Abstracts
Eva Miranda
Title:
From contact and cosymplectic manifolds to hybrid computers
Abstract:
In [1], we established an association between a universal Turing machine and a dynamical system defined by a Reeb vector field. The Etnyre-Ghrist correspondence [2], connecting Reeb and (mirror) Beltrami vector fields, led to the intriguing concept of associating a Turing machine with a stationary solution of the Euler equations, thus yielding the design of an "abstract" Fluid computer. A key step in this construction involves upgrading an area-preserving diffeomorphism of the disc (extending one of Cris Moore's mappings of the square Cantor set [3]) to the time-one-map of a 3D Reeb vector field on a solid torus—a fundamental unit which we call a 'flubit.'
This construction can be extended to different geometries, such as cosymplectic manifolds, allowing us to associate a universal Turing machine with the 'Reeb vector field' of the cosymplectic structure. Notably, this vector field also possesses a mirror, a harmonic vector field, as established in [4].
Both the contact and cosymplectic situations can be generalized to a broader framework. In this talk, we will revisit the constructions presented in [1], taking it as the starting point for the design of a hybrid computer. Drawing inspiration from Feynman's rules in quantum computing, we opt to replace qubits with 'flubits.' This shift leads us to define a computational field theory, or 'hybrid computer,' as a functor from the category of cobordisms (decorated with transverse vector fields) to the category of partial recursive functions. This construction generalizes the ideas introduced in [1] and [4]. Time permitting, we will explore the relationship between dynamical properties of the transverse vector field and computational complexity.
This talk is based on joint works with Robert Cardona, Daniel Peralta-Salas and Fran Presas and ongoing work with Søren Dyhr, Ángel González-Prieto and Daniel Peralta-Salas.
[1] R. Cardona, E Miranda, D. Peralta-Salas and F. Presas, Constructing Turing complete Euler flows in dimension 3, Proc. Natl. Acad. Sci. USA 118 (2021), no. 19, Paper No. e2026818118, 9 pp.
[2] J. Etnyre, R. Ghrist, Contact topology and hydrodynamics: I. Beltrami fields and the Seifert conjecture. Nonlinearity 13, 441 (2000).
[3] C. Moore, Generalized shifts: Unpredictability and undecidability in dynamical systems. Nonlinearity 4, 199 (1991).
[4] S. Dyhr, A. González-Prieto, E Miranda, D. Peralta-Salas, Harmonic vector fields and cosymplectic manifolds, working paper, 2024.
[5] A. González-Prieto, E Miranda, D. Peralta-Salas, The hybrid computer, working paper, 2024.
Matthias Meiwes
Title:
Homotopical orbit growth in the horseshoe map, and the Hofer metric
Abstract:
Recently, some fruitful approaches were found to express complexity of Hamiltonian or Reeb dynamics in terms of Floer theory or contact homology. One goes back to Alves and Pirnapasov, where, in a certain sense, the growth of homotopy classes of orbits in a complement of a link of periodic orbits in 3D flows is studied. In my talk, I will describe a method of how to detect homotopical orbit growth in the complement of a set of orbits in the horseshoe map. I will discuss some recent results in the symplectic dynamics context where this method is applied. One, obtained jointly with M. Alves, L. Dahinden, and A. Pirnapasov, deals with stability properties of the topological entropy of Reeb flows. Another is the following: The length of any closed monotone curve in a "large" subset of curves, open with respect to the Hofer distance, grows exponentially under iteration of a given Hamiltonian surface diffeomorphism of positive topological entropy.
Alfonso Sorrentino
Title:
The Hamilton–Jacobi equation on networks: from Aubry–Mather Theory to Homogenization.
Abstract:
Over the last few years, there has been an increasing interest in studying the Hamilton–Jacobi Equation on networks and related questions. These problems involve several subtle theoretical issues and have a significant impact on applications in various fields. While locally - i.e., on each branch of the network (arcs) - the study reduces to the analysis of 1-dimensional problems, the main difficulties arise in matching together the information converging at the juncture of two or more arcs, and relating the local analysis at a juncture with the global structure/topology of the network.
In this talk, firstly, I shall discuss several results related to the global analysis of this problem. More specifically, we developed analogs of the so-called Weak KAM theory and Aubry–Mather theory in this setting; the salient point of our approach is to associate the network with an abstract graph, encoding all of the information on the complexity of the network, and to relate the differential equation to a discrete functional equation on this graph. Then, I shall describe how to prove a Homogenization result in this context, with particular emphasis on the role of the topological complexity of the network in determining the limit problem.
Daniel Peralta-Salas
Title:
On the existence of critical compatible metrics on contact 3-manifolds.
Abstract:
In this talk I will review a recent joint work with Y. Mitsumatsu and R. Slobodeanu where we disprove the generalized Chern-Hamilton conjecture on the existence of critical compatible metrics on contact 3-manifolds. More precisely, we show that a contact 3-manifold admits a critical compatible metric for the Chern-Hamilton energy functional if and only if it is Sasakian or its associated Reeb flow is algebraic Anosov. As a corollary we prove that no contact structure on the torus admits a critical compatible metric and that critical compatible metrics can only occur when the contact structure is universally tight.
