Learn Hardcore Turkish: Umarım siz de bize katılırsınız.

Umarım means I hope and can directly introduce a clause. Turkish doesn’t need a that/ki here: Umarım siz de bize katılırsınız is complete. You can say Umarım ki for emphasis or style, but many consider the ki redundant; both forms are common and acceptable.

Why is katılırsınız used (the -r aorist) instead of a clear future like katılacaksınız?

Katılırsınız is the aorist (general/habitual), but with verbs of wishing or thinking (like umarım), Turkish often uses the aorist to mean a polite, non-committal future: I hope you’ll join. Katılacaksınız is a definite, scheduled future (You are going to join), which can sound too firm with umarım. The aorist here keeps the hope soft and less imposing.

Could I say katılıyorsunuz instead?

Not for a future hope. Katılıyorsunuz is present continuous (you are joining/attending now) or a near-term arrangement being carried out. With a hope about a future action, prefer katılırsınız (soft future) or katılacaksınız (definite future) depending on nuance.

What does siz de mean? Does de modify you or us?

De is the additive clitic meaning also/too, and it modifies the word it follows. In siz de, it means you too (as well as others). If you said bize de, it would mean to us too (as well as to others). So word placement changes what is being marked as “also.”