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# Cornea, Octavian

Full Professor

Faculty of Arts and Science - Department of Mathematics and Statistics

514 343-7526

### Affiliations

• Membre Centre de recherches mathématiques
• Membre CRM  Centre de recherches mathématiques

### Student supervision Expand allCollapse all

Croisements de lignes de flot entre fonctions de Morse et décomposition en cône itéré Theses and supervised dissertations / 2020-08
Fontaine, Paul
Abstract
This master’s thesis introduces a new way to sudy Morse functions on a compact manifold. More specifically, crossings between flows of pseudo-gradients associated to Morse functions allow one to define geometric realisations of morphisms between the Morse complexes. This new class of morphisms leads to the definition of a triangulated category. The main question is to determine if every object of this category admits an iterated cone decomposition. Such a decomposition would greatly simplify the study of the dynamic of a Morse function by interpreting it as many perfect Morse functions. A second topic concerns the global genericity condition to which this category is subject. We study a way, through deformation of Morse functions, to avoid such a constraint.

Groupes de cobordisme lagrangien immergé et structure des polygones pseudo-holomorphes Theses and supervised dissertations / 2018-12
Perrier, Alexandre
Abstract
In this thesis, we shall study Floer theory for Lagrangian immersions. In the first chapter, we prove a decomposition theorem for pseudo-holomorphic disks with boundary on a given generic Lagrangian immersion. We apply this result to the computation of certain Floer complexes. We conclude with work in progress on the computation of the obstruction of the surgery of two transverse Lagrangian submanifolds. In the second chapter, we consider surfaces. We show that a cobordism group, whose relations are given by unobstructed immersed lagrangian cobordisms, is isomorphic to the Grothendieck group of the derived Fukaya category. We also compute the immersed Lagrangian cobordism group.

Fukaya categories of Lagrangian cobordisms and duality Theses and supervised dissertations / 2018-11
Campling, Emily
Abstract
We introduce a new type of duality structure for A_∞-categories called a relative weak Calabi-Yau pairing which generalizes Kontsevich and Soibelman's notion of a weak (proper) Calabi-Yau structure. We prove the existence of a relative weak Calabi-Yau pairing on Biran and Cornea's Fukaya category of Lagrangian cobordisms Fuk_cob(C x M). Here M is a symplectic manifold which is closed or tame at infinity. This duality structure on Fuk_cob(C x M) extends the relative Poincaré duality satisfied by Floer complexes for pairs of Lagrangian cobordisms. Moreover, we show that the relative weak Calabi-Yau pairing on Fuk_cob(C x M) satisfies a compatibility condition with respect to the usual weak Calabi-Yau structure on the monotone Fukaya category of M. The construction of the relative weak Calabi-Yau pairing on Fuk_cob(C x M) is based on counts of curves in C x M satisfying an inhomogeneous nonlinear Cauchy-Riemann equation. In order to prove the existence of this duality structure and to verify its properties, we extend the methods of Biran and Cornea to establish regularity and compactness results for the relevant moduli spaces. We also consider the implications of the existence of the relative weak Calabi-Yau pairing on Fuk_cob(C x M) for the cone decomposition in the derived Fukaya category of M associated to a Lagrangian cobordism, and we present an example involving Lagrangian surgery.

Complexes de type Morse et leurs équivalences Theses and supervised dissertations / 2017-04
Morin, Audrey
Abstract
In this thesis, we study aspects of Morse theory and the chain complexes that derive from it : the Morse complex, the Milnor complex and the Barraud-Cornea complex. Using different techniques from differential topology and Morse theory, which will be presented in the first chapters, we carefully build these complexes before proving their equivalence. This thesis synthesises and compares three points of view in Morse theory in a document accessible to beginning graduate students.

Complexe de Morse et bifurcations Theses and supervised dissertations / 2015-01
Duquerroix, Florian
Abstract
Let (ft,Xt)t∈J be a family of pairs, where J is an interval, ft is a smooth real-valued Morse function defined on a smooth compact manifold V , and Xt is a pseudo-gradient field associated to ft. The purpose of this master thesis is to study the bifurcations undergone by the associated Morse complexes. Two ways are used to reach this result : the direct study of the bifurcations and the continuation method. We prove that the two methods produce the same results from a functorial point of view.

