In most shoes, the technology used in the mid sole is called Ethylene Vinyl Acetate. It is an elastic substance. Adidas stepped their quality up with a new type of material. They invented the Boost. Boost is made up of hundreds of nuggets of springy Thermoplastic Polyurethane. It absorbs the stress when walking on rough ground and the immediately goes back to it normal state. This is proven to make you run faster and for a longer time.
article: http://www.telegraph.co.uk/technology/news/9929446/Tech-Talk-Adidas-Energy-Boost-running-shoes.html
This is related to engineering because they had to find a way to engineer a material that would be soft and comfortable but at the same time bounce back.
