3rd TAGRA

Fractal structures occur naturally in mathematics, physics, engineering, and many complex systems. Since classical calculus is formulated primarily on smooth spaces, a natural question arises: How can calculus be developed directly on a fractal set? In this talk, we introduce fractal calculus, a local framework for differentiation and integration on fractal supports. The approach is based on a fractal measure and an associated staircase coordinate, which encode the intrinsic geometry of the underlying set. This leads to the (F^\alpha)-derivative and (F^\alpha)-integral, where the order (\alpha) has a geometric interpretation related to the dimension of the fractal support. We emphasize the distinction between fractal calculus and conventional fractional calculus. While fractional calculus is commonly associated with nonlocal operators and memory effects, fractal calculus is designed to describe local differential and integral operations with respect to the geometry of a fractal set. The classical calculus is recovered in the limiting case (alpha=1). Several examples and applications will be presented, including fractal differential equations and models arising in mathematical physics and applied mathematics. We conclude by discussing current challenges and future directions in the development of analysis and differential equations on fractal structures.