Cobordismes lagrangiens et uniréglage Theses and supervised dissertations / 2014-11
Létourneau, Vincent
Abstract
In this dissertation we study the following question: do Lagrangian cobordisms preserve uniruling? In the two first chapters, the necessary pseudoholomorphic curves theory is quickly presented. We first study in detail the proof that the spaces of simple $J$-holomorphic curves is a manifold of finite dimension. We then present the necessary results to produce the appropriate compactification of these spaces to get to the definition of Gromov-Witten invariants. In the third chapter then some results on the property of uniruling are presented: what are its consequences, how can it be obtained. In the fourth chapter quantum homology is defined, in particular for Lagrangian cobordism, and its ring and module structures are studied which are finally used in the last chapter to present examples of cobordisms which preserves uniruling.

Exact Lagrangian cobordism and pseudo-isotopy Theses and supervised dissertations / 2014-09
Suárez López, Lara Simone
Abstract
In this thesis we study the properties of Lagrangian submanifolds of a symplectic manifold by using the relation of Lagrangian cobordism. More precisely, we are interested in determining when an elementary Lagrangian cobordism is trivial. Using techniques coming from Floer homology and the s-cobordism theorem, we show that under some topological assumptions, an exact Lagrangian cobordism is a Lagrangian pseudo-isotopy. This is a weaker version of a conjecture proposed by Biran and Cornea, which states that any exact Lagrangian cobordism is Hamiltonian isotopic to a Lagrangian suspension.

Source spaces and perturbations for cluster complexes Theses and supervised dissertations / 2012-11
Charest, François
Abstract
We define objects made of marked complex disks connected by metric line seg- ments and construct two sequences of moduli spaces of these objects, referred as the ⊗ version (nonsymmetric) and the • version (symmetric). This allows choices of coherent perturbations over the corresponding versions of the Floer trajectories proposed by Cornea and Lalonde ([CL]). These perturbations are intended to lead to an alternative geometric description of the (obstructed) A∞ and L∞ structures studied by Fukaya, Oh, Ohta and Ono ([FOOO2],[FOOO]) and Cho ([Cho]). Given a Pin± monotone lagrangian submanifold L ⊂ (M, ω) with mini- mal Maslov number ≥ 2, we define an A∞ -algebra structure from the critical points of a generic Morse function on L. We express this structure as a cochain complex extending the pearl complex introduced by Oh ([Oh]) and further ex- plicited by Biran and Cornea ([BC]), equipped with its quantum product. This could also be seen as an alternative geometric description of a Fukaya cate- gory of (M, ω) with L as its only object, a hamiltonian relative version appear- ing in [Sei]. Using spaces of quilted clusters, we verify, using more general quilted cluster spaces, that this defines a functor from a homotopy category of Pin± monotone lagrangian submanifolds hL mono,± (M, ω) to the homotopy category of cochain complexes hK(Λ-mod) where Λ is an appropriate Novikov ring.

Quelques propriétés des sous-variétés lagrangiennes monotones : Rayon de Gromov et morphisme de Seidel Theses and supervised dissertations / 2012-08
Charette, François
Abstract
We present in this thesis a few properties of monotone Lagrangian submanifolds. We first solve a conjecture of Barraud and Cornea in the monotone setting by showing that the relative Gromov radius of two Hamiltonian-isotopic Lagrangians gives a lower bound on the Hofer distance between them. The general non-monotone case remains open to this day. We define all the structures relevant to state and prove the conjecture. We then define a new version of a Lagrangian Seidel morphism through the recently introduced Lagrangian cobordisms of Biran and Cornea. We show that this new version is chain-homotopic to various other versions appearing in the litterature. That all these previous versions are the same is folklore but did not appear in the litterature. We conclude with a conjecture claiming that an exact triangle obtained by Lagrangian surgery is isomorphic to an exact triangle of Seidel involving the symplectic Dehn twist.

