Symplectic topology Symplectic manifolds arise both from mathematical physics as phase-spaces for hamiltonian systems and from algebraic geometry as smooth quasi-projective varieties over the complex numbers. Thurston proved that if K is a hyperbolic knot then the manifolds obtained by increasingly large surgeries on K converge in the Gromov-Hausdorff sense to the complement of K. Roberta Guadagni obtained her Ph. In his thesis, Siegel gives a combinatorial proof that the action is well-defined. Permanent link to this document https: Mohammad works at the Simons Center for Geometry and Physics in Stony Brook, and has been supported as a fellow of the collaboration in The tangent spaces of a symplectic manifold can be made into complex vector spaces this involves a choice J for how i will act on tangent vectors, but the choice is in some ways inessential.

Keywords Four-manifolds Lefschetz fibrations Seibergâ€”Witten invariants pseudo-holomorphic curves Lagrangian submanifolds Hilbert schemes. Dmitry spent a year as a postdoc at Uppsala University before joining the collaboration in as a visiting assistant professor at UC Berkeley. This is a technical summary aimed at other mathematicians, and full of geometers’ jargon. We have not proved that any large class of geometric mirror pairs obey HMS equivalently, core HMS , but we overcome one major difficulty in doing so, by showing that it suffices to prove core HMS. I am interested particularly in near-symplectic geometry and in maps from 4-manifolds to surfaces Lefschetz fibrations and their generalisations. His research focuses on the geometric properties of J-holomorphic curves in symplectic manifolds.

In a little more detail: I am interested in the structures in Floer homology arising from Lagrangian correspondences between symplectic manifolds.

In the summer ofColumbia ran an internal “research experience for Columbia undergraduates. This theory, of what I call Lagrangian matching invariants, is an example of a much more general formalism in symplectic topology.

## Papers by undergraduate advisees

Geometric mirror symmetry explain how to construct mirror pairs X,Y. His research focuses on Lagrangian Floer theory. Symplectic pdrutz probe symplectic manifolds X using pseudo-holomorphic curvesi.

The second aspect concerns symplectic geometry, particularly symplectic Floer homology. Atanas Atanasov’s senior thesis. Justin Hilburn obtained his Ph. The two aspects come together by means of a sort of topological field theory for 3- and 4-manifolds singularly fibred by surfaces, based on the idea that Lagrangian correspondences between symplectic manifolds in this case, symmetric products of Riemann surfaces can serve as boundary conditions for pseudo-holomorphic curves.

# PERUTZ, TIMOTHY – Mathematics – CNS Directory

Dingxin Zhang obtained his Ph. But symplectic topology refers specifically to the study of global aspects of symplectic manifolds. Instead, it’s about looking-glass worlds like Lewis Carroll’s, in which things one knows peruyz reappear, but transformed into quite different forms.

Jingyu Zhao obtained her Ph. Permanent link to this document https: Roberta joins the collaboration in as a postdoctoral fellow at the University of Pennsylvania. He has held postdoctoral positions at University of Miami, Vienna, and most recently University of Alberta.

Download Email Please enter a valid email address. I was assisted by Thomas Peters; without him, the projects would probably not have peeutz a success.

# Perutz : Lagrangian matching invariants for fibred four-manifolds: II

Hansol Hong obtained his Ph. Specialized structures on manifolds pperutz manifolds, framed manifolds, etc. Another thing you can do with pseudo-holomorphic curves is use them to set up Floer cohomology HF L,L’ for pairs L,L’ of Lagrangian submanifolds in a symplectic manifold. You have partial access to this content. Sheridan, we prove that homological mirror pairs are Hodge-theoretic mirror pairs.

Kontsevich’s homological mirror symmetry philosophy proposes that, for mirror pairs, there should be a derived Morita equivalence of the Fukaya category F X and the derived category of coherent sheaves DCoh Y. Here is an individual listing: His research concerns partially wrapped Fukaya categories and their relations to other invariants. Her research focuses on equivariant structures in Floer theory. His research focuses on Lagrangian Floer theory and its applications to homological mirror symmetry for orbifolds.

Kyler Siegel’s senior thesis. Symplectic manifolds arise both from mathematical physics as phase-spaces for hamiltonian systems and from algebraic geometry as smooth quasi-projective varieties over the complex numbers. Bibliographic information can be found at MathSciNet reviews will appear in due course. The invariants are derived from moduli spaces of pseudo-holomorphic sections of relative Hilbert schemes of points on the fibres, subject to Lagrangian boundary conditions.

MR Digital Object Identifier: I have been impressed at how they have risen to the challenge. Google Scholar Project Euclid.