**Abstract：**

It has been a long standing problem since early 1980s posed by the late J.L.Lions that concerning the Uniform Controllability of Wave Equations in highly oscillating (hetrogeneous) medium. The identification of the limit of (optimal) controls and the (optimal) control for the homogenized limiting problem; as well as counterexamples to the uniform controllability were known since late 1980s and early 1990s. In this lecture, I will discuss my recent joint work with Zhongwei Shen on this problem. We obtain the sharp convergence rate for solutions of wave equations in a bounded domain with rapidly oscillating periodic coefficients. The results are used to prove the exact boundary controllability that is uniform in ε - the scale of the microstructure, for

the projection of solutions to the subspace generated by the eigenfunctions with eigenvalues less that Cε^{−2/3}. I shall discuss also the related eigenvalues and eigenfunctions estimates which are uniform in ε.

**Biography:**

Fanghua Lin is a Silver Professor of Mathematics at the Courant Institute of Mathematical Sciences, NYU. He holds a B.S. from Zhejiang University and a Ph.D. from the University of Minnesota. Professor Lin's research interests include classical and applied analysis, partial diffe rential equations, geometric measure theory, and calculus of variations. In addition, he has published more than 180 original research articles in top journals of mathematics, including Annals of Mathematics, Acta Mathemtica, Journal of the American Mathematical Society, and Commutations in Pure and Applied Mathematics.

Professor Lin is a Fellow of the American Mathematical Society and the American Academy of Arts and Sciences. He has been awarded the Alfred P. Sloan Research Fellowship in 1989, the Outstanding Young Researcher award from NSF-China in 1998, the Chang Jiang Chair Professorship from the Ministry of Education of China in 1999, an Oversea Assessor for Chinese Academy of Sciences in 2002, the M. Bocher Prize of American Mathematical Society in 2002, the S.S. Chern Prize at ICCM in 2004 and a 1000-talent (short) program in 2007.

*Seminar by the NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai*