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I am mainly interested in the mathematical foundation of string theory. String theory is a physical theory which some physicists are trying to use to unify all the fundamental interactions in the universe. The basic assumption of string theory is that the fundamental constituents of our universe are strings (one-dimesional objects), not particles (zero-dimensional objects). Up to now, as a physical theory, string theory is still a theory without experimental proof, though it is consistent with all the existing experimental results. But the mathematics underlying string theory seems to be related to all branches of mathematics. Some very abstract mathematical theories now find their use in string theory. More interestingly (at least to mathematicians), intuition of string theorists predicted surprising mathematical theorems, and techniques used by string theorists supplied powerful tools to solutions of mathematical problems. Therefore, no matter whether string theory as a physical theory is correct or not, the mathematics underlying string theory will always be interesting to mathematicians.
Since I am mainly interested in using the mathematics underlying string theory to solve mathematical problems, my research must be mathematically rigorous. Though the physical intuition is very important and powerful, most of the arguments used by physicists do not solve mathematical problems at all. In most cases, these arguments are more or less the same as mathematicians' discussions on some beautiful conjectures. To prove these mathematical conjectures completely, we have to insist the mathematical standard of rigor. In physics, all the theories, no matter whether they are rigorous or not, will have to be tested eventually by experiments. Though the mathematical rigor might be useful when a theory is first proposed or when a theory cannot be tested by experiments yet, physicists will almost always ignore this issue whenever there is a hope to compare their theories with experimental results. But in mathematics, a mathematical problem is solved if and only if the proof is rigorously written down. For example, in the original paper by Wiles proving Fermat's Last Theorem, there is a gap which was filled after one year by Taylor and Wiles. Without this last step by Taylor and Wiles, Fermat's Last Theorem would not be considered as solved.
Certainly solving mathematical problems rigorously using ideas from physics does not mean filling details or reformulating the existing physical theories mathematically. Instead, it means creating and developing new mathematical theories so that the mathematical problems can be reformulated, studied and finally solved in the framework of the new theories. These new mathematical theories might also be useful in physics
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