Complexity of random smooth functions on compact manifolds Liviu Nicolaescu 15B5242C1053C6558K0558J5060D0560G1560G60random Morse functionsKac-Rice formulaspectral geometryrandom matrices We prove a universal result relating the expected distribution of critical values of a random linear combination of eigenfunctions of the Laplacian on an arbitrary compact Riemann $m$-dimensional manifold to the expected distribution of eigenvalues of a $(m+1)\times(m+1)$ random symmetric Wigner matrix. We then prove a central limit theorem describing what happens to the expected distribution of critical values when the dimension of the manifold is very large. Indiana University Mathematics Journal 2014 text pdf 10.1512/iumj.2014.63.5321 10.1512/iumj.2014.63.5321 en Indiana Univ. Math. J. 63 (2014) 1037 - 1065 state-of-the-art mathematics http://iumj.org/access/