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:____3____0__**
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 homomorphis***m**

__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
ChassÃ©**, *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 **i**n 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 topolo**gy:*

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-holomor**phic 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 f**ew 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 reall**y 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, AstÃ©risque, 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.**

** **

ZOOM link:

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

Meeting ID: 935 8672 0072

Passcode: 807693

(pdf's and video)

(pdf's and video)

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.

Plan of the course.

References (to be completed):

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.

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.

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

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.

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).

P. Biran, O. Cornea - A Lagrangian Pictionary, Kyoto Math. Journal, to appear.

P. Biran, O. Cornea and E. Shelukhin - Lagrangian shadows and triangulated categories, AstÃ©risque, 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.