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Pietro Majer

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University of Zagreb

University of Florida

Austrian Academy of Sciences

Thomás A S Haddad

Universidade de São Paulo

Klaniczay Gabor

Central European University

Petros Bouras-Vallianatos

National & Kapodistrian University of Athens

University of Zurich, Switzerland

Charalambos GASPARIS

National Hellenic Research Foundation

Matthias M Tischler

ICREA

The University Of Jordan

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Papers by Pietro Majer

Variational methods on manifolds with boundary

SIGLEITItal

Infinite dimensional Grassmannians

arXiv: Algebraic Topology, 2003

We study the analytic and homotopy properties of some infinite dimensional Grassmannians, useful ... more We study the analytic and homotopy properties of some infinite dimensional Grassmannians, useful for developing a Morse theory for infinite dimensional manifolds. We study the space of Fredholm pairs of a Hilbert space, we determine its homotopy type, and we define a determinant bundle over it. We study the space of compact perturbations of a given closed linear subspace, and the related concept of essential Grassmannian.

Gap phenomenon for autonomous functionals

Stable foliations and CW-structure induced by a Morse–Smale gradient-like flow

Journal of Topology and Analysis, 2021

We prove that a Morse–Smale gradient-like flow on a closed manifold has a “system of compatible i... more We prove that a Morse–Smale gradient-like flow on a closed manifold has a “system of compatible invariant stable foliations” that is analogous to the object introduced by Palis and Smale in their proof of the structural stability of Morse–Smale diffeomorphisms and flows, but with finer regularity and geometric properties. We show how these invariant foliations can be used in order to give a self-contained proof of the well-known but quite delicate theorem stating that the unstable manifolds of a Morse–Smale gradient-like flow on a closed manifold [Formula: see text] are the open cells of a CW-decomposition of [Formula: see text].

Monotone Paths in Random Hypergraphs

The Electronic Journal of Combinatorics, 2012

We determine the probability thresholds for the existence of monotone paths, of finite and infini... more We determine the probability thresholds for the existence of monotone paths, of finite and infinite length, in random oriented graphs with vertex set $\mathbb N^{[k]}$, the set of all increasing $k$-tuples in $\mathbb N$. These graphs appear as line graph of uniform hypergraphs with vertex set $\mathbb N$.

Ljusternik-Schnirelman theory with local Palais-Smale condition and singular dynamical systems

Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 1991

L'accès aux archives de la revue « Annales de l'I. H. P., section C » (http://www.elsevier.com/lo... more

Ordinary di erential operators in Hilbert spaces and Fredholm pairs

Ordinary differential equations in Banach spaces and the spectral flow

A 2. Fredholm properties of F A Chapter 5. Spectral flow 1. Essentially hyperbolic operators 2. T... more A 2. Fredholm properties of F A Chapter 5. Spectral flow 1. Essentially hyperbolic operators 2. The spectral flow in Banach spaces 3. The Fredholm index and the spectral flow Appendix A. The Cauchy problem Appendix B. Fredholm operators Appendix C. Spectral decomposition Appendix D. Continuous sections of linear maps Appendix. Bibliography

On the best estimate of the number of orthogonal geodesic chords on a manifold with boundary homeomorphic to an annulus

ABSTRACT

Gap Phenomenon for Some Autonomous Functionals

Journal of Convex Analysis

We give an example of an autonomous functional F (u) = Ω f (u, Du)dx (Ω open subset of IR 2 , u :... more We give an example of an autonomous functional F (u) = Ω f (u, Du)dx (Ω open subset of IR 2 , u : Ω → IR 2 in the Sobolev space W 1,1) which is sequentially weakly lower semicontinuous in W 1,p for every p ≥ 1 but does not agree with the relaxation of the same functional restricted to smooth functions when p < 2. A Lavrentiev phenomenon occurs for a related boundary problem.