Éclatement et contraction lagrangiens et applications Theses and supervised dissertations / 2010-08
Rieser, Antonio P.
Abstract
Given a symplectic manifold (M,ω) and a Lagrangian submanifold L, we construct versions of the symplectic blow-up and blow-down which are defined relative to L. Furthermore, if M admits an anti-symplectic involution ϕ, i.e. a diffeomorphism such that ϕ2 = Id and ϕ*ω = —ω , and we blow-up an appropriately symmetric configuration of symplectic balls, then we show that there exists an antisymplectic involution on the blow-up ~M as well. We derive a homological condition for real Lagrangian surfaces L = Fix(ϕ) which determines when the topology of L changes after a blow down, and we then use these constructions to study the real packing numbers for real Lagrangian submanifolds in (ℂP²,ℝP²).

Structures quantiques de certaines sous-variétés lagrangiennes non-monotones Theses and supervised dissertations / 2010-06
Ngô, Fabien
Abstract
Let (M,ω) be a closed connected symplectic maniflod. We consider lagrangian submanifolds α : L →֒ (M,ω). If α is monotone, i.e. there exists η > 0 such that ημ = ω, Paul Biran and Octav Cornea defined a relative version of quantum homology. In this relative setting they deformed the boundary operator of the Morse complex as well as the intersection product by means of pseudoholomorphic discs. We note (QH(L,Λ), ∗) the quantum homology of L endowed with the quantum product. The main goal of this dissertation is to generalize their construction to a larger class of spaces. Namely, we consider : either the so called almost monotone lagrangian submanifolds, i.e. α is C1-close to a monotone lagrangian embedding, or the toric fibers of toric Fano manifolds. In those cases, we are able to generalize the constructions made by Biran and Cornea. However, in those non necessarily monotone cases, QH(L) will depend on some choices, but in a way irrelevant for the applications we have in mind. In the almost monotone case, we are mainly interested in displaceability, uniruling and ernegy estimates for hamiltonian diffeomorphsims. Finally, we end by an application, that combine the two approaches, concerning the dynamics of hamiltonian that displace all non-monotone toric fibers of CPn.

Homologie de morse et théorème de la signature Theses and supervised dissertations / 2009
St-Pierre, Alexandre
Abstract
Mémoire numérisé par la Division de la gestion de documents et des archives de l'Université de Montréal.

Invariants spectraux en homologie de Floer lagrangienne Theses and supervised dissertations / 2007
Leclercq, Rémi
Abstract
Thèse numérisée par la Direction des bibliothèques de l'Université de Montréal.

Opération d'intersection généralisée en théorie de Morse Theses and supervised dissertations / 2007
Charette, François
Abstract
Mémoire numérisé par la Direction des bibliothèques de l'Université de Montréal.

Quelques propriétés du complexe de Morse-Novikov Theses and supervised dissertations / 2004
Rousseau, Olivier
Abstract
Mémoire numérisé par la Direction des bibliothèques de l'Université de Montréal.

### Selected publications Expand allCollapse all

#### Lagrangian Cobordism I

P. Biran and O. Cornea, Lagrangian Cobordism I (electronic), (2012), , Journal of the American Mathematical Society

#### Lagrangian Topology and Enumerative Geometry

P.Biran and O.Cornea, Lagrangian Topology and Enumerative Geometry 16, 963-1052 (2012), , Geometry and Topology

#### Spectral sequences in Conley's theory

O. Cornea, K. A. de Rezende and M.R. da Silveira, Spectral sequences in Conley's theory 30, 1009-1054 (2010), , Ergodic Theory & Dynamical Systems

#### Rigidity and Uniruling for Lagrangian Manifolds

P. Biran and O. Cornea, Rigidity and Uniruling for Lagrangian Manifolds 13, 2881-2989 (2009), , Geometry and Topology

#### A Lagrangian Quantum Homology

P. Biran and O. Cornea, A Lagrangian Quantum Homology CRM Proceedings and Lecture Notes, 1-44 (2009), , New Perspectives and Challenges in Symplectic Field Theory,

#### Lagrangian Intersections and the Serre Spectral Sequence

J.F. Barraud and O. Cornea, Lagrangian Intersections and the Serre Spectral Sequence 166, 657-722 (2007), , Annals of Mathematics

