Alexander Gratherdieck: Math s Great Mind. expected from mathematical scholars around the world. He seemed to be successful in every

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1 Castellano 1 Alec X. Castellano Professor Petersen Math 101 March Alexander Gratherdieck: Math s Great Mind Gratherdieck s contributions to the mathematical society help set the bar for what is expected from mathematical scholars around the world. He seemed to be successful in every aspect of math he tackled and his ideas and applications are still in use today. He revolutionized category theory and set the status quo for how algebraic geometry is studied around the world. Growing up in unstable environments must have helped him concentrate on what was important to his life considering his immediate success after reaching a stable lifestyle. Although, he died in seclusion away from most of his collogues his ideas and theories are greatly accepted around the world. He is known to be one of the most brilliant minds to ever study mathematics. Alexander was the product of an affair between a Jewish revolutionary and his French born mother. He was born on March 28, 1928 in Germany but was raised by his mother in France while his father fought in the Spanish Civil War. He was put into the hands of Lutheran Pastor Wilhelm Heydorn who raised him in A boarding school while his mother attempt to escape capture from occupying Germans during WW2. Alexandre spent the rest of his childhood with Wilhelm due to the fact that his mother later died of Tuberculous in a Nazi concentration camp and he never had a close relationship with his father. It is here where he discovered his interest in mathematical applications and ideas. He began studying mathematics at a pacifist university named College Cevenol in France. It was obvious that he was much more passionate about his studies than the rest of his class and he was encouraged to apply to other prestige

2 Castellano 2 universities to continue his studies. After dropping out a few times to concentrate on independent work, he graduated and found himself in the doctorates program at the University of Nancy where he wrote his dissertation on functional analysis. His abstract ideas famed him popularity and he began his career as a research professor at the Insitut des Hautes Estudes Scientifiques. He was offered positions at many other universities including Harvard, but turned them down due to the lack of respect he would have encountered because of the his peasant background. He later left Insitut des Estudes for lack of educational freedom to research and continued to work as a research professor at the University of Montpellier. Here is where his motivation earned a name for himself and he began publishing works on category theory, helping to set the modern foundation for Algebraic Geometry. He retired early and spent the rest of his life in seclusion in the Pyrenees mountains on the border of Spain. He died November 13, 2014 and his studies are still known to be the most influential and widely accepted works in mathematics today. Alexander Grothendieck was the leader in introducing the modern Theory of Algebraic Geometry; this included commutative algebra, homological algebra, sheaf theory, and category theory. He created a format in technical work through his theory of schemes. This in turn led to the study of algebraic number theory, algebraic topology and representation theory. He introduced and revolutionized the use of geometric objects, such as algebraic curves and surfaces through the study of algebraic equations for those objects; quadratic curves, cubic curves, and sextic curves were all finalized by Alexander. An algebraic curve over a field uses the equation, where is a polynomial in and with coefficients. A nonsingular algebraic curve is an algebraic curve over a variable which has no singular points. A point on an algebraic curve is simply a solution of the equation of the curve. A rational point is a point on the

3 Castellano 3 curve, where two variables are in the same field (Adamek). bivariate quadratic curve can be written as The general Understanding of geometric objects, such as algebraic curves and surfaces was one of Alexanders major areas of study. Emphasis on universal properties across varied mathematical structures brought category theory into mainstream as an organizing principal for math in general. It creates common language for describing similar structures and techniques seen in many different math systems. His notion of abelian category is now the basis of study homological algebra. A category has two functions; the ability to compose arrows associatively or the placement of an arrow to identify each object (Douglas). Some of Alexandre s most famous work was on functional analysis which analyzes vector spaces endowed in topology. In particular, infinite-dimensional spaces in contrast of linear algebraic expressions with finite-dimensional spaces (Balakrishnan). The extension of theory of measure, integration, and probability to infinite dimensional spaces. Linear algebra also plays part in functional analysis to help set boundaries and possible extensions of space. Many different theorems have been developed to work with functional analysis. One in particular is

4 Castellano 4 specter theorem which is basically conditions under which an operator or a matrix that can be diagonalized. In general, the spectral theorem identifies as a class of linear operators that can be modeled by multiplication for infinite or known dimensional spaces. There are many theorems known as spectral theorem, but one in particular has many applications in functional analysis. This equation is to find the length or vastness of a particular specter (Berezansky). Let A be the operator of multiplication by t on L 2 [0, 1], that is Let A be a bounded self-adjoin operator on a Hilbert space H. Then there is a measure of space (X, Σ, μ) and a real-valued measurable function fon X and a unitary operator U:H L 2 μ(x) such that where T is the multiplication operator: and This is the beginning of the research area of functional analysis which involves many different aspects of geometric algebra. One may spend their entire careers fully understanding the theory s and equations related to functional analysis because of its wide variety of abstract solutions and infinite dimensional possibilities. Category theory and functional analysis are responsible for many advances in technology and how we live our daily lives. Category theory helps set the basis of algebraic expressions whose expressions are taught in beginning, intermediate, and advanced mathematics. They help us measure and estimate how we build, construct, and develop infrastructures around the world.

5 Castellano 5 There really is no limit to how we use category theory, it defines a proved way of geometrics that gives developed minds a proven basis on what to rely on when using mathematics for any given reason. Functional analyses helped set the basis for black hole theory, space travel, and many other abstract ideas that we have proven over time. It allows us as a species to literally think outside the box and develop ways of thinking that we have not ever imagined probable, and prove it. As ridiculous as it may have sounded when Alexandre was first constructing his own research on the idea, it is now a proved and accepted mathematical category that NASA and many other space operators use when constructing research and experiments. In the future I can see functional analysis being put to much greater use and opening new parts of the galaxy that we have not yet discovered. Alexandre was a great mind that only comes around every other century or so. When the next genius of these standards comes around it is simply unimaginable what his theory s will construct for future societies.

6 Castellano 6 Works Cited Abramsky, S. & Duncan, R., 2006, A Categorical Quantum Logic, Mathematical Structures in Computer Science, 16 (3): Adamek, J. et al., 1990, Abstract and Concrete Categories: The Joy of Cats, New York: Wiley. Arnold, Douglas N. "Functional Analysis." Penn State Topology (1977): n. pag. Print. Balakrishnan, A. V. Applied Functional Analysis ed.2. New York: Springer-Verlag, Berezansky, Y. M.; Us, G. F.; and Sheftel, Z. G Functional Analysis Vol 1. Boston, MA: Birkhäuser, Conner, JJ O. "Alexander Grothendieck." Grothendieck Biography. University St Andrews, Nov Web. 05 Apr