This is a graduate-level topic course on symplectic field theory (SFT).
Teaching assistants: 姚一晨.
Final Score = 60% Homework (15% for each) + 40% Final Exam.
Main reference: Lectures on Symplectic Field Theory by Chris Wendl.
September 28, 2025,
October 30, 2025,
November 27, 2025,
December 29, 2025,
Preparation materials for final exam:
Final exam (January 19, 2026):
September 15, 2025,
Topics: Introduction to this course, recollection of background on symplectic geometry (including Arnold conjecture on fixed points of Hamiltonian diffeomorphisms).
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September 18, 2025,
Topics: Recollection of background on contact geometry (including contact manifolds arising from energy level sets).
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September 22, 2025,
Topics: Symplectic embedding, Liouville domain, symplectic cobordism, complexification (of a vector space equipped with a complex structure), bi-graded algebra of (p,q)-type linear maps.
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September 25, 2025,
Topics: Bundle-valued forms, connections on a vector bundle, parallel transport, existence of an affine connection that preserves almost complex structure, section-description of \bar{\partial}_J u.
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September 28, 2025,
Topics: section-description of \bar{\partial}_J, differentiation (linearization) of \bar{\partial}_J (both local and global computation), Cauchy-Riemann type operator.
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October 9, 2025,
Topics: Recollection of basic concepts in Sobolev space: L^p space and L^p norm, convolution and mollifer, weak derivative, W^{k,p} space and W^{k,p} norm, Fr\'echet space and Hilbert space, Morrey's inequality (-> Sobolev embedding theoerm for p>n), Gagliardo-Nirenberg-Sobolev inequality (-> Sobolev embedding theoerm for n>p).
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October 13, 2025,
Topics: Applications of Solobev embedding theorems (resulting in more embedding theorems), compactness of the embeddings, useful property one: W^{k,p} space is closed under the multiplication, useful property two: composition preserves the W^{k,p} norm, a local expression of a J-holomorphic curve (in terms of \partial_{\bar z}) and its extended version (defined over the complex plane).
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October 16, 2025,
Topics: A further simplification of the extended (loca) Cauchy-Riemann equation via the T-operator, regularity of the solution of the (local) Cauchy-Riemann equation, renormalization trick.
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October 20, 2025,
Topics: Fredholm operator (definition + a useful lemma to verify Fredholm property), perturbations of a Fredholm operator, invariant property of the Fredholm index, Fredholm map (between Banach manifolds), implicit function theorem (in the Banach manifold setting, regular value (Smale-Sard theorem), proof of D_u (linearization of \bar{\partial}_J) is Fredholm.
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October 23, 2025,
Topics: Universal moduli space (which implies manifold structure for generic almost complex structure), identification between paths of symplectic matrices and paths of symmetric matrices, operator A = -Jd/dt - S(t) where S(t) is a circle of symmetric matrices, Fredholm property of operator A.
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October 27, 2025,
Topics: Morse theory via Fredholm linearization, a short summary of moduli spaces in different settings, an \R-parametrized operators on W^{1,2}(S^1, \R^{2n}) from Reed dynamics, spectral flow in finite-dimensional setting, spectral flow for parametrized operator -J\partial/\partial_t - S_s(t), the spectral flow approach to Conley-Zehnder index.
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October 30, 2025,
Topics: Geometric interpretation of spectral flow, Carleman Similarity Principle (including two steps of "coordinate" transformation and one step of PDE-solving).
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November 3, 2025,
Topics: Applications of Carleman Similarity Principle (including unique continuation, finiteness of critical points in a compact space), the dichotomy (local) behavior of an intersection of two J-holomorphic curves, local behavior of a self-intersection of a J-holomorphic curve.
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November 6, 2025,
Topics: Structure (covering) theorem on J-holomorphic curves, simple = somewhere injective, Cauchy-Riemann operator (revisited, as a generalization of Cauchy-Riemann equation), first Chern number (for complex vector bundles over closed Riemann surfaces).
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November 10, 2025,
Topics: Index formula of complex-linear Cauchy-Riemann operator (as one version of the Riemann-Roch Theorem), proof of index formula (from the case - line bundle over S^2), variants of index formula (a generic path of almost complex structures; counting with marked points).
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November 13, 2025,
Topics: Energy in symplectic geometry (in three cases: closed surface, Floer trajectory in Hamiltonian Floer homology, symplectization), Hamiltonian structure on odd-dimensional manifolds, mapping torus of a symplectomorphism.
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November 17, 2025,
Topics: Periodic closed Hamiltonian orbits via mapping torus, stable framing, symplectization of a stable Hamiltonian structure, symplectic cobordism with stable boundary, being stable is equivalent to the existence of a tamed almost complex structure on symplectization, energy on symplectization.
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November 24, 2025,
Topics: Dependence of the rescaled energy on the target range parameter, rescaled energy on a symplectic cobordism with stable boundary, (statement of) removal of singularities, monotonicity lemma, quantum property of J-holomorphic curve (with its half proof presented).
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November 27, 2025,
Topics: Quantum property of J-holomorphic curve (with its other half proof presented), diameter of a J-holomorphic annulus, proof of removal of singularities.
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December 1, 2025,
Topics: Energy control --> constant J-holomorphic curve, convergence to critical points (in Morse theory setting).
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December 8, 2025,
Topics: Exponential decay in Morse setting (with proof), equivalence between finite energy condition and exponential decay in Hamiltonian Floer setting (only statement).
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December 11, 2025,
Topics: Proof of "asymptotic behavior --> exponential decay" (left with the Heinz trick).
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December 15, 2025,
Topics: The Heinz trick, bubble point (equivalence to divergence of L^1-norm), Hofer's cute lemma, renormalization trick.
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December 18, 2025,
Topics: Born of bubble spheres, bubble tree, the Gromov compactness and its proof.
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December 22, 2025,
Topics: Finiteness of pseudo-holomorphic representable homology classes under energy bound, broken (both Floer and Morse) flowlines (as a limit), proof for the existence of Morse broken flowlines.
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December 26, 2025,
Topics: Finiteness of relative homotopy classes represented by Floer cylinders (in Hamiltonian Floer homology) under energy bound, origin of Novikov ring in Hamiltonian Floer homology, C^1-bound of a J-holomorphic map into (non-compact) completion of a symplectic cobordism under energy control.
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December 29, 2025,
Topics: Finish the proof of C^1-bound from energy control (in the setting of non-compact target), discussion of the asymptotic behavior near puncture points when C^0-bound fails (within non-compact target), moduli space of pointed Riemann surfaces.
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January 5, 2026,
Topics: Geometric structure of the moduli space of pointed Riemann surfaces, dimension formula, nodal Riemann surface, moduli space of equivalence classes of nodal Riemann surfaces \overline{\cal M}_{g,l}, examples for (g,l) = (0,3), (0,4) and (0,5).
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January 8, 2026,
Topics: Smoothen the nodal points by circle compactification, sequential compactness of \overline{\cal M}_{g,l}, Gromov compactness based on pointed Riemann surfaces, definition of holomorphic buildings.
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January 12, 2025,
Topics: stability of holomorphic buildings, SFT-compactness (statement), definition of cylindrical contact homology (under h-admissible condition).
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January 15, 2025 (FINAL CLASS),
Topics: Construction and invariant properties of cylindrical contact homology, infinitely many contact structures on \T^3, neck-stretching technique, obstructing Lagrangian embeddings (of split tori) in E(a,b).