everything.study
PMATH 965

Topics in Geometry and Topology

Graduate

Winter 2026 · Toric Varieties

Readings

  • Toric varieties by Cox–Little–Schenck
    unspecified
  • There are no required textbooks for
    unspecified
Fall 2026 · Algebraic Stacks

Topics

  • Algebraic spaces
  • Deligne–Mumford stacks
  • Artin stacks
  • Moduli stack of curves
  • Equivariant geometry of schemes
  • Cohomology on stacks and group cohomology
  • Possible applications to motivic integration

Readings

  • Course material is posted on LEARN.
    unspecified
Fall 2026 · Spin Geometry

Topics

  • Clifford algebras
  • Spin and spin-c structures
  • Spinor bundles
  • Dirac operators and curvature identities
  • Index theory for Dirac-type operators
  • Heat-kernel methods
  • Atiyah–Singer index theorem
  • Geometric applications and possible Hodge-theory examples
Winter 2025 · Topics in Geometry - Harmonic Maps

Readings

  • Selected topics in harmonic maps
    unspecified
  • Calculus of variations and harmonic maps
    unspecified
  • Harmonic mappings of Riemannian manifolds
    unspecified
  • A report on harmonic maps
    unspecified
  • Another report on harmonic maps
    unspecified
  • Harmonic morphisms between Riemannian manifolds
    unspecified
  • There is no required text. I will be preparing my lectures mostly from the following sources:
    unspecified
Fall 2024 · Rational homotopy in geometry

Topics

  • Models in rational homotopy theory
  • Differential forms and homotopy information modulo torsion
  • Characteristic classes and homological algebra
  • Geodesics on Riemannian manifolds
  • Topological constraints on complex structures
  • Homotopy types of almost complex manifolds
Fall 2023 · Mirror symmetry for toric varieties and other GIT quotients

Topics

  • Geometric invariant theory and toric varieties
  • Batyrev–Borisov and Berglund–Hübsch mirror symmetry
  • Quantum cohomology and stable maps
  • I-functions and J-functions
  • Mirror theorem for toric complete intersections
  • Abelian/non-Abelian correspondence
  • Applications to the Fanosearch programme
Fall 2023 · Selected topics on Riemann surfaces

Topics

  • Connections on vector and principal bundles
  • Nirenberg integrability theorem
  • Coulomb gauge and Uhlenbeck compactness
  • Hitchin–Kobayashi correspondence on Riemann surfaces
  • Compactification of moduli spaces of stable rank-two bundles
Fall 2022 · A second course in Riemannian Geometry

Topics

  • available at: URL to be determined. 1

Readings

  • M. P. do Carmo, Riemannian geometry, translated from the second Portuguese edition by Francis Flaherty, Mathematics: Theory & Applications, Birkh ̈ auser Boston, Inc., Boston, MA, 1992. MR1138207 I will likely change notation from do Carmo, and I will certainly change the sign and normalization conventions for curvature to the standard ones. Other useful references are:
    unspecified
  • S. Gallot, D. Hulin and J. Lafontaine, Riemannian geometry , third edition, Universitext, Springer-Verlag, Berlin, 2004. MR2088027
    unspecified
  • J. Jost, Riemannian geometry and geometric analysis , seventh edition, Universitext, Springer, Cham, 2017. MR3726907
    unspecified
  • J. M. Lee, Introduction to Riemannian manifolds , second edition, Graduate Texts in Mathematics, 176, Springer, Cham, 2018. MR3887684
    unspecified
  • There is no required textbook for
    unspecified
Winter 2021 · Geometric Invariant Theory, Symplectic Reduction and Moduli Spaces

Topics

  • Algebraic and Lie groups acting on varieties
  • Affine and projective GIT quotients
  • Semistability and moduli spaces
  • Sheaf cohomology, line bundles and divisors
  • Vector bundles on curves
  • Symplectic structures and Hamiltonian actions
  • Moment maps and symplectic quotients
  • Relationship between GIT and symplectic reduction
Winter 2021 · Symplectic Geometry

Topics

  • in Geometry and Topology (PMATH 965) - Symplectic Geometry
Winter 2020 · Gauge Theory

Topics

  • Vector and principal bundles
  • Connections, curvature and gauge groups
  • Covariant derivatives and holonomy
  • Sheaves and sheaf cohomology
  • Characteristic classes and Chern–Weil theory
  • Flat and metric connections
  • Yang–Mills, anti-self-dual and Hermitian–Einstein equations
  • Moduli of Hermitian–Einstein connections
  • Possible stable holomorphic bundles and Kobayashi–Hitchin correspondence
Fall 2020 · Deformation Theory with a Viewpoint Toward Moduli Spaces

Topics

  • Deformation theory and local moduli spaces
  • Schlesinger’s criterion as named in the source
  • Cotangent complex
  • Variations of Hodge structures
  • Applications to Hilbert schemes, curves, K3 surfaces and abelian varieties
Winter 2019 · Special Holonomy

Readings

  • Lee; Riemannian Manifolds: An Introduction to Curvature ; Springer
    unspecified
  • Boothby; An Introduction to Di↵erentiable Manifolds and Riemannian Geometry ; Academic Press
    unspecified
  • Huybrechts; Complex Geometry ; Springer.
    unspecified
Sources