacher Dirichlet, he showed that Riemann-integrable functions are "representable" by Fourier series. Dirichlet has shown this for continuous, piecewise-differentiable functions (thus with countably many non-differentiable points). Riemann gave an example of a Fourier series representing a continuous, almost nowhere-differentiable function, a case not covered by Dirichlet. He also proved the Riemann-Lebesgue lemma: if a function is representable by a Fourier series, then the Fourier coefficients go to zero for large n.Riemann's essay was also the starting point for Georg Cantor's work with Fourier series, which was the impetus for set theory.He also worked with hypergeometric differential equations in 1857 using complex analytical methods and presented the solutions through the behavior of closed paths about singularities (described by the monodromy matrix). The proof of the existence of such differential equations by previously known monodromy matrices is one of the Hilbert problems.

Number theory

He made some famous contributions to modern analytic number theory. In a single short paper, the only one he published on the subject of number theory, he investigated the zeta function that now bears his name, establishing its importance for understanding the distribution of prime numbers. The Riemann hypothesis was one of a series of conjectures he made about the function's properties.

In Riemann's work, there are many more interesting developments. He proved the functional equation for the zeta function (already known to Leonhard Euler), behind which a theta function lies. Through the summation of this approximation function over the non-trivial zeros on the line with real portion 1/2, he gave an exact, "explicit formula" {\displaystyle \pi (x)}. Riemann knew of PafnutyChebyshev's work on the Prime Number Theorem. He had visited Dirichlet in 1852.

List of terms associated with Riemann's name

· Riemann hypothesis

· The Riemann zeta function

· Riemann integral

· The multiple Riemann integral

· Riemann derivative

· Riemannian geometry

· Riemann surface

· Riemann sphere

· Spherical Riemann geometry

· Riemann curvature tensor

· Riemann's mapping theorem

· Riemann's theorem on conditionally converging series

· Riemann's removable singular point theorem

· Cauchy - Riemann conditions

Used sources

1. Bernhard Riemann at the Mathematics Genealogy Project

2. The Mathematical Papers of Georg Friedrich Bernhard Riemann

3. Riemann'spublicationsatemis.de

4. Bernhard Riemann - one of the most important mathematicians

5. https://www.geni.com/people/Ida-Schilling/6000000025101232998

6. DetlefLaugwitz: Bernhard Riemann1826-1866. Birkhдuser, Basel 1996, ISBN 978-3-7643-5189-2

7. http://people-archive.ru/character/bernhard-riman

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