Learn Turkish: Patron gelmezse toplantı başlamaz. - If the boss doesn't come, the meeting doesn't start.

It combines the verb root gel (come), the negative aorist suffix -mez (doesn't), and the conditional suffix -se (if). Together, they form the negative conditional aorist, meaning 'if he doesn't come'.

Why is the negative suffix -mez on the first verb but -maz on the second verb?

This is due to vowel harmony. The root gel has a front vowel (e), so it takes -mez to become gelmezse. The root başla ends in a back vowel (a), so it takes -maz to become başlamaz. Both are the exact same negative aorist suffix.

Where is the suffix for 'he' or 'it' on these verbs?

There isn't one! In Turkish, the third-person singular (he, she, it) doesn't use a personal suffix. The endings -mezse and -maz inherently apply to a third-person subject. The nouns patron and toplantı simply specify exactly who or what those subjects are.

I remember that verbs ending in vowels do weird things in the aorist tense. Does başla change here?

Not in the negative! While the positive aorist can be tricky with vowel-ending roots, the negative aorist is very regular. You just take the verb root başla and attach -maz directly to the end. No letters are dropped or altered.