Ivan Smith
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Previous Organizations / Institutes
The organizations / institutes from the scholar's other articles in the past ten years
Research Subjects
Ivan Smith has a total of 22 co-authors. Here are the top co-authors by number of shared publications:
- Tobias Ekholm
Department of Mathematics, Uppsala University, Box 480, 751 06 Uppsala, Sweden
6 shared publications - Ailsa Keating
University of Cambridge
3 shared publications - Michael Wemyss
University of Glasgow, UK
3 shared publications - Nick Sheridan
School of Mathematics, University of Edinburgh, Edinburgh, UK
3 shared publications - Denis Auroux
Department of Mathematics Harvard University 02138 Cambridge MA USA
2 shared publications - Jonathan David Evans
Lancaster University, UK
2 shared publications
Latest Publications
- Open Access
- 22 July 2026
- Published by Cambridge University Press (CUP) in Compositio Mathematica
- Vol. 162 (4) , 1046-1051
Corrigendum: Symplectomorphisms and spherical objects in the conifold smoothing (Compositio Math. 160 (2024), 2738–2773)
This note corrects two errors in the paper Symplectomorphisms and spherical objects in the conifold smoothing [Compositio Math. 160 (2024), 2738–2773]. Neither affects the main results of the paper (in particular, neither affects any result stated in the Introduction of the paper). We apologise to our readers for these inaccuracies.
- 22 July 2026
- Published by Cambridge University Press (CUP) in Compositio Mathematica
- Vol. 162 (4) , 1046-1051
Open AccessCorrigendum: Symplectomorphisms and spherical objects in the conifold smoothing (Compositio Math. 160 (2024), 2738–2773)
This note corrects two errors in the paper Symplectomorphisms and spherical objects in the conifold smoothing [Compositio Math. 160 (2024), 2738–2773]. Neither affects the main results of the paper (in particular, neither affects any result stated in the Introduction of the paper). We apologise to our readers for these inaccuracies.
- Open Access
Splitting symplectic monodromy
We discuss some examples in which symplectic monodromy (provably or conjecturally) splits off the symplectic mapping class group, hoping to illustrate different techniques and inputs to the arguments. Along the way we formulate several open questions and conjectures.
Open AccessSplitting symplectic monodromy
We discuss some examples in which symplectic monodromy (provably or conjecturally) splits off the symplectic mapping class group, hoping to illustrate different techniques and inputs to the arguments. Along the way we formulate several open questions and conjectures.
- Open Access
Orbifold Floer spectral invariants, symmetric product links and Weyl laws
We explain a strategy, based on spectral invariants on symmetric product orbifolds, for proving the smooth closing lemma for Hamiltonian diffeomorphisms of a symplectic manifold when the orbifold quantum cohomologies of its symmetric products possess suitable idempotents. We relate the existence of such idempotents to the manifold containing a sequence of Lagrangian links, whose number of components tends to infinity, satisfying a number of properties. Orbifold Floer cohomology for global quotient orbifolds is used axiomatically, and is constructed in a companion paper. We illustrate this strategy by giving a new proof of the smooth closing lemma for area-preserving diffeomorphisms of the 2-sphere. The construction of suitable Lagrangian links in higher dimensions remains an intriguing open problem.
Open AccessOrbifold Floer spectral invariants, symmetric product links and Weyl laws
We explain a strategy, based on spectral invariants on symmetric product orbifolds, for proving the smooth closing lemma for Hamiltonian diffeomorphisms of a symplectic manifold when the orbifold quantum cohomologies of its symmetric products possess suitable idempotents. We relate the existence of such idempotents to the manifold containing a sequence of Lagrangian links, whose number of components tends to infinity, satisfying a number of properties. Orbifold Floer cohomology for global quotient orbifolds is used axiomatically, and is constructed in a companion paper. We illustrate this strategy by giving a new proof of the smooth closing lemma for area-preserving diffeomorphisms of the 2-sphere. The construction of suitable Lagrangian links in higher dimensions remains an intriguing open problem.