#### Quantization of the Serre Spectral Sequence

J.F. Barraud and O. Cornea, Quantization of the Serre Spectral Sequence 5, 249-280 (2007), , Journal of Symplectic Geometry

#### Homotopical dynamics in Symplectic Topology

J.F. Barraud and O. Cornea, Homotopical dynamics in Symplectic Topology Springer , 109-148 (2006), , Morse theoretical Methods in Non-Linear Analysis and Symplectic Topology,

#### Cluster Homology: an overwiev of the construction and results

O. Cornea and F. Lalonde, Cluster Homology: an overwiev of the construction and results 12, 1-12 (2006), , ERA - AMS

#### New obstructions to the thickenig of CW complexes

O. Cornea, New obstructions to the thickenig of CW complexes 132, 2769-2781 (2004), , Proc. AMS

#### Rigidity and Glueing for Morse and Novikov complexes

O. Cornea and A. Ranicki, Rigidity and Glueing for Morse and Novikov complexes 5, 343-394 (2003), , Journal of the European Math. Soc.

#### Lusternik Schnirelmann Category

O. Cornea, G. Lupton, J.Oprea and D.Tanre, Lusternik Schnirelmann Category , (2003), , AMS Surveys and Monographs Series

#### Homotopical Dynamics II: Hopf invariants, smoothings and the Morse complex

O. Cornea, Homotopical Dynamics II: Hopf invariants, smoothings and the Morse complex 35, 549-573 (2002), , Ann. Scient. Ec. Norm. Sup.

#### Homotopical Dynamics IV: Hopf invariants and Hamiltonian flows

O. Cornea, Homotopical Dynamics IV: Hopf invariants and Hamiltonian flows 55, 1033-1088 (2002), , Communications on Pure and Applied Math

#### Homotopical Dynamics III: real singularities and Hamiltonian flows

O. Cornea, Homotopical Dynamics III: real singularities and Hamiltonian flows 109 , 183-204 (2001), , Duke Mathematical Journal

#### Homotopical Dynamics: Suspension and Duality

O. Cornea, Homotopical Dynamics: Suspension and Duality 20 , 379-391 (2000), , Ergodic Theory & Dynamical Systems

#### Cone-decompositions and degenerate critical points

Cornea, Octavian, Cone-decompositions and degenerate critical points 77, 437--461 (1998), , Proc. London Math. Soc. (3)

#### Rational category and cone length of Poincaré complexes

Cornea, Octavian, Félix, Yves et Lemaire, Jean-Michel, Rational category and cone length of Poincaré complexes 37, 743--748 (1998), , Topology

#### Lusternik-Schnirelmann-categorical sections

Cornea, Octavian, Lusternik-Schnirelmann-categorical sections 28, 689--704 (1995), , Ann. Sci. École Norm. Sup. (4)

#### Strong LS category equals cone-length

Cornea, Octavian, Strong LS category equals cone-length 34, 377--381 (1995), , Topology

#### There is just one rational cone-length

Cornea, Octavian, There is just one rational cone-length 344, 835--848 (1994), , Trans. Amer. Math. Soc.

#### Cone-length and Lusternik-Schnirelmann category

Cornea, Octavian, Cone-length and Lusternik-Schnirelmann category 33, 95--111 (1994), , Topology

#### The genus and the fundamental group of high-dimensional manifolds

Cornea, Octav, The genus and the fundamental group of high-dimensional manifolds 41, 169--178 (1989), , Stud. Cerc. Mat.

#### Weak discontinuities and discrete approximations

Cornea, Octav, Weak discontinuities and discrete approximations 37, 12--17 (1988), , An. Univ. Bucure\c sti Mat.

#### Some separation properties of hypersurfaces

Cornea, Octav, Some separation properties of hypersurfaces 40, 289--296 (1988), , Stud. Cerc. Mat.

#### Common properties of closed functions and Darboux functions

Cornea, Octav, Common properties of closed functions and Darboux functions 39, 18--21 (1987), , Stud. Cerc. Mat.