We use Luttinger surgery to show that there are no Lagrangian Klein bottles in S^2× S^2 in the ℤ_2-homology class of an S^2-factor if the symplectic area of that factor is at least twice that of the other.

We use Luttinger surgery to show that there are no Lagrangian Klein bottles in S-2 x S(2 )in the Z(2)-homology class of an S-2-factor if the symplectic area of that factor is at least twice that of the other.

Suppose you have a family of Lagrangian submanifolds $L_t$ and an auxiliary Lagrangian $K$. Suppose that $K$ intersects some of the $L_t$ more than the minimal number of times. Can you eliminate surplus intersection (surplusection) with all fibres by performing a Hamiltonian isotopy of $K$? Or will any Lagrangian isotopic to $K$ surplusect some of the fibres? We argue that in several important situations, surplusection cannot be eliminated, and that a better understanding of surplusection phenomena (better bounds and a clearer understanding of how the surplusection is distributed in the family) would help to tackle some outstanding problems in different areas, including Oh's conjecture on the volume-minimising property of the Clifford torus and the concurrent normals conjecture in convex geometry. We pose many open questions.

This is a series of three lectures I gave at the Korea Institute of Advanced Study in June 2019 at a workshop about "Algebraic and Symplectic Aspects of Degenerations of Complex Surfaces". I focus on the symplectic aspects, in particular on the case of cyclic quotient surface singularities. These notes have been available on a public Git repository since 2019, and I noticed that people occasionally cited them in the years since. For that reason, I decided to post them on arXiv for a more permanent record; I have made some small corrections and annotations but otherwise they are unchanged. These notes are a purely expository account of stuff I was thinking about 2016-2019, and are largely self-aggrandising.

This paper has been written to illustrate the power of techniques from tropical geometry and mirror symmetry for studying the KSBA moduli space of surfaces on or near the Noether line. We focus on the moduli space of octic double planes (K^2 = 2, p_g = 3) and use methods from tropical and toric geometry to classify the strata corresponding to normal KSBA-stable surfaces, focusing on the non-Gorenstein case.

We compute symplectic cohomology for Milnor fibres of certain compound Du Val singularities that admit small resolution by using homological mirror symmetry. Our computations suggest a new conjecture that the existence of a small resolution has strong implications for the symplectic cohomology and conversely. We also use our computations to give a contact invariant of the link of the singularities and thereby distinguish many contact structures on connected sums of S^2 × S^3.

We compute the wrapped Fukaya category $\mathcal{W}(T^*S^1, D)$ of a cylinder relative to a divisor $D= \{p_1,\ldots, p_n\}$ of $n$ points, proving a mirror equivalence with the category of perfect complexes on a crepant resolution (over $k[t_0,\ldots, t_n]$) of the singularity $uv=t_0t_1\ldots t_n$. Upon making the base-change $t_i= f_i(x,y)$, we obtain the derived category of any crepant resolution of the $cA_{n}$ singularity given by the equation $uv= f_0\ldots f_n$. These categories inherit braid group actions via the action on $\mathcal{W}(T^*S^1,D)$ of the mapping class group of $T^*S^1$ fixing $D$. We also give a geometric model of the derived contraction algebra of a $cA_n$ singularity in terms of the relative Fukaya category of the disc.

Suppose you have a nonorientable Lagrangian surface L in a symplectic 4-manifold. How far can you deform the symplectic form before the smooth isotopy class of L contains no Lagrangians? I solve this question for a particular Lagrangian Klein bottle. I also discuss some related conjectures.

This is a book aimed at graduate students and researchers in symplectic geometry, based on a course I taught in 2019. The primary message is that the base of a Lagrangian torus fibration inherits an integral affine structure, which you can use to"read off"a lot of interesting geometry of the total space. Topics covered include: action-angle coordinates, symplectic reduction, toric manifolds, visible and tropical Lagrangians, almost toric systems, Milnor fibres of cyclic quotient singularities, mutation of polygons, non-toric blow-up, an almost toric view on Lisca's classification of fillings of lens spaces, resolutions of cusp singularities, Markov triples and Vianna tori. The book ends with a short list of open problems. Throughout there is an emphasis on examples and there are some exercises with solutions.

