Lectures on Lagrangian Floer Theory by Professor Kaoru Ono

# Lectures on Lagrangian Floer Theory by Professor Kaoru Ono ## The purpose of this lecture series is to introduce Lagrangian Floer theory. This is one of the main tools of current research in symplectic geometry. For example, it provides a source of symplectic invariants and plays an essential role in the homological mirror symmetry. The course will start with basics on Floer theory, and move on to a general theory of $A_{\infty}$-structures, (weak) Maurer-Cartan equation, potential function, bulk deformations, etc., which is then applied to Lagrangian torus fibers, followed by related topics. ### Lectures * Time: Saturday 11:00-12:30, ==14:00-15:25, 15:35-17:00==:zap: * Dates: April 14-28, May 19-June 2 * Location: R440 Astro-Math Building, NTU ### Discussions in April * Time: Monday 10:30-12:00 * Dates: April 16, 23 * Location: R202 Astro-Math Building, NTU ### ==Discussions in May==:zap: * Time: Monday 15:30-17:00 * Date: May 21 * Location: R440 Astro-Math Building, NTU ### Prerequisites * Differential manifolds (differential forms, de Rham cohomology, Frobenius theorem for integrability of distributions, etc) * Algebraic topology ((co)chain complex and (co)homology, etc) * Characteristic classes (Chern classes, Stiefel-Whiney classes in relation to orientability, spin condition for vector bundles, etc) [1, 2] * Statement of Riemann-Roch/Atiyah-Singer index theorem [1, 2] * Notion and some properties of Fredholm operators [3, 4] * Morse theory [5, 6, 8] ### References **Characteristic classes and index theory** [1] Nash, Differential Topology and Quantum Field Theory, Academic Press 1991, especially Chap. II, III, IV. [2] Eguchi, Gilkey, Hanson, Gravitation, Gauge Theories and Differential Geometry, Phys. Rep. 66 (1980) 213-393 especially Chap. 6, 7, 8. **Functional analysis** [3] Lang, Real and Functional Anaysis, GTM, Springer 1993. [4] Kirillov, Gvishiani, Theorems and Problems in Functional Analysis, PBM, Springer 1982. **Morse theory** [5] Milnor, Morse Theory, AM-51, Princeton University Press 1963. [6] Nicolaescu, An Invitation to Morse Theory, Universitext, Springer 2011. **Holomorphic curve techniques in symplectic geometry** [7] Audin, Lafontaine eds, Holomorphic Curves in Symplectic Geometry, Birkhauser. [8] Audin, Damian, Morse Theory and Floer Homology, Universitext, Springer. [9] McDuff, Salamon, J-holomorphic Cruves and Symplectic Topology, AMS. [10] Oh, Symplectic Topology and Floer Homology, 2 volumes, Cambridge. [11] Fukaya, Oh, Ohta, Ono, Lagrangian Intersection Floer Theory - Anomaly and Obstruction, AMS/IP. --- ## Youtube livestream links | Date | Link| Note| | -------- | -------- |--------| |4/14 Part I|https://youtu.be/SfngHuEruxU || |4/14 Part II|https://youtu.be/0A49ZHsPrt0 || |4/16|https://youtu.be/GLXbVMJJ4XI|[note0416.pdf](https://drive.google.com/file/d/0B_futVPGLz-9S0w0bUdwRmI2bmxrWVNnN2daaFNJUW1Hd1Rz/view?usp=sharing)| |4/21 Part I|https://youtu.be/g6Ot2KoNfcA || |4/21 Part II|https://youtu.be/4njvwwmEzbM || |4/23|https://youtu.be/C_kOOaNr8hM|| |4/28 Part I|https://youtu.be/gWVbPY-F0G4 || |4/28 Part II|https://youtu.be/EryHexbjE5s || |5/19 Part I|https://youtu.be/xQo4HOyIuEs || |5/19 Part II|https://youtu.be/FN8_1dKBtas || |5/21|https://youtu.be/hliLtueo1Sw || |5/26 Part I|https://youtu.be/OeeQ9I8UOmo|| |5/26 Part II|https://youtu.be/w-7bFOMUH9g || |6/02 Part I|https://youtu.be/Z7zk8a-OaHY || |6/02 Part II|https://youtu.be/aNLpoF3sTmg || ## Related Activities ### Lectures on sigma-models and branes * Speaker: Professor Siye Wu (NTHU) * Date/Time: Sunday May 20, 10:30-12:00, 14:00-15:30; Monday May 21, 10:30-12:00; ==Friday May 25, 16:00-17:00==:zap: * Location: R617, Astro-Math Building * Organizer: Jih-Hsin Cheng (Academia Sinica) --- ## Speaker: Professor Kaoru Ono (RIMS, University of Kyoto) The research focus of Professor Kaoru Ono lies in symplectic geometry, in particular Floer theory and holomorphic curves in symplectic manifolds. He has made many fundamental and important contributions to the field. For example, in a joint work with K. Fukaya, he constructed Floer cohomology of Hamiltonian diffeomorphisms on arbitrary closed symplectic manifolds as well as Gromov-Witten invariants, and hence settled the Betti number version of the Arnold's conjecture. Using Novikov-Floer cohomology, he also proved the $C^1$-flux conjecture. In recent years, he has been collaborating with Fukaya, Oh and Ohta in Floer theory for Lagrangian submanifolds and its implications in symplectic geometry and homological mirror symmetry. He was awarded the Autumn Prize of the Mathematical Society of Japan (2005) and the Inoue Prize for Science (2006). --- ### Contact For questions, please contact River Chiang (NCKU), Nan-Kuo Ho (NTHU), or Mai-Lin Yau (NCU). --- ## Lunch box sign-up Please sign up [here](https://docs.google.com/spreadsheets/d/1MIHt7Mw4RD3sFEc_8_p7IsswkbHvxthPIxUjZS7yGdA/edit?usp=sharing) by 16:00 each Friday. ## Sponsor: NCTS