Geometry & Topology
Modern geometry and topology study spaces under a wide range of equivalences — diffeomorphism, homeomorphism, homotopy, symplectomorphism, quasi-isometry. Central problems include the classification of manifolds in low dimensions, the geometry of minimal surfaces and Einstein metrics, and the symplectic and contact topology that underlies classical mechanics. The area is in continuous exchange with algebra (via K-theory and algebraic topology), analysis (via geometric PDE), and mathematical physics (mirror symmetry, gauge theory).
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math.DG Differential Geometry
Riemannian geometry, curvature, Einstein metrics, geometric flows.
math.AT Algebraic Topology
Homotopy, (co)homology, spectra, stable homotopy theory.
math.GT Geometric Topology
Low-dimensional topology, knots, 3- and 4-manifolds.
math.GN General Topology
Point-set topology, continuum theory, dimension theory.
math.MG Metric Geometry
Coarse geometry, Gromov-Hausdorff, CAT(0) and CAT(κ) spaces.
math.SG Symplectic Geometry
Symplectic manifolds, Floer homology, symplectic field theory.
Active research areas
The most active problem clusters right now.
Ricci flow and geometric flows
Perelman-style techniques, mean curvature flow, inverse mean curvature flow.
Search arXiv →Symplectic and contact topology
Floer homologies, holomorphic curves, fillings of contact manifolds.
Search arXiv →4-manifold topology
Seiberg-Witten, Heegaard-Floer, the Kronheimer-Mrowka program.
Search arXiv →Stable homotopy and higher categories
Chromatic homotopy theory, TMF, ∞-operads.
Search arXiv →Geometric measure theory
Minimal surfaces, rectifiability, varifolds, Plateau’s problem.
Search arXiv →Landmark results
Major results shaping the field.
- 2002–2003
Perelman’s Poincaré proof
Ricci flow with surgery settles the Poincaré and Thurston geometrization conjectures.
- 2006–2014
Mirzakhani on moduli of surfaces
Volumes of moduli spaces and applications (Fields Medal 2014).
- 2007
Taubes on Weinstein conjecture
Existence of Reeb orbits on contact 3-manifolds.
- 2011
Kronheimer–Mrowka
Instanton-based invariants distinguishing smooth 4-manifolds.
Leading journals
Where current research in this area is published.