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 – on the theory of infinite-dimensional representations of so-called semisimple groups – have been very influential. They were one of the main sources of inspiration in the development of the Langlands programan ambitious mathematical research roadmap that relates areas such as algebra, number theory, analysis or geometry.
Harish-Chandra Mehrotra was born in 1923 in Kanpur, northern India. After graduating from Allahabad University, he continued his studies in physics at the Indian Institute of Science in Bangalore under the guidance of CV Raman –Nobel Prize in Physics in 1930– and Homi J. Bhabha.
He emigrated to Cambridge (United Kingdom) in 1945, just before Bhabha founded the prestigious Tata Institute of Fundamental Research –where the following generations of Indian mathematicians and physicists would be trained. Shortly after, he moved to the United States, where he spent the rest of his career. Although he always had the desire to have played a greater role in promoting science in India, his ill health and premature death prevented this.
He obtained his doctorate in physics while at the University of Cambridge, under the direction of Paul Dirac –also Nobel Prize winner in Physics, in 1933–, but, upon completion, he decided to change his research topic to mathematics. This transfer between disciplines is more common than you might think. Fundamental physics, since the middle of the 20th century, needs increasingly advanced mathematics for its formalization, which is why many physicists who are approaching the frontier of mathematical research decide to go one step further and modify their specialization.
In the case of Harish-Chandra, the logical rigor required in mathematical proofs was also an argument in favor of the leap. While theoretical physicists rely on intuition, which guides their work and allows them to anticipate phenomena before they are verified experimentally, – of which, according to Harish-Chandra, it lacked–, he felt more confident when he could back up his results with a rigorous mathematical argument. Without the required intuition, he thought his physical research could lead him astray.
During his thesis, Harish-Chandra studied the so-called Poincare group, which is an algebraic construction that captures the space-time symmetries of Einstein’s theory of relativity. Specifically, a symmetry group of a given object – for example, a square – is the set of operations that can be performed on it, without altering its properties. For example, when you rotate the square 90 degrees, it returns to have the same appearance, so this operation is part of the group of symmetries of the square.
When considering the Euclidean plane – that is, the plane together with the notion of distance – any rotation belongs to its group of symmetries, since, when the plane rotates, the distance between any pair of points does not change. However, when you stretch it, the distance between the points will become greater, so this operation does not fall within the group of symmetries of the plane.
In Einstein’s theory of relativity, space-time is considered together with the so-called Minkowski metric –instead of the usual distance–. And the Poincaré group is the group of space-time symmetries, that is, all transformations that do not change the Minkowski metric.
In his doctoral thesis, Harish-Chandra studied representations of the infinite-dimensional Poincaré group. A representation of a group in a linear space consists of assigning, to each element of the group, a linear transformation of said space – that is, a function that meets certain properties -, which allows the use of linear algebra techniques in the study of the groups. Linear space can be finite or infinite; In the case of the Poincaré group, it admits certain representations of infinite dimension, which cannot be decomposed as finite representations. Although these are more difficult to handle than finite ones, they are a necessary tool to study certain groups.
As a mathematician, Harish-Chandra studied the infinite representations of other groups, more specifically the so-called semisimple groups. For these works, he was a strong candidate for the Fields Medal in 1958. However, it seems that a member of the jury did not want the medal to be given to two mathematicians from the French Bourbaki group – he considered both Harish-Chandra and René Thom to be. , who received the award that year. Harish-Chandra was never a member of this group, but his style, with its attention to detail and rigor, was perhaps thought to be close to that associated with the famous collective.
After working at Columbia University (New York, USA), in 1963 he was appointed permanent professor at the Institute for Advanced Studies in Princeton (USA). Since 1969 he had several heart attacks, possibly aggravated by his intense work schedule – which extended his work day until late at night. His doctor insisted that he take annual vacations, which he took advantage of to encourage his love of painting – in particular, he was a great admirer of Gaugin – and he finally died in 1983 of a heart attack while walking in the afternoon on the last day of a conference. in honor of Émile Borel, his friend and collaborator.
Tomás Gómez de Quiroga He is a scientific researcher at the Higher Council for Scientific Research at the Institute of Mathematical Sciences (ICMAT)
Coffee and Theorems is 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 advances in this discipline, share meeting points between mathematics and other social and cultural expressions and remember those who marked their development and knew how to transform coffee into theorems. The name evokes the definition of the Hungarian mathematician Alfred Rényi: “A mathematician is a machine that transforms coffee into theorems.”
Editing and coordination: Ágata A. Timón G Longoria (ICMAT).
You can follow SUBJECT in Facebook, x and instagramor sign up here to receive our weekly newsletter.
#years #mathematician #explored #infinite #symmetries