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infinity-stacks, 'dérivateurs', cohomological formalism of toposes as an inspiration for a new homotopic algebra
Tame topology.
Yoga of anabelian geometry and Galois-Teichmüller theory.
Schematic point of view, or "arithmetics" for regular polyhedra and regular configurations of all sorts.
He wrote that the central theme of the topics above is that of topos theory, while the first and last were of the least importance to him.

Here the usage of yoga means a kind of 'meta-theory' that can be used heuristically. The word yoke, meaning a linkage, is derived from the same Indo-European root.

Life

Childhood and studies
Born to a Russian Jewish father and German Protestant mother in Berlin, he was a displaced person during much of his childhood due to the upheavals of World War II. Alexander lived with his father, Alexander Shapiro, and his mother, Hanka Grothendieck, both of whom were socialist revolutionaries. Until 1933 they lived together in Berlin. At the end of that year, Shapiro moved to Paris, and Hanka followed him the next year. They left Alexander with a family in Hamburg where he went to school. During this time, his parents fought in the Spanish Civil War. In 1939 Alexander came to France and lived in various camps for displaced persons with his mother. His father was sent to Auschwitz where he died in 1942. After the war, young Grothendieck studied mathematics in France, initially at Montpellier. He had decided to become a math teacher because he had been told that mathematical research had been completed early in the 20th century and there were no more open problems. However, his talent was noticed, and he was encouraged to go to Paris in 1948. Initially, Grothendieck attended Henri Cartan's Seminar at École Normale Superieure, but lacking the necessary background to follow the high powered seminar, he moved to University of Nancy where he wrote his dissertation under Laurent Schwartz in functional analysis, from 1950. At this time he was a leading expert in the theory of topological vector spaces. However he set this subject aside by 1957 in order to work in algebraic geometry and homological algebra.

Politics and retreat from scientific community
Grothendieck's radical left-wing and pacifist politics were doubtless born by his family history and his wartime experiences. He gave lectures on category theory in the forests surrounding Hanoi while the city was being bombed, to protest against the Vietnam war. He retired from scientific

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