Geometry & Dynamics Seminar
2010-2011
| 27.10.2010, 13:00 (Wednesday) | Andrey Mironov (Sobolev Institute of Mathematics, Novosibirsk) | ||
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Orthogonal Curvilinear Coordinate Systems Corresponding to Singular Spectral Curves | ||
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| Abstract: | We study the limiting case of the Krichever construction of orthogonal curvilinear coordinate systems when the spectral curve becomes singular. We show that when the curve is reducible and all its irreducible components are rational curves, the construction procedure reduces to solving systems of linear equations and to simple computations with elementary functions. We also demonstrate how well-known coordinate systems, such as polar coordinates, cylindrical coordinates, and spherical coordinates in Euclidean spaces, fit in this scheme. |
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| 3.11.2010, 13:00 (Wednesday) | Liat Kessler (Technion) | ||
| Title: | Determining whether two sequences of sizes of symplectic blow ups of the complex projective plane yield symplectomorphic manifolds. |
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| Location: | Amadu bldg., room 814, Technion | ||
| Abstract: | For a symplectic manifold that is obtained by a sequence of symplectic blow ups from CP^2, we give an algorithm that identifies the homology classes of minimal area among the homology classes of embedded symplectic spheres with self intersection -1. This enables us to determine given two sequences of symplectic blow ups of CP^2 whether they yield the same symplectic manifold. We deduce results on counting toric actions on symplectic four-manifolds. |
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| 17.11.2010, 13:00 (Wednesday) | Mark Branson (Technion) | ||
| Title: | The Action-Maslov Homomorphism on Monotone Symplectic Manifolds | ||
| Location: | Schreiber bldg., room 210, Tel-Aviv University | ||
| Abstract: | The action-Maslov homomorphism is a useful tool for understanding several diverse properties of the Hamiltonian group. We will discuss the construction of quantum homology of a monotone symplectic manifold and the relationship between quantum homology and the action-Maslov homomorphism. The structure of quantum homology gives restrictions on the possible Seidel elements, which in turn gives restrictions on the action-Maslov homomorphism. No prior knowledge of symplectic geometry will be required. |
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| 24.11.2010, 13:00 (Wednesday) | Tony Rieser (Technion) | ||
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| Location: | Amadu bldg., room 814, Technion | ||
| Abstract: | Abstract | ||
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| 15.12.2010, 13:00 (Wednesday) | Frol Zapolsky (IHES) | ||
| Title: | Spectral Invariants on Cotangent Bundles and Applications | ||
| Location: | Amadu bldg., room 814, Technion | ||
| Abstract: | I'm going to define certain functionals on the Hamiltonian group and on the set of compactly supported continuous functions of a cotangent bundle (for a wide class of bases), starting from spectral invariants arising from Lagrangian Floer homology. These functionals generalize Viterbo's symplectic homogenization, and yield more or less standard applications to bounded cohomology, Hofer's geometry, Aubry-Mather theory and symplectic rigidity. |
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| 22.12.2010, 13:00 (Wednesday) | Lev Buhovsky (University of Chicago) | ||
| Title: | On the uniqueness of Hofer's geometry | ||
| Location: | Schreiber bldg., room 210, Tel-Aviv University | ||
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In this talk we address the question whether Hofer's metric is unique among the Finsler-type bi-invariant metrics on the group of Hamiltonian diffeomorphisms. The talk is based on a recent joint work with Yaron Ostrover. |
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| 29.12.2010, 13:00 (Wednesday) | Gabi Ben Simon (ETH-Zürich) | ||
| Title: | Cohomological invariants and order invariants of Teichmuller spaces. | ||
| Location: | Schreiber bldg., room 210, Tel-Aviv University | ||
| Abstract: | I will present two different invariants for Teichmuller spaces. The first one is cohomological and it is due to Burger-Iozzi -Wienhard. The second one is an "order invariant" which is due to Hartnick and Ben Simon. Both have a generalization (which I will only mention) in the theory of higher Teichmuller theory. The ideas are strongly related to group actions on the circle. |
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| 29.12.2010, 14:15 (Wednesday) | Paul Biran (ETH-Zürich and Tel-Aviv University) | ||
| Title: | Enumerative & categorical aspects of Lagrangian cobordisms. | ||
| Location: | Schreiber bldg., room 210, Tel-Aviv University | ||