Inclusions of Banach spaces

Ricerche di Matematica

A survey on non-autonomous basins in several complex variables

Consider a holomorphic automorphism which acts hyperbolically on some invariant compact set. Then... more Consider a holomorphic automorphism which acts hyperbolically on some invariant compact set. Then for every point in the compact set there exists a stable manifold, which is a complex manifold diffeomorphic to real Euclidean space. If the point is fixed, then the stable manifold is even biholomorphic to complex Euclidean space. In fact, it is known that the stable manifold of a generic point is biholomorphic to Euclidean space, and it has been conjectured that this holds for every point. In this article we survey the history of this problem, addressing both known results and the techniques used to obtain those results. Moreover, we present a list of seemingly simpler open problems and prove several new results, all pointing towards a positive answer to the conjecture discussed above.

A non-squeezing theorem for convex symplectic images of the Hilbert ball

Calculus of Variations and Partial Differential Equations, 2015

We prove that the non-squeezing theorem of Gromov holds for symplectomorphisms on an infinite-dim... more We prove that the non-squeezing theorem of Gromov holds for symplectomorphisms on an infinite-dimensional symplectic Hilbert space, under the assumption that the image of the ball is convex. The proof is based on the construction by duality methods of a symplectic capacity for bounded convex neighbourhoods of the origin. We also discuss some examples of symplectomorphisms on infinite-dimensional spaces exhibiting behaviours which would be impossible in finite dimensions.

Truncation of nonlinearities in some supercritical elliptic problems

Two variational methods on manifolds with boundary

Topology, 1995

Infinite paths and cliques in random graphs

Fundamenta Mathematicae, 2012

We study the thresholds for the emergence of various properties in random subgraphs of (N, <). In... more We study the thresholds for the emergence of various properties in random subgraphs of (N, <). In particular, we give sharp sufficient conditions for the existence of (finite or infinite) cliques and paths in a random subgraph. No specific assumption on the probability is made. The main tools are a topological version of Ramsey theory, exchangeability theory and elementary ergodic theory. Contents 14 6. Infinite cliques 15 Appendix A. A topological Ramsey theorem 17 Appendix B. Exchangeable measures 21 References 25

On the effect of the domain on the number of orthogonal geodesic chords

Differential Geometry and its Applications, 1997

In this paper we study the multiplicity of the orthogonal geodesic chords on a convex Riemannian ... more

On the existence of infinitely many periodic solutions to some problems of n-body type

Communications on Pure and Applied Mathematics, 2010

ABSTRACT We prove the existence of infinitely many periodic solutions with prescribed period to a... more

Infinite paths in random shift graphs

We determine the probability thresholds for the existence of monotone paths, of finite and infini... more

Ramsey-type results on random graphs

Variational methods on manifolds with boundary

SIGLEITItal

Infinite dimensional Grassmannians

arXiv: Algebraic Topology, 2003

We study the analytic and homotopy properties of some infinite dimensional Grassmannians, useful ... more We study the analytic and homotopy properties of some infinite dimensional Grassmannians, useful for developing a Morse theory for infinite dimensional manifolds. We study the space of Fredholm pairs of a Hilbert space, we determine its homotopy type, and we define a determinant bundle over it. We study the space of compact perturbations of a given closed linear subspace, and the related concept of essential Grassmannian.

Gap phenomenon for autonomous functionals

Stable foliations and CW-structure induced by a Morse–Smale gradient-like flow

Journal of Topology and Analysis, 2021

We prove that a Morse–Smale gradient-like flow on a closed manifold has a “system of compatible i... more We prove that a Morse–Smale gradient-like flow on a closed manifold has a “system of compatible invariant stable foliations” that is analogous to the object introduced by Palis and Smale in their proof of the structural stability of Morse–Smale diffeomorphisms and flows, but with finer regularity and geometric properties. We show how these invariant foliations can be used in order to give a self-contained proof of the well-known but quite delicate theorem stating that the unstable manifolds of a Morse–Smale gradient-like flow on a closed manifold [Formula: see text] are the open cells of a CW-decomposition of [Formula: see text].

Monotone Paths in Random Hypergraphs

The Electronic Journal of Combinatorics, 2012

We determine the probability thresholds for the existence of monotone paths, of finite and infini... more We determine the probability thresholds for the existence of monotone paths, of finite and infinite length, in random oriented graphs with vertex set $\mathbb N^{[k]}$, the set of all increasing $k$-tuples in $\mathbb N$. These graphs appear as line graph of uniform hypergraphs with vertex set $\mathbb N$.

