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Professor Schwarz's earlier papers gave certain calculations and definitions that are now part of standard introductions to topology and geometry. Some of his work from this initial period concerned counting the number of closed geodesics on a Riemannian manifold. Among other things, he noticed that the (unparametrized) loop space of a manifold is an orbifold and not itself a manifold, thereby correcting a fallacy in earlier work by Bott and Morse. In another paper, he established that the fundamental group of a negatively curved manifold has exponential growth, a result which was rediscovered by and is sometimes attributed to Milnor. And he had a series of papers defining and discussing a certain topological invariant of a fiber bundle called the genus . The same notion was rediscovered by Smale, who applied it to the computational complexity of algorithmic problems, through the notion of topological complexity.
The thrust of Professor Schwarz's work in mathematical physics has been to apply topology to physics . In this aim he anticipated some of the work of the physicist Edward Witten. His most important result, joint with Belavin, Polyakov, and Tyupkin, was the discovery of instantons, which are solutions to classical gauge field equations that are localized in four Euclidean dimensions >. Although they are Euclidean objects, instantons manifest themselves in 4-dimensional Minkowskian spacetime as avenues for quantum tunneling. In this guise, they were used by t'Hooft to arrive at one of the notorious predictions of a Grand Unified Theory of physics (excluding only gravity): that the proton must be unstable. Finally, instantons reappeared in pure mathematics in Donaldson theory, a body of ideas which applies gauge field theory techniques to demonstrate many surprising non-trivialities about smooth 4-manifolds.
Another notorious prediction of Grand Unified Theories due to t'Hooft is the existence of magnetic monopoles. They are examples of solitons, which are like instantons but localized in one fewer dimension. Professor Schwarz also noticed that magnetic monopoles must exist for topological rather than merely computational reasons [3]. His argument fortified the classical philosophical speculation that particles in nature could be akin to knots in a cord; they could be indivisible and indestructible because of topology. It is yet possible that elementary particles such as quarks and electrons are kinds of solitons.
The thrust of Professor Schwarz's work in mathematical physics has been to apply topology to physics . In this aim he anticipated some of the work of the physicist Edward Witten. His most important result, joint with Belavin, Polyakov, and Tyupkin, was the discovery of instantons, which are solutions to classical gauge field equations that are localized in four Euclidean dimensions >. Although they are Euclidean objects, instantons manifest themselves in 4-dimensional Minkowskian spacetime as avenues for quantum tunneling. In this guise, they were used by t'Hooft to arrive at one of the notorious predictions of a Grand Unified Theory of physics (excluding only gravity): that the proton must be unstable. Finally, instantons reappeared in pure mathematics in Donaldson theory, a body of ideas which applies gauge field theory techniques to demonstrate many surprising non-trivialities about smooth 4-manifolds.
Another notorious prediction of Grand Unified Theories due to t'Hooft is the existence of magnetic monopoles. They are examples of solitons, which are like instantons but localized in one fewer dimension. Professor Schwarz also noticed that magnetic monopoles must exist for topological rather than merely computational reasons [3]. His argument fortified the classical philosophical speculation that particles in nature could be akin to knots in a cord; they could be indivisible and indestructible because of topology. It is yet possible that elementary particles such as quarks and electrons are kinds of solitons.
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Universeno. 384 (2023): 384-384
UNIVERSEno. 5 (2023): 231-231
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