We give a construction of Lagrangian torus fibrations with controlled discriminant locus on certain affine varieties. In particular, we apply our construction in the following ways. We find a Lagrangian torus fibration on the 3-fold negative vertex whose discriminant locus has codimension 2; this provides a local model for finding torus fibrations on compact Calabi-Yau 3-folds with codimension 2 discriminant locus. Then, we find a Lagrangian torus fibration on a neighbourhood of the one-dimensional stratum of a simple normal crossing divisor (satisfying certain conditions) such that the base of the fibration is an open subset of the cone over the dual complex of the divisor. This can be used to construct an analogue of the non-archimedean SYZ fibration constructed by Nicaise, Xu and Yu.

The Milnor fibre of a ℚ-Gorenstein smoothing of a Wahl singularity is a rational homology ball B p,q . For a canonically polarised surface of general type X, it is known that there are bounds on the number p for which B p,q admits a symplectic embedding into X. In this paper, we give a recipe to construct unbounded sequences of symplectically embedded B p,q into surfaces of general type equipped with non-canonical symplectic forms. Ultimately, these symplectic embeddings come from Mori's theory of flips, but we give an interpretation in terms of almost toric structures and mutations of polygons. The key point is that a flip of surfaces, as studied by Hacking, Tevelev and Urzúa, can be formulated as a combination of mutations of an almost toric structure and deformation of the symplectic form.

Let X be a minimal surface of general type with positive geometric genus ($b_+ > 1$) and let $K^2$ be the square of its canonical class. Building on work of Khodorovskiy and Rana, we prove that if X develops a Wahl singularity of length $\ell$ in a Q-Gorenstein degeneration, then $\ell \leq 4K^2 + 7$. This improves on the current best-known upper bound due to Lee ($\ell \leq 400(K^2)^4$). Our bound follows from a stronger theorem constraining symplectic embeddings of certain rational homology balls in surfaces of general type. In particular, we show that if the rational homology ball $B_{p,1}$ embeds symplectically in a quintic surface, then $p \leq 12$, partially answering the symplectic version of a question of Kronheimer.

Let $G$ be a compact Lie group and $\mathbf{k}$ be a field of characteristic $p \geq 0$ such that $H^* (G)$ does not have $p$-torsion. We show that a free Lagrangian orbit of a Hamiltonian $G$-action on a compact, monotone, symplectic manifold $X$ split-generates an idempotent summand of the monotone Fukaya category $\mathcal{F}(X; \mathbf{k})$ if and only if it represents a non-zero object of that summand (slightly more general results are also provided). Our result is based on: an explicit understanding of the wrapped Fukaya category $\mathcal{W}(T^*G; \mathbf{k})$ through Koszul twisted complexes involving the zero-section and a cotangent fibre; and a functor $D^b \mathcal{W}(T^*G; \mathbf{k}) \to D^b\mathcal{F}(X^{-} \times X; \mathbf{k})$ canonically associated to the Hamiltonian $G$-action on $X$. We explore several examples which can be studied in a uniform manner including toric Fano varieties and certain Grassmannians.

We study Lagrangian embeddings of a class of two-dimensional cell complexes L-p,L-q into the complex projective plane. These cell complexes, which we call pinwheels, arise naturally in algebraic geometry as vanishing cycles for quotient singularities of type (1/p(2)) (pq - 1, 1) (Wahl singularities). We show that if a pinwheel admits a Lagrangian embedding into CP2 then p is a Markov number and we completely characterise q. We also show that a collection of Lagrangian pinwheels L-pi,L-qi, i = 1, ... , N, cannot be made disjoint unless N <= 3 and the p(i) form part of a Markov triple. These results are the symplectic analogue of a theorem of Hacking and Prokhorov, which classifies complex surfaces with quotient singularities admitting a Q-Gorenstein smoothing whose general fibre is CP2.