Ljusternik-Schnirelman theory with local Palais-Smale condition and singular dynamical systems

Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 1991

L'accès aux archives de la revue « Annales de l'I. H. P., section C » (http://www.elsevier.com/lo... more

Ordinary di erential operators in Hilbert spaces and Fredholm pairs

Ordinary differential equations in Banach spaces and the spectral flow

A 2. Fredholm properties of F A Chapter 5. Spectral flow 1. Essentially hyperbolic operators 2. T... more A 2. Fredholm properties of F A Chapter 5. Spectral flow 1. Essentially hyperbolic operators 2. The spectral flow in Banach spaces 3. The Fredholm index and the spectral flow Appendix A. The Cauchy problem Appendix B. Fredholm operators Appendix C. Spectral decomposition Appendix D. Continuous sections of linear maps Appendix. Bibliography

On the best estimate of the number of orthogonal geodesic chords on a manifold with boundary homeomorphic to an annulus

ABSTRACT

Gap Phenomenon for Some Autonomous Functionals

Journal of Convex Analysis

We give an example of an autonomous functional F (u) = Ω f (u, Du)dx (Ω open subset of IR 2 , u :... more We give an example of an autonomous functional F (u) = Ω f (u, Du)dx (Ω open subset of IR 2 , u : Ω → IR 2 in the Sobolev space W 1,1) which is sequentially weakly lower semicontinuous in W 1,p for every p ≥ 1 but does not agree with the relaxation of the same functional restricted to smooth functions when p < 2. A Lavrentiev phenomenon occurs for a related boundary problem.

Inclusions of Banach spaces

Ricerche di Matematica

A survey on non-autonomous basins in several complex variables

Consider a holomorphic automorphism which acts hyperbolically on some invariant compact set. Then... more Consider a holomorphic automorphism which acts hyperbolically on some invariant compact set. Then for every point in the compact set there exists a stable manifold, which is a complex manifold diffeomorphic to real Euclidean space. If the point is fixed, then the stable manifold is even biholomorphic to complex Euclidean space. In fact, it is known that the stable manifold of a generic point is biholomorphic to Euclidean space, and it has been conjectured that this holds for every point. In this article we survey the history of this problem, addressing both known results and the techniques used to obtain those results. Moreover, we present a list of seemingly simpler open problems and prove several new results, all pointing towards a positive answer to the conjecture discussed above.

A non-squeezing theorem for convex symplectic images of the Hilbert ball

Calculus of Variations and Partial Differential Equations, 2015

We prove that the non-squeezing theorem of Gromov holds for symplectomorphisms on an infinite-dim... more We prove that the non-squeezing theorem of Gromov holds for symplectomorphisms on an infinite-dimensional symplectic Hilbert space, under the assumption that the image of the ball is convex. The proof is based on the construction by duality methods of a symplectic capacity for bounded convex neighbourhoods of the origin. We also discuss some examples of symplectomorphisms on infinite-dimensional spaces exhibiting behaviours which would be impossible in finite dimensions.

Truncation of nonlinearities in some supercritical elliptic problems

Two variational methods on manifolds with boundary

Topology, 1995

Infinite paths and cliques in random graphs

Fundamenta Mathematicae, 2012

We study the thresholds for the emergence of various properties in random subgraphs of (N, <). In... more We study the thresholds for the emergence of various properties in random subgraphs of (N, <). In particular, we give sharp sufficient conditions for the existence of (finite or infinite) cliques and paths in a random subgraph. No specific assumption on the probability is made. The main tools are a topological version of Ramsey theory, exchangeability theory and elementary ergodic theory. Contents 14 6. Infinite cliques 15 Appendix A. A topological Ramsey theorem 17 Appendix B. Exchangeable measures 21 References 25

On the effect of the domain on the number of orthogonal geodesic chords

Differential Geometry and its Applications, 1997

In this paper we study the multiplicity of the orthogonal geodesic chords on a convex Riemannian ... more

On the existence of infinitely many periodic solutions to some problems of n-body type

Communications on Pure and Applied Mathematics, 2010

ABSTRACT We prove the existence of infinitely many periodic solutions with prescribed period to a... more

Infinite paths in random shift graphs

We determine the probability thresholds for the existence of monotone paths, of finite and infini... more

Ramsey-type results on random graphs