FLOERTHINLOWDIM · Floer theoretical invariants of low dimensional manifolds

Floer theoretical invariants of low dimensional manifolds

The researcher’s main area of research is symplectic topology, a branch of mathematics that studies geometry and topology of symplectic manifolds, a class of spaces that has been originally motivated by classical mechanics. Recently, the field has seen an explosion of activity due to its relevance to string theory. Roughly speaking, the target space of strings forms a symplectic manifold and the symplectic topology of this target space is the focus of study in understanding the physics of the corresponding string theory. It is also these type of considerations that led physicists to discover surprising dualities. The one which is relevant here is mirror duality/symmetry. Mirror symmetry is an area of both mathematics and physics that consists of largely conjectural evidences. These conjectures are striking as they predict a very broad correspondence between symplectic geometry and algebraic geometry. There are intriguing examples of calculations showing that questions from one of these fields can be interpreted and solved in the other one. Mathematicians are led to study symplectic manifolds because they provide a template to study the quantum invariants of low-dimensional manifolds. Low-dimensional topology is the study of spaces that have dimension less than five. Historically, higher dimensions have been easier to understand due to powerful techniques of algebraic topology which are no longer available in low dimensions. Quantum invariants such as Seiberg-Witten theory and Heegaard Floer theory have been elucidating our understanding of low-dimensional spaces. Although, Seiberg-Witten theory was first discovered in gauge theory, it has recently been understood that an equivalent theory which might legitimately be considered as its “symplectic shadow”, namely Heegaard Floer theory, is computationally much more manageable and leads to a much more concrete understanding of these invariants. The researchers current study focusses on understanding the target symplectic manifolds that arise in Heegaard Floer theory in great detail in order to build a mirror theory. More concretely, this means reconstructing Heegaard Floer theory in the realm of algebraic geometry. Work Performed and Results Obtained Tim Perutz and the researcher constructed invariants for 3-manifolds with boundary which extends Ozsvath-Szabo’s Heegaard Floer homology for closed 3-manifolds. These invariants take the shape of functors between extended Fukaya categories of g-fold symmetric products of the boundary surfaces. These invariants satisfy a TQFT-style gluing law, and the closed 3-manifold invariants are Heegaard Floer cohomology groups. This project branches out in many directions; Tim Perutz and the researcher are expecting to be on a long term collaboration pursuing further ramificaitions of this project. This project was one of the proposed projects in the initial grant application. During the first period, the researcher has completed two manuscripts on this project. One of these has been published in an internationally prestigious journal: Proceedings of National Academy of Sciences (PNAS). The other one has been completed and is available as an electronic preprint. It is submitted to one of the top journals in mathematics: Memoirs of the AMS. The researcher has also worked on other problems that were not included in the grant application. These problems emerged and were solved during the current period under revision. They are nonetheless in the general research area of the researcher. The researcher produced two publications during the first period. These have been published in internationally respected journals: Commentarii Mathematici Helvetici and Algebraic and Geometric Topology. In the second period, the researcher produced 3 other manuscripts. Two of these has been accepted for publication in an internationally respected journals : Selecta Mathematica and Compositio Mathematica. The other is submitted to a top level journal. Only the last one of these has some relation to the proposed project of the grant under review. It can be seen as a continuation of the study done during the first period of the grant. Academic Activities The grant has benefited the career of the researcher through increased time for research, resources for equipment, for the researcher’s academic travel and to invite academic visitors to the host institution. The first has been achieved through a reduced teaching load since the beginning of the project (as stated in the proposal) which have enabled the work above to be carried out. The equipment purchased consists of a laptop, some associated supplies (such as a software license), and a number of academic books. The resources for travel has been used to attend travel to the following conferences in Europe, North America and Asia : January 2011, Workshop on Symplectic and Contact Geometry, Uppsala (Sweden) June 2011, Conference on Low-dimensional manifolds and high-dimensional categories, UC Berkeley (CA) June 2011, Workshop on Homological Invariants in low-dimensional topology, Simons Center, Stony Brook (NY) Nov 2011, Workshop on Circle-valued Morse theory and Alexander invariants, University of Tokyo (Japan) Mar 2012, Geometric structures on manifolds, Banff (Canada) May 2012, Gokova Geometry/Topology Conference, Gokova (Turkey) July 2012, CAST Conference, Budapest (Hungary) July 2012, Workshop on Holomorphic curves, Stanford University, Palo Alto (CA) Sept 2013, Workshop on Mirror Symmetry and Cluster Algebras, Leeds (UK) Oct 2013, Nantes-Orsay Seminar on symplectic and contact geometry, Nantes (FR) May 2014, Locally free geometry seminar, University of Amsterdam, Amsterdam (Netherlands) July 2014, Geometry conference, National Cheng Kung University, Tainan (Taiwan) Bibliography: 1. Lekili, Yanki ; Perutz, Timothy. Fukaya categories of the torus and Dehn surgery. Proc. Natl. Acad. Sci. USA 108 (2011), no. 20, 8106–8113. 2. Lekili, Yanki ; Perutz, Timothy. Arithmetic mirror symmetry for the 2-torus. Preprint available on arXiv:1211.4632, submitted to Mem. of AMS. 3. Lekili, Yanki Planar open books with four binding components. Algebr. Geom. Topol. 11 (2011), no. 2, 909–928. 4. Lekili, Yanki ; Maydanskiy Maksim. The symplectic topology of some rational homology balls. Comment. Math. Helv., 89 (2014) 571–596. 5. Lekili, Yanki ; Pascaleff James. Equivariant Fukaya categories and representation theory, preprint (44 pages) submitted to Compositio Mathematica (2014). 6. Lekili, Yanki ; Evans, Jonathan. Floer cohomology of the Chiang Lagrangian, (40 pages) accepted by Selecta Mathematica. (2014). 7. Lekili, Yanki ; Polishchuk Alexander. A modular compactification of M1,n from A-infty -structures, preprint (42 pages) submitted Duke Math. Jour. (2014).

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