What are the well-known applications of fractal mathematics?

On my web site you will find a "Chronicle of Books." The titles of these books give you an indication of the number and variety of applications. There are too many to be all listed here but here are some examples: fractal antenna, fractal wall along highways, modern cement, chemical engineering, geophysics ...

What are the mathematical tools at our disposal to understand and use fractal theory (functional analysis, algebra ...)?

The functional analysis and algebra are relatively not used. Fractal analysis has now its own tools, developed in books mentionned above

do Fractals fit market moves better than stochastic volatility models, jump models, or more generally Levy processes?

In no way Hable Lévy processes go against fractals: they are themselves fractal! I introduced them in 1963 as a fractal models of prices with jumps. Later, I introduced two other models yet more efficient: infinite memory fractal and multifractal .

Why is this technique rarely used by trading desks?

This technique is still not well known and continues to face reluctance of supporters of techniques based on Bachelier’s ancient theory of 1900. However I’ve been told that my ideas are more and more used by trading desks. Unfortunately everyone keeps secret his recipes.

Doesn’t fractal theory assume that derivatives cost is too high to allow hedging with the model?

The usual formulas of derivatives cost are broadly known to be inappropriate. But why worry in advance about prices given by fractal theory? Young researchers should rather go beyond fears of expensive prices and contribute to develop the theory.