- Open Access
- 1 November 2024
- Published by Cambridge University Press (CUP) in Compositio Mathematica
- Vol. 160 (11) , 2738-2773
Symplectomorphisms and spherical objects in the conifold smoothing
Let $X$ denote the ‘conifold smoothing’, the symplectic Weinstein manifold which is the complement of a smooth conic in $T^*S^3$ or, equivalently, the plumbing of two copies of $T^*S^3$ along a Hopf link. Let $Y$ denote the ‘conifold resolution’, by which we mean the complement of a smooth divisor in $\mathcal {O}(-1) \oplus \mathcal {O}(-1) \to \mathbb {P}^1$ . We prove that the compactly supported symplectic mapping class group of $X$ splits off a copy of an infinite-rank free group, in particular is infinitely generated; and we classify spherical objects in the bounded derived category $D(Y)$ (the three-dimensional ‘affine $A_1$ -case’). Our results build on work of Chan, Pomerleano and Ueda and Toda, and both theorems make essential use of working on the ‘other side’ of the mirror.- 1 November 2024
- Published by Cambridge University Press (CUP) in Compositio Mathematica
- Vol. 160 (11) , 2738-2773
Open AccessSymplectomorphisms and spherical objects in the conifold smoothing
Let $X$ denote the ‘conifold smoothing’, the symplectic Weinstein manifold which is the complement of a smooth conic in $T^*S^3$ or, equivalently, the plumbing of two copies of $T^*S^3$ along a Hopf link. Let $Y$ denote the ‘conifold resolution’, by which we mean the complement of a smooth divisor in $\mathcal {O}(-1) \oplus \mathcal {O}(-1) \to \mathbb {P}^1$ . We prove that the compactly supported symplectic mapping class group of $X$ splits off a copy of an infinite-rank free group, in particular is infinitely generated; and we classify spherical objects in the bounded derived category $D(Y)$ (the three-dimensional ‘affine $A_1$ -case’). Our results build on work of Chan, Pomerleano and Ueda and Toda, and both theorems make essential use of working on the ‘other side’ of the mirror. - Open Access
- 7 October 2024
- Published by Springer Nature in Geometric and Functional Analysis
- Vol. 34 (6) , 1647-1733
Gromov-Witten Invariants in Complex and Morava-Local K-Theories
Given a closed symplectic manifold X, we construct Gromov-Witten-type invariants valued both in (complex) K-theory and in any complex-oriented cohomology theory $\mathbb{K}$ which is Kp(n)-local for some Morava K-theory Kp(n). We show that these invariants satisfy a version of the Kontsevich-Manin axioms, extending Givental and Lee’s work for the quantum K-theory of complex projective algebraic varieties. In particular, we prove a Gromov-Witten type splitting axiom, and hence define quantum K-theory and quantum $\mathbb{K}$ -theory as commutative deformations of the corresponding (generalised) cohomology rings of X; the definition of the quantum product involves the formal group of the underlying cohomology theory. The key geometric input of these results is a construction of global Kuranishi charts for moduli spaces of stable maps of arbitrary genus to X. On the algebraic side, in order to establish a common framework covering both ordinary K-theory and Kp(n)-local theories, we introduce a formalism of ‘counting theories’ for enumerative invariants on a category of global Kuranishi charts.- 7 October 2024
- Published by Springer Nature in Geometric and Functional Analysis
- Vol. 34 (6) , 1647-1733
Open AccessGromov-Witten Invariants in Complex and Morava-Local K-Theories
Given a closed symplectic manifold X, we construct Gromov-Witten-type invariants valued both in (complex) K-theory and in any complex-oriented cohomology theory $\mathbb{K}$ which is Kp(n)-local for some Morava K-theory Kp(n). We show that these invariants satisfy a version of the Kontsevich-Manin axioms, extending Givental and Lee’s work for the quantum K-theory of complex projective algebraic varieties. In particular, we prove a Gromov-Witten type splitting axiom, and hence define quantum K-theory and quantum $\mathbb{K}$ -theory as commutative deformations of the corresponding (generalised) cohomology rings of X; the definition of the quantum product involves the formal group of the underlying cohomology theory. The key geometric input of these results is a construction of global Kuranishi charts for moduli spaces of stable maps of arbitrary genus to X. On the algebraic side, in order to establish a common framework covering both ordinary K-theory and Kp(n)-local theories, we introduce a formalism of ‘counting theories’ for enumerative invariants on a category of global Kuranishi charts. - Open Access
Quantitative Heegaard Floer cohomology and the Calabi invariant
We define a new family of spectral invariants associated to certain Lagrangian links in compact and connected surfaces of any genus. We show that our invariants recover the Calabi invariant of Hamiltonians in their limit. As applications, we resolve several open questions from topological surface dynamics and continuous symplectic topology: we show that the group of Hamiltonian homeomorphisms of any compact surface with (possibly empty) boundary is not simple; we extend the Calabi homomorphism to the group of Hameomorphisms constructed by Oh-M\"uller; and, we construct an infinite dimensional family of quasimorphisms on the group of area and orientation preserving homeomorphisms of the two-sphere. Our invariants are inspired by recent work of Polterovich and Shelukhin defining and applying spectral invariants for certain classes of links in the two-sphere.
