Title: Quasi-periodic solutions of KdV and Hyperelliptic curves
Speaker: Yuancheng Xie (Ohio State University)
Abstract: Abel and Galois told us that writing down a formula for the solutions of an algebraic equation of degree greater than five in terms of roots of the coefficients is impossible in general. Liouville showed we couldn’t even find solutions for an ODE as simple as the following one by quadratures: $\frac{dy}{dx} = x^2 + y^2$. Along this line, we shouldn't expect to have a formula for any nonlinear PDEs in general.
However, in 1967 four mathematicians from Princeton C.S. Gardner, J.M. Greene, M.D. Kruskal and R.M. Miura found a way to write down infinitely many exact solutions for the Kortwege-de Vries equation ($u_t = 6uu_x - u_{xxx}$) for rapid decaying initial data. Since then mathematicians found hundreds of other examples of this kind and applied them to algebraic geometry, representation theory, combinatorics, mathematical physics etc. In this talk I will sketch a nowadays classical method for writing down a quasi-periodic solution of KdV in terms of Riemann theta functions of a hyperelliptic curve.
