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

ZOOM link:

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

Meeting ID: 935 8672 0072
Passcode: 807693

The course will end with two lectures - January 11, 12 - on the usual timetable, 9:00-11:00 MTL hour.

Videos and pdfs of the notes.

STUDENT MICRO-CONFERENCE ON LAGRANGIAN SUBMANIFOLDS  15 December, 2020.
(pdf's and video)

This will consist of four 20 minutes talks followed by 10min of discussion (on the model of the short research talks at the symplectic zoominar):

9:00 - 9:30           Dominique Rathel-Fournier, Lagrangian cobordism groups of surfaces

Abstract: I will give a survey of what is known about the Lagrangian cobordism groups of symplectic surfaces and their relationship with the Fukaya category. I will describe how immersed Lagrangian cobordisms naturally appear in this context, and briefly explain how to deal with them.

9:30 - 10:00         Pierre-Alexandre Mailhot,  From Hamiltonian isotopies to Lagrangian cobordims : an extension of the Calabi homomorphism

Abstract: In recent works, Jake Solomon introduced a functional that extends the Calabi homomorphism to exact Lagrangian paths.  By the Lagrangian suspension construction, we can associate an elementary Lagrangian cobordism to every exact Lagrangian path. This fact suggests the existence of a functional Cal_cob that further extends the Calabi homomorphism to Lagrangian cobordisms. The goal of the talk is to give a construction of that functional. We will prove its invariance under Hamiltonian isotopy using its first variation. The latter will yield a characterization of the critical points of Cal_cob for elementary cobordisms. We will then evaluate Cal_cob on some examples of Lagrangian cobordisms including the trace of the surgery of two curves on the torus. In view of that calculation, we will find a sharp lower bound for Cal_cob in terms of the shadow of the cobordism.

10:00 - 10:30       Filip Brocic, Relative Gromov width

Abstract: Barraud and Cornea conjectured in 2003 that every closed Lagrangian in C^n has finite relative Gromov width. Biran and Cornea proved that conjecture holds in monotone setting. There are also some positive results by Charette for orientable surfaces in C^2 and for Lagrangians with non-positive sectional curvature by Borman and McLean. Most suprisnigly it turned out that conjecure is not true. Rizzel proved that flexible Lagrangians constructed by Ekholm, Eliashber, Murphy and Smith have infinite width. In this talk I will give overview of this results and sketch some proofs if time allows.

10:30 - 11:00       Jean-Philippe Chassé, Metric constraints & shadow metrics

Abstract. Biran, Cornea and Shelukhin have recently introduced families of metrics defined on large collections of Lagrangian submanifolds ---  so-called weighted fragmentation (pseudo)metrics. However, precisely because of their generalness, it has been quite hard to understand their precise behavior so far. I will present a conjecture of Cornea relating a special case of these metrics, the shadow metrics, with the set-theoretic Hausdorff distance, when one looks at a subspace of Lagrangian submanifolds respecting certain metric constraints. I will then explain how one proves the conjecture using Groman and Solomon's reverse isoperimetric inequality for $J$-holomorphic curves.

MICRO-CONFERENCE ON MONOTONE LAGRANGIANS 7-8 December, 2020.
(pdf's and video)

December 7, 8:30 - 9:45      -  Felix Schlenk (Neuchatel)How to distinguish Lagrangian tori

Abstract: One technique to distinguish monotone Lagrangian submanifolds up to ambient symplectomorphism is by counting the holomorphic discs of Maslov index two with
boundary on the Lagrangian. Another, more elementary technique is by studying a symplectic invariant on "neighbours" of the Lagrangians. This technique of versal deformations was introduced by Chekanov  in his construction of exotic Lagrangian tori in R^{2n}. I will outline this construction, and if time permits shall explain how it can be used to tell apart Vianna's tori in Del Pezzo surfaces.

December 7, 10:00 - 11:15   -  Renato Vianna (UFRJ):  Monotone Lagrangian tori in Del Pezzo surfaces

Abstract: We will introduce the concept of almost toric fibrations developed by Symington, given an specific example in the projective plane, where we can see how to interpolate from the monotone Clifford torus to the monotone Chekanov torus. We will discuss the wall-crossing phenomena. Then we will show how infinitely many monotone Lagrangian tori arise as fibres of ATF and how the Maslov index 2 holomorphic disks they bound can distinguish them.

December 8, 8:30 - 9:45    -    Jonny Evans (Lancaster): What to do when you first meet a Lagrangian submanifold

Abstract: Whenever I meet a new Lagrangian submanifold for the first time, there are some things I make sure I find out about it. I will illustrate this with a worked example: the Chiang Lagrangian in CP^3 which I worked out with YankI Lekili back in 2013-14: https://arxiv.org/abs/1401.4073 . Along the way, we will see some useful lemmas for understanding holomorphic discs.

Plan of the course.

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:

V. I Arnold, Lagrange and Legendre cobordisms I, II, Funktional. Anal. i Prilozhen 14 (1980).

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

Yu.V. Chekanov - Lagrangian embeddings and Lagrangian cobordism, Topics in Singularity theory, Amer. Math. Soc. Transl. Ser. 2, vol 180, (1997), 13-23.

Yu. Chekanov, F. Schlenk - Notes on monotone Lagrnagian twist tori, ERA AMS, 17 (2010) 104-121.

M. Damian - On the topology of monotone Lagrangian submanifolds, Ann. ENS, (2015).

J. Evans, J. Kedra - Remarks on monontone Lagrangians in C^n, Mathematical Research Letters (2014) 1241-1255.

K. Fukaya - Application of Floer Homology of Lagrangian Submanifolds to Symplectic Topology, in Morse Theoretic Methods in Non-Linear Analysis and Symplectic Topology, NATO Science Series (P.Biran, O.Cornea, F. Lalonde eds.) (2006).

K.
Fukaya, Yong-Geun Oh, H. Ohta, K. Ono - Lagrangian Intersection Floer Theory: Anomaly and Obstruction, Part I and II , AMS.

F. Lalonde, J-C. Sikorav - Sous-varie'te's Lagrangiennes et Lagrangiennes exactes des fibre's cotangents, Comment. Math. Helv. 66 (1991) 18-33.

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.

Ch.Yu Mak, W. Wu - Dehn twists exact sequences through Lagrangian cobordism, Compositio Math. , 154,  (2018)  2485--2533.

L. Poleterovich - The surgery of Lagrange submanifolds, Geom. Funt. Anal. 1 (1991), 198-210.

M.Pozniak - PhD Thesis, Warwick (1994), (ETH web page).

P. Seidel - Fukaya categories and Picard-Leschetz Theory, EMS (2008).

R. Viana- Infinitely many exotic monotone Lagrangian tori in CP2, Journal of Topology 9 (2016) 535-551.

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, Kyoto Math. Journal, to appear.
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.