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When someone starts a new endeavor, how can I give him my best wish?

I wish you best luck in pursuing the Ph.D.

I hope you best luck in pursuing the Ph.D.

I wish everything would go smoothly for your research.

Is the subjunctive mood right?

I hope everything will go smoothly for your research.

I am a bit confused here because of the following usage of wish:

I wish the sun could rise from the West.

In this construction, the clause cannot be true but is wishful thinking. When I wish someone good luck, I hope I do not mean that it is wishful thinking.

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Hope and wish are synonymous but have different root meanings and inferences, therefore they cannot be used interchangeably all the time.

I think the best way to distinguish them is that "a hope" is a feeling, whereas "a wish" is an expression of your hopes. In idiomatic use, the difference between the two has blurred a little, but traditionally, you make a wish while you have a hope.

Here is an example where they are interchangeable:

I hope for good weather
I wish for good weather

In your example though, they are not interchangeable:

I wish you best luck in pursuing the Ph.D.

This is fine and idiomatic.

I hope you best luck in pursuing the Ph.D.

This is not idiomatically correct. It should be:

I hope you have good luck in pursuing the Ph.D.

Your last example has a different idiomatic meaning:

I wish everything would go smoothly for your research.

This would be taken as a lament - when we use "wish" about something that is already happening it suggests that we are wishing for a reversal of fortune or a change in present conditions. I would assume from this statement that things are not going smoothly at present.

A more idiomatic way to express a desire that things continue to go well would be:

I hope your research continues to go smoothly.

OR

I wish you well in your continued research.

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