  This paper investigates the interference nulling capability of reconfigurable
intelligent surface (RIS) in a multiuser environment where multiple
single-antenna transceivers communicate simultaneously in a shared spectrum.
From a theoretical perspective, we show that when the channels between the RIS
and the transceivers have line-of-sight and the direct paths are blocked, it is
possible to adjust the phases of the RIS elements to null out all the
interference completely and to achieve the maximum $K$ degrees-of-freedom (DoF)
in the overall $K$-user interference channel, provided that the number of RIS
elements exceeds some finite value that depends on $K$. Algorithmically, for
any fixed channel realization we formulate the interference nulling problem as
a feasibility problem, and propose an alternating projection algorithm to
efficiently solve the resulting nonconvex problem with local convergence
guarantee. Numerical results show that the proposed alternating projection
algorithm can null all the interference if the number of RIS elements is only
slightly larger than a threshold of $2K(K-1)$. For the practical sum-rate
maximization objective, this paper proposes to use the zero-forcing solution
obtained from alternating projection as an initial point for subsequent
Riemannian conjugate gradient optimization and shows that it has a significant
performance advantage over random initializations. For the objective of
maximizing the minimum rate, this paper proposes a subgradient projection
method which is capable of achieving excellent performance at low complexity.