Open AccessQuantitative Heegaard Floer cohomology and the Calabi invariant
We define a new family of spectral invariants associated to certain Lagrangian links in compact and connected surfaces of any genus. We show that our invariants recover the Calabi invariant of Hamiltonians in their limit. As applications, we resolve several open questions from topological surface dynamics and continuous symplectic topology: we show that the group of Hamiltonian homeomorphisms of any compact surface with (possibly empty) boundary is not simple; we extend the Calabi homomorphism to the group of Hameomorphisms constructed by Oh-M\"uller; and, we construct an infinite dimensional family of quasimorphisms on the group of area and orientation preserving homeomorphisms of the two-sphere. Our invariants are inspired by recent work of Polterovich and Shelukhin defining and applying spectral invariants for certain classes of links in the two-sphere.
- Open Access
- 12 March 2023
- Published by Springer Nature in Selecta Mathematica
- Vol. 29 (2) , 1-62
Double bubble plumbings and two-curve flops
We discuss the symplectic topology of the Stein manifolds obtained by plumbing two 3-dimensional spheres along a circle. These spaces are related, at a derived level and working in a characteristic determined by the specific geometry, to local threefolds which contain two floppable $$(-1,-1)$$ -curves meeting at a point. Using contraction algebras we classify spherical objects on the B-side, and derive topological consequences including a complete description of the homology classes realised by graded exact Lagrangians.- 12 March 2023
- Published by Springer Nature in Selecta Mathematica
- Vol. 29 (2) , 1-62
Open AccessDouble bubble plumbings and two-curve flops
We discuss the symplectic topology of the Stein manifolds obtained by plumbing two 3-dimensional spheres along a circle. These spaces are related, at a derived level and working in a characteristic determined by the specific geometry, to local threefolds which contain two floppable $$(-1,-1)$$ -curves meeting at a point. Using contraction algebras we classify spherical objects on the B-side, and derive topological consequences including a complete description of the homology classes realised by graded exact Lagrangians. - Open Access
Double bubble plumbings and two-curve flops
We discuss the symplectic topology of the Stein manifolds obtained by plumbing two 3-dimensional spheres along a circle. These spaces are related, at a derived level and working in a characteristic determined by the specific geometry, to local threefolds which contain two floppable $(-1,-1)$-curves meeting at a point. Using contraction algebras we classify spherical objects on the B-side, and derive topological consequences including a complete description of the homology classes realised by graded exact Lagrangians.
Open AccessDouble bubble plumbings and two-curve flops
We discuss the symplectic topology of the Stein manifolds obtained by plumbing two 3-dimensional spheres along a circle. These spaces are related, at a derived level and working in a characteristic determined by the specific geometry, to local threefolds which contain two floppable $(-1,-1)$-curves meeting at a point. Using contraction algebras we classify spherical objects on the B-side, and derive topological consequences including a complete description of the homology classes realised by graded exact Lagrangians.
