Lagrangian topology for the impatient (course MAT 6359 A - U de M - Fall 2020)

Resource page


Schedule: Mondays 9:00 -11:00; Tuesdays 9:00 - 11:00

IMPORTANT: No classes on Monday, October 12 and Monday, October 19.

ZOOM link:

https://umontreal.zoom.us/j/93586720072?pwd=Q3pSd051MW5ITEtTRXRyY21NVWRoZz09

Meeting ID: 935 8672 0072
Passcode: 807693




Plan of the course.


Videos and pdfs of the notes.




References (to be completed):

A few books on symplectic topology:

M. Audin, M. Damian - Morse theory and Floer homology, Springer.
D. McDuff, D. Salamon - Introduction to Symplectic Topology, Oxford U. Press.
D. McDuff, D. Salamon - J-holomorphic  curves and Symplectic Topology, AMS.
L, Polterovich - The Geometry of the Group of Symplectic Diffeomorphisms, Springer.

A few books on algebraic topology and homological algebra:

A. Dold - Lectures on Algebraic Topology, Springer.
E. Spanier - Algebraic Topology, McGraw-Hill.
R.M. Switzer, Algebraic Topology - Homotopy and Homology, Springer.
Ch. Weibel - An introduction to homological algebra, Cambridge U. Press.

The basics for Morse theory, manifolds:

M.W. Hirsch - Differentiable Topology, Springer.
J. Milnor - Topology from the differentiable viewpoint, Univ. Press of Virginia.
J. Milnor - Morse Theory, Princeton U. Press.
J.Milnor - Lectures on the h-cobordism theorem, Princeton U. Press.


More specialized books and papers:

Lagrangian Floer theory:


M. Akaho, D. Joyce - Immersed Lagrangian Floer Theory, JDG (arxiv).
J.-F. Barraud, O. Cornea - Lagrangian intersections and the Serre spectral sequence,  Annals of Mathematics (2007).
P.Biran , O. Cornea - Quantum structures for Lagrangian submanifolds, preprint 2007.
P.Biran, O.Cornea - Rigidity and uniruling for Lagrangian manifolds,  Geometry and Topology 13 (2009) 2881-2989
K. Fukaya, Yong-Geun Oh, H. Ohta, K. Ono - Lagrangian Intersection Floer Theory: Anomaly and Obstruction, Part I and II , AMS.
Y.-G. Oh. Floer cohomology of Lagrangian intersections and pseudo-holomorphic disks. I. Comm. Pure Appl. Math., 46(7):949--993, 1993.
Y.-G. Oh. Addendum to: ”Floer cohomology of Lagrangian intersections and pseudo-holomorphic disks. I.” [Comm. Pure Appl. Math. 46 (1993), no. 7, 949-993]. Comm. Pure Appl. Math., 48(11):1299-1302, 1995.
Y-G Oh, Floer cohomology, spectral sequences, and the Maslov class of Lagrangian embeddings, Internat. Math. Res. Notices (1996) 305-346.
Y-G Oh, Relative Floer and quantum cohomology and the symplectic topology of La- grangian submanifolds, from: “Contact and symplectic geometry (Cambridge, 1994)”, (C B Thomas, editor), Publ. Newton Inst. 8, Cambridge Univ. Press (1996) 201-267.

M.Pozniak, PhD Thesis, Warwick (1994), (ETH web page).
P. Seidel - Fukaya categories and Picard-Leschetz Theory, EMS (2008).
F. Zapolsky - The Lagrangian Floer-quantum-PSS package and canonical orientations in Floer theory, (2015)  arXiv:1507.02253


Structure of J-holomorphic disks:

U. Frauenfelder, Gromov convergence of pseudoholomorphic disks, JFPTA, 3, (2008)
215-271
L. Lazzarini, Relative frames on J-holomorphic curves, JFPTA, 9,  (2011)
213- 256



Algebraic structures:


A.Bondal, M. Kapranov - Ehanced Triangulated Categories, Mat. USSR Sbornik (1991).
B. Keller - Introduction to A_{\infty} -algebras and modules (ArXiv).
B. Keller - On differential graded categories (ArXiv).
S. Schwede - Topological Triangulated Categories, ArXiv 2012.


Persistence (just a few on this):

S. Barannikov, The Framed Morse complex and its invariants (1994).
L. Polterovich, D. Rosen, K. Samvelyan, J. Zhang, Topological Persistence in Geometry and Analysis (2019).
O. Cornea, A. Ranicki, Rigidity and gluing for the Morse and Novikov complexes JEMS (2002).


Some other papers that I know really well (on arxiv or my web page):

P. Biran, O. Cornea - A Lagrangian Pictionary, preprint.
P. Biran, O. Cornea and E. Shelukhin - Lagrangian shadows and triangulated categories, Astérisque, to appear.
P. Biran, O. Cornea - Lagrangian Cobordism and Fukaya Categories,  GAFA.
P. Biran, O. Cornea - Lagrangian Cobordism I,  J. Amer. Math. Soc.
O. Cornea, F. Lalonde - Cluster Homology: an overwiev of the construction and results, ERA - AMS.




Interesting talks -  live and recordings - at the Symplectic Zoominar.