@MISC{Lin_researchstatement, author = {Jiayuan Lin}, title = {RESEARCH STATEMENT}, year = {} }

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Abstract

My primary research interests in algebraic geometry lie in the Minimal Model Program and its applications, moduli spaces of stable maps (curves), and moduli spaces of branchvarieties. Recently I have also paid close attention to the new advances in computational algebraic geometry. 1 Minimal Model Program and its applications In many branches of mathematics, classifications among objects up to certain relations are central themes. For example, topologists can classify topological spaces up to homeomorphism or up to the weaker relation of homotopy. Similarly, in algebraic geometry, we classify algebraic varieties up to either isomorphism or a weaker relation, birational equivalence (two varieties X and Y are birationally equivalent if there exist rational maps f: X − → Y and g: Y − → X such that g ◦f and f ◦ g are identity maps on some open subsets U ⊂ X and V ⊂ Y). Among algebraic varieties in the same birational equivalence class, we want to single out some “good ” representatives. Such good representatives are called minimal models. It is well known that every surface has a minimal model. Is there a minimal model for every higher dimensional algebraic variety? The answer was unknown for a long period of time even for threefolds. At first people tried to find a minimal model in the smooth category, but this turned out to be impossible. Gradually people realized that one can only