- Open Access
- 11 November 2021
- Published by Springer Nature in Communications in Mathematical Physics
- Vol. 388 (3) , 1181-1203
Floer Theory of Higher Rank Quiver 3-folds
We study threefolds Y fibred by $$A_m$$ -surfaces over a curve S of positive genus. An ideal triangulation of S defines, for each rank m, a quiver $$Q(\Delta _m)$$ , hence a $$CY_3$$ -category $$\mathcal {C}(W)$$ for any potential W on $$Q(\Delta _m)$$ . We show that for $$\omega $$ in an open subset of the Kähler cone, a subcategory of a sign-twisted Fukaya category of $$(Y,\omega )$$ is quasi-isomorphic to $$(\mathcal {C},W_{[\omega ]})$$ for a certain generic potential $$W_{[\omega ]}$$ . This partially establishes a conjecture of Goncharov (in: Algebra, geometry, and physics in the 21st century, Birkhäuser/Springer, Cham, 2017) concerning ‘categorifications’ of cluster varieties of framed $${\mathbb {P}}GL_{m+1}$$ -local systems on S, and gives a symplectic geometric viewpoint on results of Gaiotto et al. (Ann Henri Poincaré 15(1):61–141, 2014) on ‘theories of class $${\mathcal {S}}$$ ’.- 11 November 2021
- Published by Springer Nature in Communications in Mathematical Physics
- Vol. 388 (3) , 1181-1203
Open AccessFloer Theory of Higher Rank Quiver 3-folds
We study threefolds Y fibred by $$A_m$$ -surfaces over a curve S of positive genus. An ideal triangulation of S defines, for each rank m, a quiver $$Q(\Delta _m)$$ , hence a $$CY_3$$ -category $$\mathcal {C}(W)$$ for any potential W on $$Q(\Delta _m)$$ . We show that for $$\omega $$ in an open subset of the Kähler cone, a subcategory of a sign-twisted Fukaya category of $$(Y,\omega )$$ is quasi-isomorphic to $$(\mathcal {C},W_{[\omega ]})$$ for a certain generic potential $$W_{[\omega ]}$$ . This partially establishes a conjecture of Goncharov (in: Algebra, geometry, and physics in the 21st century, Birkhäuser/Springer, Cham, 2017) concerning ‘categorifications’ of cluster varieties of framed $${\mathbb {P}}GL_{m+1}$$ -local systems on S, and gives a symplectic geometric viewpoint on results of Gaiotto et al. (Ann Henri Poincaré 15(1):61–141, 2014) on ‘theories of class $${\mathcal {S}}$$ ’. - Open Access
Complex cobordism, Hamiltonian loops and global Kuranishi charts
Let $(X,\omega)$ be a closed symplectic manifold. A loop $\phi: S^1 \to \mathrm{Diff}(X)$ of diffeomorphisms of $X$ defines a fibration $\pi: P_{\phi} \to S^2$. By applying Gromov-Witten theory to moduli spaces of holomorphic sections of $\pi$, Lalonde, McDuff and Polterovich proved that if $\phi$ lifts to the Hamiltonian group $\mathrm{Ham}(X,\omega)$, then the rational cohomology of $P_{\phi}$ splits additively. We prove, with the same assumptions, that the $\mathbb{E}$-generalised cohomology of $P_{\phi}$ splits additively for any complex-oriented cohomology theory $\mathbb{E}$, in particular the integral cohomology splits. This class of examples includes all complex projective varieties equipped with a smooth morphism to $\mathbb{CP}^1$, in which case the analogous rational result was proved by Deligne using Hodge theory. The argument employs virtual fundamental cycles of moduli spaces of sections of $\pi$ in Morava $K$-theory and results from chromatic homotopy theory. Our proof involves a construction of independent interest: we build global Kuranishi charts for moduli spaces of pseudo-holomorphic spheres in $X$ in a class $\beta \in H_2(X;\mathbb{Z})$, depending on a choice of integral symplectic form $\Omega$ on $X$ and ample Hermitian line bundle over the moduli space of one-pointed degree $d = \langle \Omega,\beta\rangle$ stable genus zero curves in $\mathbb{CP}^d$.Open AccessComplex cobordism, Hamiltonian loops and global Kuranishi charts
Let $(X,\omega)$ be a closed symplectic manifold. A loop $\phi: S^1 \to \mathrm{Diff}(X)$ of diffeomorphisms of $X$ defines a fibration $\pi: P_{\phi} \to S^2$. By applying Gromov-Witten theory to moduli spaces of holomorphic sections of $\pi$, Lalonde, McDuff and Polterovich proved that if $\phi$ lifts to the Hamiltonian group $\mathrm{Ham}(X,\omega)$, then the rational cohomology of $P_{\phi}$ splits additively. We prove, with the same assumptions, that the $\mathbb{E}$-generalised cohomology of $P_{\phi}$ splits additively for any complex-oriented cohomology theory $\mathbb{E}$, in particular the integral cohomology splits. This class of examples includes all complex projective varieties equipped with a smooth morphism to $\mathbb{CP}^1$, in which case the analogous rational result was proved by Deligne using Hodge theory. The argument employs virtual fundamental cycles of moduli spaces of sections of $\pi$ in Morava $K$-theory and results from chromatic homotopy theory. Our proof involves a construction of independent interest: we build global Kuranishi charts for moduli spaces of pseudo-holomorphic spheres in $X$ in a class $\beta \in H_2(X;\mathbb{Z})$, depending on a choice of integral symplectic form $\Omega$ on $X$ and ample Hermitian line bundle over the moduli space of one-pointed degree $d = \langle \Omega,\beta\rangle$ stable genus zero curves in $\mathbb{CP}^d$.