Short answers, pulled from the story.
Scribes in ancient Mesopotamia and Egypt began recording geometric principles during the 2nd millennium BC. The Rhind Papyrus dates from between 2000 and 1800 BC and contains practical formulas for calculating areas and volumes.
Euclid revolutionized geometry around 300 BC when he wrote his Elements. This text introduced mathematical rigor through an axiomatic method that remains the standard format for definitions, axioms, theorems, and proofs today.
Nikolai Ivanovich Lobachevsky lived from 1792 to 1856 and discovered non-Euclidean geometries without introducing contradictions despite centuries of assumptions about space. János Bolyai lived from 1802 to 1860 while Carl Friedrich Gauss lived from 1777 to 1855 and also contributed to these discoveries.
Differential geometry uses techniques of calculus and linear algebra to study problems involving curvature and has applications in physics among other fields. Albert Einstein postulated that the universe is curved making differential geometry crucial to general relativity.
Alexander Grothendieck introduced scheme theory between the late 1950s and mid-1970s allowing topological methods including cohomology theories in purely algebraic contexts. This development enabled algebraic geometry to become autonomous with Hilbert's Nullstellensatz establishing correspondence between algebraic sets and ideals of polynomial rings.