Riemann

Few Mathematicians have impacted the world of mathematics like Riemann did. Of all the published volume he has one of the smallest amount of published material.

Riemann's life
His life was spent in Gootingen. Even though he did his Ph.D under Gauss. He wasn't of much influence. Riemann was quite a loner and resisted attempts to befriended even his contemporaries like esenstein etc. His days was spent in relative poverty as most of the mathematician in that era recieved meager pay. The only other source of money was tutoring and license to brew wine.
Most of his early schooling was done under his father. It was only after he was 15 year old that he moved to a proper school. But whatever it may be it does mean that his father taught him well. He got married to a sister's friend though he nutured a desire to get married to weber's niece for a long time. His last days were spent in Italy and he seems to have like. He seems to have impacted the world in Didekind a contemporary of Riemann was the first person to have written the biography of Riemann. Throughtout his life Riemann was plagued with ill health. He lived from - to -

One of the profound influence
Riemann's Contribution
Complex Analysis
3 people who have profoundly impacted the study of complex numbers are Dedekind

Riemann Geometry (Source: Making the invisible Visible)
To appreciate this lets go back to the time of Euclid. His 12 Volume Elements had been the only text book on Geometry and had influenced thinking for more than 2000 years. It was only in renaissance that people began to think differently. Basically the 5 postulates of Euclid's element are

They make sense for plane geometry. But in the era
With Riemann Geometry you can have sum of the angles of triangle greater than 180 degree
Visualizing complex numbers is two dimension can be by Riemann Sphere Concept. Complex numbers can be mapped using Riemann S

Zeta function
One of the great problem in mathematics which interested for a long time was a prime number theorem and Riemann's description was one of the turning point to have finally formalized this as a theorerm.

Riemann Spehere.
Riemann was not a prolific publisher like Gauss or his contemporaries and add to that his frail health.
Riemann Surface
To understand the concept of integral is
Riemann Integral
The very first thing one gets introuduced in the calculus I course is Riemann Integral which tells how to integrate
Cauchy Riemann Equation for Analiticity

Summary

Biblography

Zero
Riemann Turning points in conception of Mathematics
Life by the numbers
Mathematics and its history


The chapter is a aptly named Geometry, Physics, Philospohy. It is Reimann's genius that encompasses all these 3 beautiful subject and his profound contributions to these fields.
Physics as a field theory. Natural Sciences. Riemann's work in Natural philosphy in his own words influenced by (page 285) by Herbart.