Short answers, pulled from the story.
The first known mathematical tablet Plimpton 322 dates back to 1800 BC. This artifact was discovered in the ruins of ancient Mesopotamia and reveals that Babylonian scribes were already manipulating complex fractions and solving quadratic equations. The Babylonians developed a sexagesimal numeral system that remains in use today for measuring angles and time.
Euclid of Alexandria published his seminal work Elements around 300 BC. This work transformed mathematics from a collection of empirical recipes into a rigorous system of logical deduction known as the axiomatic method. Euclid's work remained the primary textbook for geometry for over two thousand years.
The Persian mathematician Al-Khwarizmi introduced systematic methods for transforming equations in the 9th century. He coined the term algebra from the Arabic word al-jabr meaning the reunion of broken parts. The Islamic Golden Age preserved and expanded upon Greek knowledge adding the decimal point to the numeral system.
Isaac Newton and Gottfried Leibniz simultaneously and independently discovered calculus in the 17th century. Newton developed calculus to explain the movement of the planets under the influence of gravity while Leibniz focused on the geometric interpretation of rates of change. Leonhard Euler unified these innovations into a single corpus in the 18th century.
Georg Cantor's work on infinite sets challenged the very nature of mathematical truth in the late 19th century. His diagonal argument demonstrated that there are different sizes of infinity which led to intense controversy. Kurt Gödel further transformed the field in the early 20th century by proving that any consistent axiomatic system powerful enough to describe arithmetic will contain true propositions that cannot be proved within the system.
The Fields Medal was established in 1936 by Canadian John Charles Fields. It is considered the mathematical equivalent of the Nobel Prize and is awarded every four years to up to four individuals under the age of 40. The Abel Prize was instituted in 2002 and the Chern Medal for lifetime achievement was introduced in 2009.