Alexander Grothendieck Kyle Aguilar Alexander Grothendieck's Life Alexander Grothendieck was born on March 28, 1928. His parents names were Alexander Schapiro, a Russian Jew who fought in numerous revolutions against Russian Tsars, and Johanna Grothendieck, a German who met Schapiro at Berlin. The family lived during the Hitler Era and there was a law passed stating if civil servants were not of Aryan descent, they were forced to retire. Because of this, in 1933, the father returned to Paris and Grothendieck was arranged to live with a pastor named Wilhelm Heydorn and his wife in Hamburg from 1934-1939. During this time, Grothendieck went to elementary school and would often study by himself in the gymnasium. Meanwhile, his parents traveled back to Spain to support the Republicans during the Spanish Civil War from 1936-1939. When the parents returned, a law passed stating that anyone of German descent were to be sent to internment camps. Generally speaking, because of Grothendieck's race and the time era, he and his parents were sent to internment camps in France. The mom and dad were separated, the mom staying with Grothendieck. Grothendieck was able to continue his education at a place that was five kilometers near the camp and also received private tutoring. During Grothendieck's studies, he was able to get into numerous colleges and universities, including University of Montpellier, University of Nancy, University of Sao Paulo, University of Kansas, and Institut des Hautes Etudes Scientifiques, to study mathematics. In high school, he was very disappointed because of the absence of certain math (specifically geometric) definitions. He did not like that there was an absence of length of a curve, area of a surface, and volume of a solid. He thought going to college would change that. However, he did not find the education very helpful. Grothendieck was exceling in mathematics with such ease, that all his professors would advise him to travel to various colleges and attend seminars. During his travels and studies, he was able to establish a number of theories that would soon help out future mathematicians as well as become one of the members of the Bourbaki group. In addition to his works, he married a woman named Mireille and had three kids named Johanna, Mathieu, and Alexandre. He also received the Fields Medal in 1966. However, the presentation was held in Moscow and being a strong pacifist, he refused to travel to Moscow and let the director of one of the schools he went to receive the medal on his behalf. In rebellion, he went to North Vietnam in 1967 and lectured at a secret location in Hanoi. Following that a few years later, he left Institut des Hautes Etudes Scientifiques, the school he was studying at during that time, because he discovered the school was receiving some funds from military sources. This led to him abandoning mathematics and changed to political protest - specifically on nuclear proliferation. Unlike his lectures in mathematics, the political protests were very ineffective. He retired at sixty years old and his research was published in his honor. He was even awarded the Crafoord Prize in 1988 but declined it completely because he did not want the money and the award was for his work earlier in his career. During his retirement, he wrote thousands and thousands of pages (some math, some non-math) of his theorems and research. Then suddenly in 1991, he left home without informing anyone to an unknown location. In fact, in 2010, he tried to erase all traces of his life. He wrote a letter to one of his students to get rid of all his files and publications, refusing any of them to be republished. At age 86, November 13, 2014, Grothendieck passed away. Today he is considered one of the greatest mathematicians of the 20th century. Alexander Grothendieck's Mathematical Works Grothendieck developed a lot of theories and proofs over the course of his life. He focused his works on proving theories and solving difficult problems that other mathematicians he worked with couldn't solve. Along the way, he was able to develop some of his own understandable concepts that helped future mathematicians. In fact, before Grothendieck made all these discoveries, his professor at the University of Montpellier and future collaborator, Elie Cartan, believed Lebesgue solved all outstanding mathematical problems. Note that Lebesgue was a mathematician who founded the theory of measure and generalized the notion of the Riemann integral. However, Grothendieck was able to formulate many theories. First, while Grothendieck was studying at the University of Nancy, Schwartz and Dieudonne were trying to develop a general theory for locally convex