The Life and Works of The Life and Works of John Von NeumannJohn Von Neumann

A presentation by Shreyak Chakraborty

John Von NeumannJohn Von Neumann

John Von Neumann was a Hungarian-American mathematician who made major contributions to a vast number of fields in mathematics especially set theory, operator theory, topology , optimization and analysis.

He is probably one of the best mathematicians of modern times.

Jean Dieudonne called him “the last of the great mathematicians”.John Von Neumann taught us that

“MATHEMATICS IS DRIVEN NOT PRIMARILY BY LOGIC BUT BY IMAGINATION”

I believe a lot many of the major modern mathematical techniques used today have been founded by ideas of John Von Neumann.

His mathematical excellence is widely known and recognized.

Born December 28, 1903

Died February 8, 1957

Residence Washington D.C, United States

Nationality Hungarian and American

Institutions University of BerlinInstitute For Advanced Study,

Princeton

Notable Awards Enrico Femi Award in 1956

Early LifeEarly Life

John Von Neumann was a child prodigy in areas of language, memorization and mathematics.

By the age of six, he could memorize telephone directories on sight and displayed prodigious mental calculation abilities!

By the age of eight, he had attained mastery in calculus and began studying advanced topics at the age of 15.

He published his first two major papers, one of which gave the modern definition of Ordinal Numbers.

He received his PhD at the age of 22 years.

From the time John started his work (in 1922), he has been recognized by the world’s leading mathematicians for his mathematical brilliance.

CareerCareerJohn Neumann had a breathtaking career with more than 32

major published papers in 1929 at the rate of one major one major paper per month!!

Such was his caliber.Between 1926 and 1930, he taught as Privatdozent at the

University of Berlin, being one of the youngest in history.In 1932, he was invited to the institute for Advanced study

and he became one of the first 4 people who were appointed faculty there.

He solved one of David Hilbert's 23 theoretical problems and collaborated on developing an algebraic ring with profound applications in quantum physics. During World War II he participated in the development of the atomic bomb. After the war he made major contributions to the development of high-speed computers; one of his computers was essential to the creation of the hydrogen bomb. As coauthor of Theory of Games and Economic Behavior (1944), he was one of the founders of game theory.

Some contributions of Some contributions of John Von NeumannJohn Von Neumann

John Neumann’s contributions are immense.He is most famously known for: Abelian Von Neumann Algebra Affiliated operator Arithmetic Logic unit (ALU) Axiom of Regularity Bounded set (topological vector space) Computer virus Density Matrix Hyperfinite Type II Factor Minimax Theorem Quantum mutual information Stochastic Computing and Stochastic Matrix Analysis

A deeper look into Neumann’s A deeper look into Neumann’s Mathematical worksMathematical worksOPERATOR THEORYJohn Neumann was the first mathematician to apply the Operator

Theory to the framework of Quantum mechanics.Operators are now invariably used in almost every branch of

mathematics and physics. Operators

Is the Schrodinger Equation and can be written as

Here are some of the concepts given by John Von Neumann:

DOUBLY STOCHASTIC MATRIX

Any square matrix the some of whose elements in each row and column equals 1 is called a doubly stochastic matrix.

A simple example is the unit matrix.

BOUNDED SET IN TOPOLOGICAL VECTOR SPACESThis concept is commonly used today in various fields of

mathematics.Any set in a topological vector space is called bounded if every

neighborhood of the null vector can be inflated or expanded to include the set.

VON NEUMANN CELLULAR AUTOMATIONThe technique of Cellular Automation is widely used today in

computational analysis and advanced mathematics.The Von Neumann automation is based on the theorem of

self replication in which a binary signal transmitted repeatedly around a loop using multiple states generates a replicated signal using a confluent cell.

This serves as a basis for logical designing today.

ARITHMETIC LOGIC UNITJohn Neumann effectively described the mechanism of

calculation using binary signals which lead to the development of the arithmetic logic unit i.e. a computer’s ability to calculate.

Illustrative simulation of the mechanism of Self-Replication Of a signal used frequently in Cellular automations.

Von Neumann Trace Inequality

For any nxn complex matrices A and B with singular values

And

Respectively, then

The Concept of an ExtractorFirst proposed by John Von Neumann, a “randomness

extractor” is a function that when applied to a high entropy system generates a random output that is shorter but uniformly distributed.

We can say that an extractor outputs a random sample from a semi-random input.

Image credit:Shreyak Chakraborty

Bounded Set mapping in a Topological Vector space

JULIA SET REPRESENTATION

One would notice that John Neumann has also contributed to the world of fractals in many ways.

An essence of Von Neumann Algebraic Geometry in n-dimensions Which is now used in many fields of Theoretical Physics.

Diagram showing an orthogonal projection of four operators. This has serious applications in Abelian-Von Neumann Algebra

ConclusionConclusionThe mathematical works of John Von Neumann have deeplyinfluenced the mathematical and scientific research in the

20th Century.

He will be remembered forever for the beauty and elegance of his equations, theorems and axioms.