We study holomorphic discs with boundary on a Lagrangian submanifold $L$ in a Kaehler manifold admitting a Hamiltonian action of a group $K$ which has $L$ as an orbit. We prove various transversality and classification results for such discs which we then apply to the case of a particular Lagrangian in $\mathbf{CP}^3$ first noticed by Chiang. We prove that this Lagrangian has non-vanishing Floer cohomology if and only if the coefficient ring has characteristic 5, in which case it generates the split-closed derived Fukaya category as a triangulated category.

We show that, for certain families phi(s) of diffeomorphisms of high-dimensional spheres, the commutator of the Dehn twist along the zero-section of T* S-n with the family of pullbacks phi(s)* gives a non-contractible family of compactly supported symplectomorphisms. In particular, we find examples: where the Dehn twist along a parametrized Lagrangian sphere depends up to Hamiltonian isotopy on its parametrization; where the symplectomorphism group is not simply connected, and where the symplectomorphism group does not have the homotopy type of a finite CW complex. We show that these phenomena persist for Dehn twists along the standard matching spheres of the A(m)-Milnor fibre. The non-triviality is detected by considering the action of symplectomorphisms on the space of parametrized Lagrangian submanifolds. We find related examples of symplectic mapping classes for T*(S-n x S-1) and of an exotic symplectic structure on T*(S-n x S-1) standard at infinity.

The Kodaira–Thurston manifold is a quotient of a nilpotent Lie group by a cocompact lattice. We compute the family Gromov–Witten invariants which count pseudoholomorphic tori in the Kodaira–Thurston manifold. For a fixed symplectic form the Gromov–Witten invariant is trivial so we consider the twistor family of left-invariant symplectic forms which are orthogonal for some fixed metric on the Lie algebra. This family defines a loop in the space of symplectic forms. This is the first example of a genus one family Gromov–Witten computation for a non-Kähler manifold.

Under certain topological assumptions, we show that two monotone Lagrangian submanifolds embedded in the standard symplectic vector space with the same monotonicity constant cannot link one another and that, individually, their smooth knot type is determined entirely by the homotopy theoretic data which classifies the underlying Lagrangian immersion. The topological assumptions are satisfied by a large class of manifolds which are realised as monotone Lagrangians, including tori. After some additional homotopy theoretic calculations, we deduce that all monotone Lagrangian tori in the symplectic vector space of odd complex dimension at least five are smoothly isotopic.

We compute the classical and quantum cohomology rings of the twistor spaces of 6-dimensional hyperbolic manifolds and the eigenvalues of quantum multiplication by the first Chern class. Given a half-dimensional totally geodesic submanifold we associate, after Reznikov, a monotone Lagrangian submanifold of the twistor space. In the case of a 3-dimensional totally geodesic submanifold of a hyperbolic 6-manifold we compute the obstruction term $m_0$ in the Fukaya-Floer $A_{\infty}$-algebra of a Reznikov Lagrangian and calculate the Lagrangian quantum homology. There is a well-known correspondence between the possible values of $m_0$ for a Lagrangian with nonvanishing Lagrangian quantum homology and eigenvalues for the action of $c_1$ on quantum cohomology by quantum cup product. Reznikov's Lagrangians account for most of these eigenvalues but there are four exotic eigenvalues we cannot account for.

We derive some restrictions on the topology of a monotone Lagrangian submanifold $L\subset\mathbf{C}^n$ by making observations about the topology of the moduli space of Maslov 2 holomorphic discs with boundary on $L$ and then using Damian's theorem which gives conditions under which the evaluation map from this moduli space to $L$ has nonzero degree. In particular we prove that an orientable 3-manifold admits a monotone Lagrangian embedding in $\mathbf{C}^3$ only if it is a product, which is a variation on a theorem of Fukaya. Finally we prove an h-principle for monotone Lagrangian immersions.