# Normal form-type arguments in the study of dispersive PDEs

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Bourgain used normal form reduction and the I-method to prove global well-posedness of one-dimensional periodic quintic NLS in low regularity. In this talk, we discuss the basic notion of normal form reduction for Hamiltonian PDEs and apply it to one-dimensional periodic NLS with general power nonlinearity. Then, we combine it with the "upside-down" I-method to obtain upperbounds on growth of higher Sobolev norms of solutions. In the case of cubic NLS, we explicitly compute the terms arising in the first few iterations of normal form reduction to improve the result. If time permits, we also discuss how one can use a normal form-type argument to prove unconidtional uniqueness of the periodic mKdV in $H^{1/2}$. The first result is a joint work with James Colliander (University of Toronto) and Soonsik Kwon (KAIST), and the second result is with Soonsik Kwon.