Geometries have experienced a significant resurgence across several scientific fields, evolving from classical Euclidean shapes to advanced, high-dimensional, and “positive” geometries that unify previously disparate concepts. This comeback is largely driven by the need to understand complex, non-linear data and the search for a “Theory of Everything” in physics. NeurIPS 2025 Conference +4

Here are the key areas where geometry has made a comeback:

1. Theoretical Physics and Cosmology

• “Positive” Geometries and Particle Physics: Researchers are replacing complex Feynman diagrams with “positive geometries”—such as the amplituhedron and cosmological polytopes—to calculate particle scattering amplitudes. These shapes, often in high dimensions, allow for simpler calculations of physical phenomena, such as interactions in the early universe.
• Quantum Geometry and Materials: In condensed matter physics”quantum geometry” (incorporating Berry curvature and quantum metric) is used to analyze electronic wavefunctions in topological materials and unconventional superconductors.
• String Theory and Manifolds: Algebraic geometry continues to be foundational in string theory, particularly regarding Calabi-Yau manifolds and mirror symmetry. AIP Publishing LLC +4

2. Biology and Biophysics

• Cell Shape and Packing: Researchers are using advanced geometry to understand how cells pack together, recently discovering a new geometric shape called the scutoid, which describes how epithelial cells reshape during organ formation.
• Biological Tissue Curvature: The “geometry of life” is being studied to understand how surface curvatures dictate cell behavior, tissue organization, and regenerative medicine.
• DNA and Protein Folding: Topology and knot theory are used to analyze how DNA is packed and unknotted, while molecular geometry helps predict protein folding mechanisms. EBSCO +2

3. Artificial Intelligence and Machine Learning

• Geometric Deep Learning: Standard AI often fails on non-Euclidean data (like graphs or 3D meshes). Geometric deep learning leverages differential geometry to allow neural networks to operate on curved manifolds, enhancing 3D computer vision and graph-based analysis.
• Information Geometry: The space of probability distributions is being treated as a Riemannian manifold, using “natural gradients” to improve machine learning efficiency. Mathematics Stack Exchange +4

4. Computer Graphics and Vision

• Discrete Differential Geometry: This field applies geometric concepts like curvature to digital surfaces, which is crucial for 3D modeling, computer-aided design (CAD), and motion planning in robotics. Reddit +3

5. Mathematics

• Enumerative Geometry: A new, high-dimensional approach to enumerative geometry has revitalized the field, allowing mathematicians to solve long-standing counting problems by applying “exotic” number systems. Quanta Magazine

These areas demonstrate that modern geometry has moved far beyond 2D/3D shapes to become a powerful, abstract tool for modeling complex, multi-dimensional structures. New Scientist