Mathematics:
* Topology: He laid the foundations for modern algebraic topology and introduced the concept of the fundamental group.
* Differential equations: He made significant contributions to the theory of differential equations, including the development of the Poincaré-Bendixson theorem and the Poincaré map.
* Dynamical systems: He pioneered the study of chaotic systems and introduced the concept of the Poincaré recurrence theorem.
* Number theory: He contributed to the theory of Diophantine equations and the study of modular forms.
Physics:
* Relativity: He developed the concept of the Poincaré group, which describes the symmetries of spacetime in special relativity.
* Electromagnetism: He made important contributions to the theory of electromagnetism and the development of the Lorentz-Poincaré model.
* Cosmology: He proposed the idea of a universe with finite volume but infinite extent, which became a significant concept in modern cosmology.
Philosophy of Science:
* Conventionalism: He argued that scientific theories are ultimately based on conventions and not on absolute truth.
* Science and philosophy: He believed that science and philosophy are inseparable and that philosophy plays an important role in guiding scientific inquiry.
Other Contributions:
* The philosophy of mathematics: He wrote extensively on the nature of mathematics and its relationship to logic and reality.
* The philosophy of time: He explored the nature of time and its role in physics and consciousness.
In addition to these specific contributions, Poincaré is also known for his brilliant mind, his exceptional mathematical intuition, and his ability to synthesize diverse fields of knowledge. He is regarded as one of the most versatile and influential scientific figures of his time, and his work continues to inspire and inform mathematicians, physicists, and philosophers today.