  We present results from two projects on lattice calculations for the
Higgs-Yukawa model. First we report progress on the search of first-order
thermal phase transitions in the presence of a dimension-six operator, with the
choices of bare couplings that lead to viable phenomenological predictions. In
this project the simulations are performed using overlap fermions to implement
the required chiral symmetry. Secondly, our study for applying finite-size
scaling techniques near the Gaussian fixed point of the Higgs-Yukawa model is
presented. We discuss the analytical formulae for the Higgs Yukawa model and
show results for a first numerical study in the pure $O(4)$ scalar sector of
the theory.
