Paul Erdos, a highly celebrated mathematician, made significant contributions to various fields of mathematics and pioneered new areas of study.
Erdos and his collaborator discovered the threshold phenomenon in the study of random graphs, marking a critical point where a graph transitions from being almost certainly disconnected to connected.
Erdos's collaborative approach, open problem posing, and emphasis on elegant proofs and the joy of mathematics have had a lasting impact on mathematicians worldwide.
Deep dives
The Career and Mathematical Achievements of Paul Erdos
Paul Erdos, a highly celebrated mathematician of the 20th century, made notable advances and pioneered new fields in mathematics. His prolific career, with over 1,500 papers and collaborations with around 500 other academics, is renowned. Erdos led an unconventional lifestyle, traveling around visiting other mathematicians rather than having a long-term position at a single university. He faced personal challenges growing up, including the trauma of losing siblings at birth and the rise of anti-Semitism. Erdos left a lasting impact on collaborative problem-solving, Ramsey theory, probabilistic number theory, and random graph theory.
The Discovery and Significance of Threshold Phenomenon
Erdos and his collaborator Alfred Renyi discovered the concept of the threshold phenomenon in their studies of random graphs. This phenomenon marks a critical point where a graph quickly transitions from being almost certainly disconnected to almost certainly connected as more edges are added randomly. This sudden change is akin to a phase transition in statistical physics. The threshold phenomenon has since been observed in various mathematical and physical contexts, representing a shift from a lack of connectivity to connectivity in a wide range of network models.
Erdos Numbers and Collaborative Mathematics
Erdos numbers are a measure of the collaborative distance between a mathematician and Paul Erdos, where, for example, a direct collaboration results in an Erdos number of one. Erdos's collaborative approach to mathematics inspired a culture of collaboration among mathematicians. He posed and conjectured numerous open problems, often attaching monetary rewards to incentivize their solutions. Erdos encouraged the pursuit of elegant and beautiful proofs, emphasizing the joy of mathematics. His influence extended to Hungary, where he is considered a founding figure of their mathematical school, and beyond to other mathematicians worldwide.
Erdos's Contributions to Number Theory and Probability
Erdos made significant contributions to number theory, including elementary proofs of mathematical theorems and probabilistic number theory. His work on random graphs and their property of being Erdos-connected provided insights into the study of network connectivity. Erdos also addressed the Ramsey theory, investigating the existence of order and regularity within mathematical structures and proposing difficult open questions. His work extended to various applications, including cryptography, epidemiology, and the study of prime numbers. Erdos's pursuit of new problems and exploration of mathematical phenomena continues to influence mathematicians today.
Personal Life and Legacy
Paul Erdos overcame personal challenges, including the loss of siblings during his birth and the rise of anti-Semitism. He led a nomadic lifestyle, traveling extensively to collaborate with mathematicians across the globe. Erdos's mathematical contributions earned him numerous honors and awards during his lifetime. His legacy includes an Erdos number system, reflecting collaborative distances between mathematicians, and a unique perspective on problem-solving and the pursuit of mathematical beauty. Erdos's impact on the field of mathematics endures through his influential work, unorthodox lifestyle, and emphasis on collaboration and joy in the pursuit of knowledge.
Paul Erdős (1913 – 1996) is one of the most celebrated mathematicians of the 20th century. During his long career, he made a number of impressive advances in our understanding of maths and developed whole new fields in the subject.
He was born into a Jewish family in Hungary just before the outbreak of World War I, and his life was shaped by the rise of fascism in Europe, anti-Semitism and the Cold War. His reputation for mathematical problem solving is unrivalled and he was extraordinarily prolific. He produced more than 1,500 papers and collaborated with around 500 other academics.
He also had an unconventional lifestyle. Instead of having a long-term post at one university, he spent much of his life travelling around visiting other mathematicians, often staying for just a few days.
With
Colva Roney-Dougal
Professor of Pure Mathematics at the University of St Andrews
Timothy Gowers
Professor of Mathematics at the College de France in Paris and Fellow of Trinity College, Cambridge
and
Andrew Treglown
Associate Professor in Mathematics at the University of Birmingham
The image above shows a graph occurring in Ramsey Theory. It was created by Dr Katherine Staden, lecturer in the School of Mathematics at the Open University.
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