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Showing 1 to 20 of 21 for “"Symplectic manifold"”.
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The Fukaya category, exotic forms and exotic autoequivalences
A symplectic manifold is a smooth manifold M together with a choice of a closed non-degenerate two-form. Recent years have seen the importance of associating an A∞-category to M, called its Fukaya category, in helping to understand symplectic properties of M and its Lagrangian submanifolds. One of …
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A classification of toric, folded-symplectic manifolds
Given a $G$-toric, folded-symplectic manifold with co-orientable folding hypersurface, we show that its orbit space is naturally a manifold with corners $W$ equipped with a smooth map $\psi: W \to \frak{g}^*$, where $\frak{g}^*$ is the dual of the Lie algebra of the torus, $G$. The map $\psi$ has …
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Quantomorphisms and Quantized Energy Levels for Metaplectic-c Quantization
… and quantized energy levels. If a symplectic manifold admits a Kostant-Souriau prequantization circle bundle, then its Poisson algebra is realized as the space of infinitesimal quantomorphisms of that circle bundle. We present a definition for a metaplectic-c quantomorphism, and …
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Rigidity of symplectic fillings, symplectic divisors and Dehn twist exact sequences
We present three different aspects of symplectic geometry in connection to complex geometry. Convex symplectic manifolds, symplectic divisors and Lagrangians are central objects to study on the symplectic side. The focus of the thesis is to establish relations of these symplectic objects to the …
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Semi-Free Hamiltonian Circle Actions on Six-Dimensional Symplectic Manifolds
Assume M is a connected, compact 6-dimensional symplectic manifold equipped with a semi-free Hamiltonian circle action such that the fixed point set consists of isolated points or compact orientable surfaces. Assume the second Betti number of M is less than 3. We give a complete list of the …
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The Novikov theory for symplectic cohomology and exact Lagrangian embeddings
Given an exact symplectic manifold, can we find topological constraints to the existence of exact Lagrangian submanifolds? I developed an approach using symplectic cohomology which provides such conditions for exact Lagrangians inside cotangent bundles and inside ALE hyperkähler spaces. For …
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Scalar Curvature Constraints on Symplectic 4-Manifolds
… interplaybetween Riemannian geometry and symplectic topology on 4-manifolds. These two branches of mathematics belong to different worlds: Where geometry concerns it- self with distances, angles, areas, curvatures, etc., all of which typically vary from one point on the manifold to the …
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Infinite staircases for Hirzebruch surfaces
… embedding function is a generalization of symplectic ball packing problems. For a symplectic manifold, the function gives the smallest amount of which the symplectic form must be scaled in order for a standard ellipsoid of a given eccentricity to embed symplectically into the manifold. …
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On the Quantum Cohomology of Fano Toric Manifolds and the Intersection Cohomology of Singular Symplectic Quotients
… the intersection cohomology of the singular symplectic reduced spaces. Let M be a closed symplectic manifold with a Hamiltonian S1-action defined on it and mu is the moment map. If 0 is a singular value of mu, the reduced space mu-1(0)/S1 is, in general, no longer an orbifold but contains …
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Symplectic circle actions with isolated fixed points
Consider a symplectic circle action on a closed symplectic manifold $M$ with non-empty isolated fixed points. Associated to each fixed point, there are well-defined non-zero integers, called \emph{weights}. We prove that the action is Hamiltonian if the sum of an odd number of weights is never …
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Quantum Steenrod operations and Fukaya categories
… introduction of mod p equivariant operations to symplectic Gromov-Witten theory has fueled exciting developments in the field. In this thesis, we develop new tools for understanding these operations and explore an application to the quantum connection. In one direction, we construct certain …
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Properties of Hamiltonian Torus Actions on Closed Symplectic Manifolds
… of certain Hamiltonian torus actions on closed symplectic manifolds. First, we will consider counting Hamiltonian torus actions on closed, symplectic manifolds M with 2-dimensional second cohomology. In particular, all such manifolds are bundles with fiber and base equal to projective spaces. We …
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Symplectic properties of Milnor fibres
We present two results relating to the symplectic geometry of the Milnor fibres of isolated affine hypersurface singularities. First, given two Lagrangian spheres in an exact symplectic manifold, we find conditions under which the Dehn twists about them generate a free non-abelian subgroup of the …
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Curvatures in generalized Kähler geometry
… con- nections of the underlying bi-Hermitian manifold. We then identify these connections as components of generalized Chern connections and as a result obtain symmetries of the generalized complex struc- tures which may be described in terms of bi-Hermitian data. We identify a second type of …
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GKM manifolds with low Betti numbers
A GKM manifold is a symplectic manifold with a torus action such that the fixed points are isolated and the isotropy weights at the fixed points are linearly independent. Each GKM manifold has a GKM graph which contains much of the topological information of the manifold, in particular the …
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Symplectic isotopy for cuspidal curves
… that if the first Chern class of a 4-dimensional symplectic manifold is sufficiently positive then the deformation is unobstructed. We prove this result when the curves have cusps and nodes, not in a prescribed position. We also prove a similar result when the curves have cusps and tacnodes in a …
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Topics in symplectic Gromov–Witten theory
… is on the Gromov--Witten theory of general symplectic manifolds. Mohan Swaminathan and I construct a framework to define a virtual fundamental class for the moduli space of stable maps to a general closed symplectic manifold. Our construction, inspired by [AMS21], works for all genera and …
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Exact Lagrangian Fillings of Legendrian links and Weinstein 4-manifolds
<p> One approach to studying symplectic manifolds with contact boundary is to consider Lagrangian submanifolds with Legendrian boundary; in particular one can study exact Lagrangian fillings of Legendrian links. There are still many open questions on the spaces of exact Lagrangian fillings of …
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Generalised cohomology and relatively exact Lagrangian submanifolds
… the topology of relatively exact Lagrangian submanifolds. One of our main goals is to study their generalised cohomology, extending known results about their singular cohomology. We do this using different (and simpler) technical set-ups to that of Cohen, Jones and Segal [18, 16]. In Chapter 2, …
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Symmetry in monotone Lagrangian Floer theory
… self-Floer theory of a monotone Lagrangian submanifold $L$ of a closed symplectic manifold $X$ in the presence of various kinds of symmetry. First we consider the group $\mathrm{Symp}(X, L)$ of symplectomorphisms of $X$ preserving $L$ setwise, and extend its action on the Oh spectral sequence …
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