The core of his work concerned the connection between space and structure, and he set himself enormous research programmes intended to reveal the underlying connections between whole areas of mathematics. His ambition was to unite algebra and geometry, arithmetic and topology in the most general terms, and in so doing to bring together concepts of symmetry and geometry.
The revolution that Grothendieck began is still continuing. Fundamental mathematical initiatives, most famously the solving of Fermat's last theorem by Andrew Wiles, would not have been possible without his work.
A native of Berlin, he was the son of an avant-garde journalist, Hanka Grothendieck, and an anarchist, Alexander "Sascha" Shapiro, who had lost an arm after being shot while escaping from prison. Separated from his parents at the age of five, the young Alexander was reunited with them in Paris just before his father was interned by the Vichy government and sent to Auschwitz, where he was killed.
Alexander was then separated once again from his mother and sent to the village of Le Chambon-sur-Lignon in the Haute-Loire, where Father André Trocmé's resistance activities embraced the protection and education of Jewish children.
From his earliest years Alexander experienced the intense psychological and creative independence of the autodidact. In 1945, he enrolled at the University of Montpellier, and three years later went to Paris to attend the elite mathematical seminar presided over by Henri Cartan at the École Normale Supérieure. There he encountered many of the ideas in areas such as topology, group theory and algebraic geometry that were to form the intellectual backbone of his future work.
His doctoral thesis at the University of Nancy (1953) combined functional analysis and algebra in a profoundly new synthesis. At the time he was living with his increasingly tubercular mother, who worked as a cleaner in order to support her son while he was engaged in research. A brief relationship between Grothendieck and the woman who ran his boarding house resulted in the birth of his first son.
For a short time Grothendieck taught in Brazil, where a colleague remarked on his natural tendency to asceticism in a lifestyle that involved a diet of bananas and milk, no rugs on the carpet because he regarded such things as useless luxuries and a penchant for wearing sandals made out of rubber tyres. He rejected anything that smacked of insincerity or taste that verged on the wrong side of sheer utility.
A dominant force wherever he taught, Grothendieck was at his most influential and productive during his time at the Institut des Hautes Études Scientifiques (IHES), south of Paris (1958-70). Though his students revered him, many of his colleagues thought him a polarising influence, resenting the turn towards pure abstraction that his mathematics demanded. Powerfully built and completely shaven-headed (like his father), he held court in front of the blackboard, chalk in hand, revealing new ways of looking at the world.
In July 1970 the radical ideology that he inherited from his father led him to form the Survival group in order to campaign against environmental degradation and global militarism. He resigned from the IHES because of the military funding of a small percentage of the institute budget.
Awarded the Fields medal in 1966, he refused to attend the ceremony in Moscow because of his critical attitude towards the Soviet government. In 1988 he rejected outright the lucrative Crafoord prize. His attempt to secure a permanent position at the prestigious Collège de France ended in failure due to his political radicalism.
However, even after leaving the IHES Grothendieck still produced important mathematics. He wrote huge unpublished papers that began circulating informally, above all the massive unpublished autobiographical manuscript Récoltes et Semailles (Harvestings and Sowings). He admired the HuaYen Buddhists for the attention they paid to the relationships between things rather than the things themselves, in the belief that whatever notions of identity and individuality we have emerge from those relationships.
Grothendieck's personal life suffered from his obsessive work ethic and increasing instability of mind. His marriage to Mireille Dufour, with whom he had a daughter and two sons, broke down; he had a son from a relationship with Justine Skalba; but then he gradually withdrew from normal life and his children. He burned many of his papers and retreated to the French Pyrenees, where for more than 20 years he lived the life of a recluse, rarely speaking to people, refusing to indulge visitors or mathematical tourists, and reportedly subject to periods of religious mania and apocalypticism.
He is survived by his five children.
Alexander Grothendieck, mathematician, born 28 March 1928; died 13 November 2014
25 November 2014 © Guardian News and Media Limited