A Hundred Years of Mathematician Exploring Infinite Symmetries | Coffee and Theorems | Science

A Hundred Years of Mathematician Exploring Infinite Symmetries |  Coffee and Theorems |  Science
Undated photo of Indian mathematician Harish-Chandra.Pratham Cbh / Wikimedia

Today, October 11, marks 100 years since the birth of one of the most important Indian mathematicians in history. Harish-Chandra. Trained as a physicist under the tutelage of two Nobel Prize winners, he began his work in mathematics studying the space-time symmetries of Einstein’s theory of relativity. His later, purely mathematical works – the theory of infinite-dimensional representations of so-called semisimple groups – were very influential. They are one of the main sources of inspiration in development Langlands ProgrammeAn ambitious mathematical research roadmap related to areas such as algebra, number theory, analysis or geometry.

Harish-Chandra Mehrotra was born in 1923 in Kanpur, North India. After graduating from Allahabad University, he continued his studies in Physics at the Indian Institute of Science, Bangalore. CV Raman -Nobel Prize in Physics in 1930- and Homi J Baba.

He moved to Cambridge (United Kingdom) in 1945, just before Baba founded the prestigious institution. Tata Institute of Basic Research The following generations of Indian mathematicians and physicists will be trained. After some time, he immigrated to America, where he spent the rest of his life. Although he always had a desire to play a greater role in promoting science in India, his poor health and untimely death prevented this.

He received his doctorate in physics while at Cambridge University under his guidance Ball track In 1933 he was awarded the Nobel Prize in Physics. This exchange between disciplines is more common than you might think. Since the middle of the 20th century, fundamental physics has increasingly required advanced mathematics for its formalization, which is why many physicists approaching the frontier of mathematical research decide to go a step further and change their specialization.

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In the Harish-Chandra case, the logical rigor required in mathematical proofs was also an argument in favor of the leap. Although theoretical physicists rely on intuition, it guides their work and allows them to anticipate phenomena before verifying them experimentally, – in which, According to Harish-Chandra, it is not-, he felt more confident when he backed up his conclusions with rigorous mathematical reasoning. Without the necessary intuition, he thought his physical research would lead him astray.

During his thesis, he studied the so-called Harish-Chandra Poincaré group, which is an algebraic construction that captures the space-time symmetries of Einstein’s theory of relativity. In particular, A balanced group A set of operations that can be performed on a given object – for example, a square – without changing its properties. For example, when you rotate the square 90 degrees, it returns with the same orientation, so this function is part of the symmetry group of the square.

When considering the Euclidean plane—that is, the plane with the concept of distance—any rotation belongs to its symmetry group because, as the plane rotates, the distance between any pair of points does not change. However, as you stretch it, the distance between the points becomes larger, so this function does not fall into the symmetry group of the plane.

In Einstein’s theory of relativity, space-time is considered one with the so-called Minkowski metric Instead of the usual distance. and the Poincaré group is the group of spacetime symmetries, i.e. all transformations that do not change the Minkowski metric.

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In her doctoral research, Harish-Chandra studied representations of the infinite-dimensional Poincaré group. The representation of a group in a linear space is, for each element of the group, a linear transformation of the specified space – that is, a function that meets certain properties -, which allows the use of linear algebraic techniques in the study. groups. Linear space can be finite or infinite; As for the Poincaré group, it admits certain representations of infinite dimension which cannot be decomposed into finite representations. Although these are more difficult to manipulate than finite ones, they are an essential tool for studying certain groups.

As a mathematician, Harish-Chandra studied infinite representations of other groups, especially so-called semisimple groups. For his work, he was a strong candidate for the Fields Medal in 1958. However, one member of the jury apparently did not want the medal to be awarded to two mathematicians from the French Bourbaki group — Harish-Chandra and that year’s recipient, René Tham. Harish-Chandra was never a member of this group, but his style, attention to detail and rigor, was thought to be perhaps associated with the famous group.

After working at Columbia University (New York, USA), in 1963 he was appointed as a permanent professor at the Institute for Advanced Study in Princeton (USA). From 1969 he suffered several heart attacks, perhaps aggravated by his intense work schedule – he extended his work days into the night. His doctor insisted that he take annual leave, which he took advantage of—especially since he was a great admirer of Gauguin. A conference day. In memory of his friend and collaborator Emily Borel.

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Tomás Gomez de Quiroga He is a Research Scientist at the Higher Council for Scientific Research at the Institute of Mathematical Sciences (ICMAT).

Coffee and principles A section dedicated to mathematics and the environment in which it is created, coordinated by the Institute of Mathematical Sciences (ICMAT), in which researchers and members of the center describe the latest developments in the field, sharing meeting points between mathematics and other communities. and remember how cultural expressions and their development signifiers and coffee can be transformed into theorems. The name evokes the Hungarian mathematician Alfred Reny’s definition: “A mathematician is a machine that turns coffee into theorems.”

Editing and Coordination: Agate A. Rudder G Longoria (ICMAT).

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