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Algebraic geometry
Keywords Birational geometry, classification theory
Office Science Building B-520 (Toyonaka Campus)

When I was a student, I thought I knew number theory, geometry, but algebraic geometry was unfamiliar for me. One may say algebra and geometry are different fields, but you know the theory of quadratic curves and are aware of efficiency of algebraic methods for solving geometric problems. The field called algebraic geometry lies on such a line. When I was studying the theory of quadratic curves, I wondered, ''Why do they only treat special equations like quadratics? There are many other equations. But how can they be treated?". When I discovered the answer might lie on this field, I decided to enter this field. The easiest non-trivial equation has the form such as "the second power of y = an equation of x of degree three", which defines the so called "an elliptic curve". The theory of elliptic curve was one of the greatest achievements of nineteenth century and keeps developing today. Recently, the famous Fermat's conjecture has been solved using this theory. The theory of quadratic and elliptic curves involve only two variables x, y. It is natural to think of the equations with many variables. In fact the algebraic geometrists are expanding the theory, curves to surfaces and higher dimensional cases these days. Two dimensional version of elliptic curves are called K3 surfaces, which can be treated only using the theory of linear algebra(!) thanks to the Torelli's theorem. These days, the 3-dimensional versions, which is called Calabi-Yau threefolds are fascinating for algebraic geometrists like me. Somehow theoretical physicists are also interested in this field. To study Calabi-Yaus by specializing these to ones with a fiber structure (on which field, I'm now working) might be one method, but I have been thinking that a new theory is needed. These days, many intriguing new theories have appeared and one may find more!