In this question we search for a big list of non elliptic operators whose Fredholm index is finite or whose Fredholm theory is extensively discussed. The main motovation is the conference linked in the head of the following post:The subject of the conference is "Fredholm Theory of Non Elliptic Operators". I read the abstract of talks in the conference but I did not underestand that what is a precise non elliptic differential operator whose index is finite. In the above MO post it is discussed that ellipticity play a crucial role for Fredholm ness but there are some sence of index theory for non eliptic operators. I would appreaciate if you give answer, 1 item per answer, which present either a precise example of non eliptic operator of finite index or contains an extended explaination of one of the abstract listed in the above conference on "Fredholm Theory for non Elliptic operators".

I ask moderators to consider this question as a wiki question.