Russian-American mathematician and Wolf Prize laureate Yakov Eliashberg delivered a public talk at IIT Madras on Thursday, as part of the inaugural annual TNQ Distinguished Lecture in Mathematics.
The talk was part of TNQ Numbers and Shapes, a new initiative by the TNQ Foundation “to advance the study of mathematics and to support mathematical research in India. This will be through a mentorship and collaboration programme that gives students exposure, guidance, and travel support to enable them to pursue cutting-edge research in pure mathematics”. The flagship event of the initiative is the lecture that will be held in Chennai every year during a week-long workshop.
The initiative will focus on geometry and topology, number theory, ergodic theory and dynamical systems, algebraic geometry, and probability and analysis.
Commemorating pure mathematics
Ahead of the lecture, TNQ Foundation head Mariam Ram said, “TNQ launched the Distinguished Lectures in the Life Sciences series in 2008, bringing together well-known life science experts to India and giving students and researchers opportunities to interact with their field’s leaders.”
Ms. Ram said she hoped the Foundation’s new mathematics programme would lead to deeper and more meaningful collaborations between mathematicians in India. “These lectures will commemorate India’s and particularly Tamil Nadu’s long history in pure mathematics,” she said, adding, “Our hope is that TNQ Numbers and Shapes will go at least a little way in inspiring students to produce beautiful mathematics.”
Professor Eliashberg and Simon Donaldson won the Wolf Prize for mathematics in 2020 “for their contributions to differential geometry and topology”.
According to the TNQ Foundation, the emergence of symplectic and contact topology — a field in which Professor Eliashberg is a leader — has been one of the most important and long-term advances in mathematical research in recent decades. He is currently the Herald L. and Caroline L. Ritch Professor of Mathematics at Stanford University.
Symplectic topology
The talk was titled “The Strange and Wonderful World of Symplectic Geometry”. At the heart of symplectic geometry is the symplectic manifold. Simply speaking, a manifold is a space that follows the rules of Euclidean geometry locally but could have a non-Euclidean structure globally. This is like how the earth seems flat to an ant crawling on it but is revealed to be spheroidal when an astronaut looks at it from outer space. (A sphere is a type of two-dimensional manifold.) Symplectic geometry is concerned with the area-related characteristics of manifolds with an even number of dimensions (2, 4, 6, 8, and so on).
Before Professor Eliashberg’s lecture, M.J. Mahan of the Tata Institute of Fundamental Research, Mumbai, introduced the mathematics that forms the foundation of topology — from Euclid’s proof of the existence of infinitely many prime numbers to the Gauss-Bonnet theorem. “Whether it’s number or shapes, there’s something underlying them that can be worked out with a paper and pencil,” he said.
Professor Eliashberg began with a brief history of the central figures of topology, including Henri Poincaré, Misha Gromov, and William Rowan Hamilton. “Solving serious mathematical problems requires many people,” he said. Then he moved to the geometrical aspects of topology and then mechanics. According to him, “symplectic geometry was born as a geometric language of classical mechanics”.
Students are taught in school to solve classical mechanics problems using Newtonian mechanics. Symplectic geometry is connected with the physics of the real world through an alternative approach called Hamiltonian mechanics. For example, say a ball is rolling down a slope. The state of this dynamical system in Hamiltonian mechanics is given by its position and momentum in three dimensions. The combination of these six variables denotes the ball’s phase space, which can be represented and analysed as a symplectic manifold.
Symplectic topology offers an alternative approach to solving problems involving systems with complicated phase spaces.
As he moved through increasingly involved ideas, Professor Eliashberg also recounted how they have been applied to study problems in thermodynamics, celestial mechanics, and chaotic systems.
Himalayan retreat
TNQ Numbers and Shapes and the Institute of Mathematical Sciences (IMSc), Chennai, organised a four-day workshop that Professor Eliashberg has been conducting since September 23. According to Dishant Pancholi of the IMSc, more than 20 students and mathematicians participated.
After the talk, Professor Eliashberg will lead a small group of Indian mathematicians at a retreat in the Himalayas dedicated to working on solutions to a problem in symplectic topology called the nearby Lagrangian conjecture.
According to a note on the TNQ website, “Various methods have been developed to address this problem, and several partial results have been obtained, but a complete resolution of the conjecture is still a long way off.”
Participants in the workshop and the retreat were selected in a process led by Professor Mahan and Professor Pancholi.
Published - September 26, 2024 09:21 pm IST
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