How might this history of interaction between Christian Europe and Islamic Middle East affect events today?Talking about the Crusades
I would argue that the Crusades affect events today because of the fact that certain groups of people in the Middle East think that they are part of a long-standing pattern in which the West attempts to dominate the Muslim world. By contrast, I think that the Crusades are barely remembered in the West and do not affect our attitudes towards the Islamic Middle East.
In statements by Islamist militants, you often hear references to Western forces that compare them to the Crusaders. This is clearly an attempt to link them to the sorts of religiously-inspired violence that typified the Crusades. This shows that the radicals, at least, are influenced by "memories" of the Crusades.
So I would say that the Crusades continue to influence events today because Islamists believe that the Crusades (as well as modern events) are part of an attempt by the West to dominate and destroy Islam. As the link below says:
The focus of much of the violence is the United States, which is often seen in the Middle East as a new Crusader force, ignorant of Islam and deeply prejudiced against Muslims.