spaces by studying Frchet spaces and their limits. However, they ran into some problems. When they consulted Grothendieck, he was able to solve all the problems in less than a year. Below is information about Frechet spaces. Frechet spaces: A Frechet space (vector space) is \a complete and metrizable space, sometimes with the restriction that it can be locally convex". As seen in this example, the space in between the two endpoints of the blue line is considered Frechet space. This means: T = (S; τ) is a topological space if and only if 8x; y 2 S such that x 6= y. So that means: 9U 2 τ : x 2 U; y2 = U and 9V 2 τ : y 2 V; x2 = V Generally speaking, this is saying that there are open sets of open space. There are no two sets of space that are the same. 2 Another definition: Note that vector spaces such as Co(X) with non-compact X occur often. V is a vector space with metric d(; ) Let d is translation invariant (meaning that the vector has translation symmetry), such that: dx(x + z; y + z) = d(x; y) for x; y; z 2 V The basis at v 2 V consists of open balls centered at v fw 2 V : d(v; w) < rg The translation invariance implies: v + Br = fv + b : b 2 Brg Br is an open ball with radius centered at 0. This will determine the topology of the whole vector space. Topology on V is locally convex if there's a local basis at every point (or by definition, 0) that has convex sets. A convex set contains elements from the vector space which all points between the two parts on the vector are part of the set. If the metric is a complete set (meaning it contains all points), then V is a Frechet space. Grothendieck published his proof of Frechet spaces in his doctorial thesis call Produits tensoriels topologiques et espaces nucleaires. It's currently published under the Bourbaki group. The paper he wrote was to demonstrate his understanding of another area of mathematics rather than just creating and finding new theories. According to sources, Grothendieck's second thesis was on the sheaf theory (showing his interest in algebraic geometry), where he was supposed to show his greatest work. However, even the best of the best make mistakes and Cartan pointed out to him one day that there were some incorrect statements Grothendieck pointed out in his thesis. The \Golden Age" of Grothendieck was his prime where he was able to make a lot more discoveries and proofs of different theories than he did in the past. These include the \theory of schemes" in the 1960s, work on the theory of topoi, algebraically proving Riemann-Roch's theorem, an algebraic definition of the fundamental group of a curve, etc. A few of these definitions of what he discovered are listed below: Riemann-Roch Theorem Definition: This theorem computes the dimension of the space of meromorphic functions with zeros and poles. If D is a divisor on compact Riemann surface of genus g then l(D) − i(D) = deg (D) + 1 − g If D = 0 and substitute, then: l(0) − i(0) = deg (0) + 1 − g l(0) = 1 and deg(0) is 0 because it's the sum of coefficients of the zero divisor. i(0) = l(K) L(K) is isomorphic to space. So: 1 − i(0) = 0 + 1 − g 3 and i(0) is equal to dimension of the space. And if we substitute D = K into the formula: l(K) − i(K) = deg (K) + 1 − g And since l(K) = i(0) = g and i(K) = l(0) = 1 then g − 1 = deg (K) + 1 − g So the formula is: deg (K) = 2g − 2 Grothendieck is also well known for his \Tohoko paper" in which he talks about numerous mathematical concepts including abelian categories, sheaves of modules, resolutions, derived functors, and the Grothendieck spectral sequence. In fact he was even rewarded the Fields medal for the advances in algebraic topology he showed in that paper. Quoting from \The Grothendieck Festschrift" it stated all the work that Grothendieck did during his golden age saying, \The mere enumeration of Grothendieck's best known contributions is overwhelming: topological tensor prod- ucts and nuclear spaces, sheaf cohomology as derived functors, schemes, K-theory and Grothendieck-Riemann- Roch, the emphasis on working relative to a base, defining and constructing geometric objects via the functors they are to represent, fibred categories and descent, stacks, Grothendieck topologies (sites) and topoi, derived categories, formalisms of local and global duality (the 'six operations'), tale cohomology and the cohomological interpretation of L-functions, crystalline cohomology, 'standard conjectures', motives and the 'yoga of weights', tensor categories and motivic Galois groups. It is difficult to imagine that they all sprang from a single mind." Collaboration With Other Scholars Grothendieck worked with numerous amounts of other scholars and mathematicians to help get where he was at.