On X one consider both symplectic and complex algebraic geometry, and likewise on Y. He also gives the first publicly available, elementary proof that a new definition of the Alexander polynomial, via grid diagrams, agrees with the classical one. Here is an individual listing: Google Scholar Project Euclid. 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. Though abstract, Hodge-theoretic mirror symmetry has very concrete implications for Gromov-Witten invariants beginning with the degree 1 curves on the quintic 3-folds.
In a little more detail: Mohammad works at the Simons Center for Geometry and Physics in Stony Brook, and has been supported as a fellow of the collaboration in His research focuses on Lagrangian Floer theory. His research focuses on the geometric properties of J-holomorphic curves in symplectic manifolds. Zack Sylvan obtained his Ph. Kyler Siegel’s senior thesis. These results lend support to the conjecture that the invariants coincide with Seiberg—Witten invariants of the underlying four-manifold, and are in particular independent of the broken fibration.
In the summer ofColumbia ran an internal “research experience for Columbia undergraduates.
Tim Perutz’s homepage
Roberta Guadagni obtained her Ph. They fit into a field peruhz which assigns Floer homology groups to three-manifolds fibred over S 1. Jingyu Zhao obtained her Ph.
We prove that under fairly mild assumptions. Core homological mirror symmetry project There are three superficially quite different formulations of mirror symmetry between X and Y: Htesis research concerns partially wrapped Fukaya categories and their relations to other invariants. The PDF is here. Download Email Please enter a valid email address. 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.
Floer homology and cohomology, symplectic aspects 57R The other project that summer was advised by Mirela Ciperiani and Kimball Martin.
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.
More like this Lagrangian matching invariants for fibred four-manifolds: Thesks second aspect concerns symplectic geometry, particularly symplectic Floer homology.
Two pressing questions about it are to show that it gives invariants of 4-manifolds, independent of the fibrations on them, and to show that it recovers Seiberg-Witten theory. His research focuses on derived algebraic geometry, rigid analytic geometry, and motivic homotopy theory.
Papers by undergraduate advisees
Article information Source Geom. Specialized structures on manifolds spin manifolds, framed manifolds, etc.
Part I—the present paper—is devoted to the symplectic geometry of these Lagrangians. MR Digital Object Identifier: Two contributions to a reading seminar organised by Yann Rollin at Imperial College in Andrew Harder obtained his Ph.
I have given several variants of a talk entitled Broken pencils and four-manifold invariants. More like this Lagrangian matching invariants for fibred four-manifolds: In his thesis, Siegel gives a combinatorial proof that the action is well-defined.
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.
Collaboration Postdocs – Simons Collaboration on Homological Mirror Symmetry
This structure is called peritz Fukaya category and may be denoted by F X. His work in algebraic geometry studies Calabi-Yau varieties, Fano varieties and Landau-Ginzburg models.
Roberta joins the collaboration in as a postdoctoral fellow at the University of Pennsylvania. Thus you may work with any convenient and reasonably large collection of Lagrangian submanifolds, and not with arbitrary Lagrangians that you have no control over. Jingyu spent a year as a postdoc at the Institute for Advanced Study before joining the collaboration in as a postdoctoral fellow at Brandeis and Harvard. 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.
T I M P E R U T Z
Research Papers My papers are available for download via the ArXiv here. But symplectic topology refers specifically to the study of global aspects of symplectic manifolds. His research focuses on Lagrangian Floer theory thesia its applications to homological mirror symmetry for orbifolds.