Which area of Economics involves heavy mathematics and also appears to be directly helping the people?
Some ideas based on the course list:


Environmental economics is doing work to try to understand and mitigate the effects of climate change. Key concepts in this debate, such as common action problems and externalities will feature in EC325 Public Economics.
Game theory is used as a basis for auction theory, which plays a major role in the design of important auctions. Auctions are used, for example, to sell licenses to companies wishing to use the radio spectrum. You can read more about spectrum auctions here.
The market design/matching markets idea is a good one with some real practical applications. For a basic intro, take a look at section 20.5 of this free book. In particular, there is a neat video by Alvin Roth, who won the Nobel prize for work in this area.
Development economics studies the process by which countries do (or don't) escape from poverty and can help us to understand what sorts of policy interventions are likely to be helpful.
Industrial economics (also known as industrial organisation) helps us to understand how competition in markets works (and when it doesn't work). The insights from this field underpin the work of organisations such as the Competition and Markets Authority (the UK's competition regulator) in ensuring that markets deliver good outcomes for consumers. You can read more about competition policy here.