In particular, he developed new tests to prove the integrability and non-integrability for systems of differential equations and discrete mappings, based on the so-called Painlevé expansions in complex time.

He showed that a torsional instability of filaments under tension can result in the formation of structures with opposite chirality for which he coined the word tendril perversion.

With colleagues, he provided a complete classification of uniform equilibria, and built the first three-dimensional theory for the nonlinear dynamics of elastic tubes conveying a fluid, studied the twining of vines, proved the existence of compact waves traveling on nonlinear rods, the inversion of curvature in bacteria, the growth of stems, the mechanics of seed expulsion, the shape and mechanics of proteins, and a full theory of growing and remodeling elastic rods suitable to describe many biological structures.

While the basic theoretical framework was understood as early as 1994, in 2005 with Martine Ben Amar, he developed a general stability method for morphoelastic solids and demonstrated that patterns and instabilities can be driven exclusively through growth.

[16] He further expanded this aspect of his research to demonstrate the occurrence of growth-induced patterns in many biological and physiological systems such as fungi, bacteria, and microbial cellular blebbing.

Together with Derek Moulton and Régis Chirat, he developed a theory to describe morphological patterns for seashells, such as spikes and commarginal ornamentation.

These ideas, first presented in the work of Kazuo Kondo in the 1940s, were known by the mechanics community but had never been used directly to build an effective theory of continuous defects.

In this fully geometric theory, first described in their 2012 paper,[21] they show that pure dislocations, disclinations, and point defects are, respectively, associated with Weitzenbock, Riemann, and Weyl manifolds.

They also introduced the concept of discombinations to describe sources of incompatibility related to multiple origins (point, lines, and edge defects).

[22] Goriely has done work in the field of materials science and renewable energy, ionic liquids, nano-particles fabrication, supercapacitors, and lithium-ion batteries.

In their 2014 paper,[23] they developed a mathematical model to predict coverage and morphology during the annealing of a thin solid film of a perovskite absorber.