| Abstract: | In this talk I will address questions on Lagrangian topology from the less traditional point of view of Lagrangian cobordisms (a notion that goes back to Arnold, and was later studies by Eliashberg, Audin and Chekanov). I will discuss which symplectic invariants of Lagrangian submanifolds "survive" under cobordisms. In particular I will focus on Floer theoretical invariants as well as enumerative invariants. Finally, I will indicate some new ideas on how to wrap Lagrangian cobordisms into meaningful categories with hopefully good algebraic properties. The talk is based on a joint work with Octav Cornea. |
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| 12.1.2011, 13:00 (Wednesday) |
Michael Khanevsky (ETH-Zürich) | ||
| Title: | Hofer’s metric on the space of diameters | ||
| Location: | Amadu bldg., room 814, Technion | ||
| Abstract: | In this talk I will address Hofer's distance between diameters in the unit disk. I will prove that distance is unbounded and show its relation to Lagrangian intersections. |
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| 16.3.2011, 13:00 (Wednesday) |
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| Title: | Polarizations and Symplectic Embeddings | ||
| Location: | Schreiber bldg., room 210, Tel-Aviv University | ||
| Abstract: | The aim of the talk will be to explain how polarizations of symplectic manifolds provide efficient tools (via Biran's decomposition theorem) to get some flexibility properties for embedding problems. For instance, I will explain that any symplectic manifold can be "covered" by a small number of standard pieces : ellipsoids. |
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| 30.3.2011, 13:00 (Wednesday) |
Egor Shelukhin (Tel-Aviv University) |
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| Title: | Moment maps and quasimorphisms |
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| Location: | Amadu bldg., room 814, Technion | ||
| Abstract: | It has long been known (and is due to many authors) that whenever a group G acts on a Hermitian symmetric space of non-compact type by Kahler isometries, one can construct a bounded two-cocycle on G by integrating the Kahler form over geodesic triangles. Similarly, Reznikov has constructed bounded two-cocycles on groups of symplectomorphisms using their action on the space of compatible almost complex structures. We show that if the action under discussion has an equivariant moment map, such a cocycle has a primitive - a quasimorphism on the universal cover of the group. This holds in the finite dimensional case - that is for Hermitian Lie groups - giving a reinterpretation of the Guichardet-Wigner quasimorphisms, and for the infinite-dimensional groups of Hamiltonian diffeomorphisms of any finite volume symplectic manifold, generalizing several previous constructions due to Barge-Ghys, Entov and Py. The moment map construction in the second case is due to Donaldson and Fujiki (for the integrable structures). We also compute the restriction of the quasimorphism to the fundamental group and determine its local type. Our construction involves a generalization of Weinstein's Action homomorphism and is related to the Barge-Ghys construction for discrete subgroups of PSL(2,R). |
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| 24.05.2011, 12:00 (Tuesday) |
Marshall Slemrod (University of Wisconsin)
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| Title: | Entropy and isometric embedding | ||
| Location: | Schreiber bldg., room 007, Tel-Aviv University | ||
| Abstract: | The problem of isometric embedding of a Riemannian Manifold into Euclidean space is a classical issue in differential geometry and nonlinear PDE. In this talk I will outline recent work my co-workers and I have done using ideas from continuum mechanics as a guide in formulating the problem and giving (we hope ) some new insight into the role of "entropy". |
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| 31.05.2011, 12:00 (Tuesday) |
Helmut Hofer (Institute for Advanced Study, Princeton) |
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| Title: | Generalizations of Fredholm Theory | ||
| Location: | Schreiber bldg., room 210, Tel-Aviv University | ||
| Abstract: | A meanwhile standard idea for producing geometric invariants (f.e Donaldson Theory, Gromov-Witten Theory, Symplectic Field Theory) consists of counting solutions of nonlinear elliptic systems associated to the geometric data. Although the idea is easy, the implementation can be very difficult and involved, due to a usually large number of technical issues, which in more classical approaches to such type of problems are more than "painful". If there weren't these inherent compactness and transversality problems the solution sets of the elliptic problems would be nice manifolds or orbifolds and the invariants would be achieved by integration of suitable differential forms over them. As it turns the arising difficulties can be overcome by a drastic generalization of nonlinear Fredholm theory and new methods for implementing it in concrete problems. |
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