A lot of it is statistical arbitrage.

That is: you look at the historical prices of all commodities over time and try to figure out which ones tend to vary together (eg copper-mining companies go up when copper prices go up... but you're looking for less obvious examples than that). Then you look at whether current prices diverge from these trends at all, and if they do you buy/sell accordingly. At least, that's the handwavey version I know -- I'm sure whatever they're doing at Rennaisance and DE Shaw is something I don't even know about.

Not a lot of it is statistical arbitrage - yes this is what Rennaisance and D.E. Shaw do, but this is not what banks (sans perhaps Goldman Sachs) devote much of their compute power to. It's just too hard to make money doing this compared to more traditional activities like market making.

As stated, risk management is a big application: VaR and market stress scenarios are computationally intensive, particularly for portfolios with path-dependent derivatives. Pricing is the other big application: it is similarly computationally intensive to value derivatives against the market-implied term structure of volatility.

How does that relate to finance? I know that protein folding involves massive computation, as it was what our university cluster was being used for 40% of the time.

Sorry, for some reason I only read the first and last sentences of your post. Maybe they are used for economic forecasting by governments?

I don't know why the grandparent mentioned it, but I do know that D.E. Shaw works on both.

There's DE Shaw, the Hedge Fund and DE Shaw Research. Different focus for each entity.

I can think only about trade forecasting, and natural resources management (like estimating oil exhaustion).

Possibly used for risk management, generating lots of different scenarios and calculating their P&L impact. This would include VaR calculation.

Some derivative pricing models require a lot of cycles.

Consolidation of multi-dimensional hierarchical data can be surprisingly compute intensive. Combine that with VaR calculations etc. then it doesn't surprise me that serious kit is required.