How to force lower case in the title?

You could use \MakeLowercase:

\documentclass{amsart}

\title{Topics on \MakeLowercase{de} R\MakeLowercase{ham} cohomology}
\author{The Author}

\begin{document}

\maketitle

\end{document}

I am not sure about using lowercase for the last name; perhaps

\title{Topics on \MakeLowercase{de} Rham cohomology}

could be better?

1. Use of \Makelowercase{text} will do the work, but lots of warning messages will come due to variation in packages.
2. use of \lowercase{text} will do the same work. Try any one of them.