Can I make potatoes sweet by cooking them? I've heard that cooking "breaks down" carbohydrates such as starch. I also know that starch is a complex carbohydrate, which is essentially made out of simple carbohydrates (sugars).

Potatoes contain a lot of starch, so I was wondering: is there a way of cooking potatoes that will break the starch down and make them sweet?

Of course, I'm not expecting to be able to make them sweet like candy, but I'm wondering if I can get some amount of sweetness out of them.

I'm just asking out of curiosity; I don't have any particular goals or dishes in mind.
A raw potato is 79% water, 17% carbohydrates (of which 88% is starch) and 2% protein. So your question is essentially: can starch be turned into sugar? 

In theory the answer is clearly 'yes': this is what plants do when they break down the starch created as the final product of photosynthesis, and what happens in the human digestive system. Starch degradation is a complicated process involving several different amylase enzymes. 

In practice, I think the answer is 'no'. In a raw potato, starch is present in many small granules. When cooked, the starch gelatinizes: the granules burst and water is absorbed. This is a irreversible chemical change that obviously has a big effect on taste and texture. But it does not produce significant amounts of sugars. 

A closer match to the starch degradation process in plants occurs in brewing. When beer is brewed from barley by heating a 'mash': the natural amylases present in the grain convert starch to sugars; these are then fermented by yeast. The exceptionally sweet-toothed might try to mimic this process with potatoes, adding something like wheat, barley or sweet potato that has natural amylase. You could then boil the potato 'proto-beer' to extract the sugar.