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,
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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