Differential Geometry and Pseudodifferential Analysis on Manifolds
Summary
Differential geometry studies smooth shapes and spaces known as manifolds, using tools such as tangent spaces, curvature and connections to describe their local and global properties. Pseudodifferential analysis extends classical differential operators by allowing more general symbolic behaviour, providing a framework to tackle questions of existence, regularity and spectral properties of solutions to partial differential equations on manifolds. Together, these fields enable the treatment of elliptic and hypoelliptic operators, the construction of parametrices (approximate inverses) and the study of heat kernels, index theory and non-commutative residues. Recent advances have focused on filtered manifolds—those equipped with a hierarchy of subbundles in the tangent bundle—where new calculi generalise the Heisenberg and Rockland frameworks. Such developments underpin applications ranging from geometric quantisation to analysis on singular spaces and foliations, and illuminate deep links between analysis, topology and mathematical physics.
Research from Nature Portfolio
No recent Nature Portfolio content available.
Differential Geometry and Pseudodifferential Analysis on Manifolds publication trend
The graph below shows the total number of articles in differential geometry and pseudodifferential analysis on manifolds across all publications each year (not limited to Nature Index journals).
Technical terms
Manifold: A space that locally resembles Euclidean space and supports smooth calculus.
Differential operator: An operator defined by derivatives acting on functions or sections of bundles.
Pseudodifferential operator: A generalisation of differential operators characterised by symbols allowing fractional order behaviour.
Filtered manifold: A manifold whose tangent bundle is endowed with a nested sequence of subbundles.
Rockland operator: A hypoelliptic differential operator satisfying a representation-theoretic nondegeneracy condition on graded groups.
Heat kernel: The fundamental solution of the heat equation on a manifold, capturing both local and global geometry.
Parametrix: An approximate inverse of an operator used to analyse solvability and regularity.
BGG sequence: A sequence of invariant differential operators arising in parabolic and conformal geometry.
K-homology: The dual theory to K-theory classifying elliptic operators modulo stable homotopy.
References
- The Heat Asymptotics on Filtered Manifolds. The Journal of Geometric Analysis (2019).
- Graded hypoellipticity of BGG sequences. Annals of Global Analysis and Geometry (2022).
- A transverse index theorem in the calculus of filtered manifolds. Journal of Functional Analysis (2024).
About these summaries
This Nature Research Intelligence Topic summary is created with the cited references and a large language model. We take care to ground generated text with facts, and have systems in place to gain human feedback on the overall quality of the process in line with our AI principles. We strive to create accurate and useful summaries for people unfamiliar with the research topic and that supports this goal. These pages are a beta release and will be updated as we learn how best to help people gain value from a research topic summary.
Turn complex research questions into confident strategic decisions
When you're under pressure to set direction, justify investment, or understand your competitive position, you need more than raw data — you need trusted insights you can act on.
Benchmark your performance against global peers using robust, methodologically sound analysis.
Combine quantitative metrics with qualitative expert insight to uncover strengths, gaps and emerging opportunities.
Gain tailored, decision-ready recommendations aligned to your strategic priorities.
Talk to us to learn more about our data dashboards and bespoke strategy reports.
Grow research skills, confidence and careers with training built for every stage of the research lifecycle.
Developed with Nature Portfolio journal Editors and internationally renowned experts. Discover three ways to learn:
Self-paced, online courses in convenient bite-sized units, covering key skills across scientific writing, publishing, grant writing, data analysis, and more.
Expert trainer-led workshops with hands-on exercises and real-time feedback across core research skills, delivered via interactive group sessions.
Editor-led workshops combining core principles in writing and publishing, personalised 1:1 feedback from Nature Portfolio Editors and hands-on exercises.
Explore course catalogues and workshop agendas, enquire about the options or request institutional pricing.