Başak Gürel
Title:
Invariant Sets and Hyperbolic Periodic Orbits
Abstract:
The presence of hyperbolic periodic orbits or invariant sets often has an effect on the global behavior of a dynamical system. In this talk we discuss two theorems along the lines of this phenomenon, extending some properties of Hamiltonian diffeomorphisms to dynamically convex Reeb flows on the sphere in all dimensions. The first one, complementing other multiplicity results for Reeb flows, is that the existence of a hyperbolic periodic orbit forces the flow to have infinitely many periodic orbits. This result can be thought of as a step towards Franks’ theorem for Reeb flows. The second result is a contact analogue of the higher-dimensional Le Calvez-Yoccoz theorem proved by the speaker and Ginzburg and asserting that no periodic orbit of a Hamiltonian pseudo-rotation is locally maximal.
The talk is based on a joint work with Erman Cineli, Viktor Ginzburg and Marco Mazzucchelli.
Simon Allais
Title:
Contact orderability and spectral selectors
Abstract:
In 2000, Eliashberg and Polterovich introduced the notion of orderability to investigate the structure of the group of contact diffeomorphisms and the structure of isotopy classes of Legendrian submanifolds. Roughly speaking, a group of contact diffeomorphisms is orderable if the relation induced by the partial order on contact Hamiltonian maps induces a partial order on the associated time-one flows. Throughout the years, orderability and non-orderability have been often studied using some sorts of spectral selectors based on Floer-like theories and generating functions. In this talk, we will explain why orderability is equivalent to the existence of spectral selectors and how these selectors can be used to derive multiple geometric properties in the orderable case: existence of Reeb chords between Legendrians, existence of time-function from the Lorentz-Finsler viewpoint, non-degeneracy of the interval topology, existence of geodesics for some natural metrics.
This is a joint work with Pierre-Alexandre Arlove.
Thomas Mark
Title:
Constraints on contact type hypersurfaces in symplectic 4-manifolds
Abstract:
In joint work with Bulent Tosun, it was shown that Heegaard Floer theory provides an obstruction for a contact 3-manifold to embed as a contact type hypersurface in standard symplectic 4-space. As one consequence, no Brieskorn homology sphere admits such an embedding (regardless of the contact structure). I will review the ideas that lead to these results, and discuss recent extensions that can obstruct suitably convex embeddings in closed symplectic 4-manifolds, particularly rational complex surfaces.
Dustin Connery-Grigg
Title:
PSS-image spectral invariants and the dynamics of low-dimensional Hamiltonian systems
Abstract:
Since their introduction by Schwarz in 2000 (following an earlier idea of Viterbo), Floer-theoretic spectral invariants have become a central tool in the modern study of symplectic topology and Hamiltonian systems. Unfortunately, given a particular Hamiltonian isotopy, it is often very difficult to compute its associated spectral invariants, and the relationship of these invariants to the underlying dynamics remains opaque. In this talk I will discuss the construction and computation of a novel class of spectral invariants for Hamiltonian systems which share the main formal properties that make classical spectral invariants useful, but which have the advantage of admitting a completely dynamical interpretation for generic Hamiltonian systems on surfaces.
Dan Cristofaro-Gardiner
Title:
Hofer-Wysocki-Zehnder's conjecture on two or infinitely many orbits
Abstract:
In their 2003 paper, Hofer, Wysocki and Zehnder conjectured that every autonomous Hamiltonian flow has either two or infinitely many simple periodic orbits on any compact star-shaped energy level; in the same paper, the authors prove this assuming in addition that the flow is nondegenerate and the stable and unstable manifolds of all hyperbolic orbits intersect transversally, a condition which holds generically. I will explain recent joint work resolving this conjecture. Our results also apply to show that every Finsler metric on the two-sphere has either two or infinitely many prime closed geodesics, answering a question attributed to Alvarez Paiva, Bangert and Long.
Rohil Prasad
Title:
On the dense existence of compact invariant sets
Abstract:
This is joint work with Dan Cristofaro-Gardiner.
We show that for any monotone-area preserving diffeomorphism of a closed surface or for any Reeb flow on a closed contact 3-manifold with torsion Chern class, there exist infinitely many distinct proper compact invariant subsets whose union is dense in the manifold. No genericity assumptions are required. The former class of systems includes all Hamiltonian diffeomorphisms of closed surfaces and the latter class of systems includes the geodesic flow of any Finsler metric on a closed surface. In particular, our methods can also show that any Finsler metric on a closed surface has infinitely many non-dense geodesics, with pairwise distinct closures, whose union is dense in the surface.
Marc Kegel
Title:
Contact surgery numbers
Abstract:
The contact surgery number of a contact 3-manifold is the minimal number of components among all surgery descriptions of that contact manifold. In this talk, I will explain general methods to give upper and lower bounds on contact surgery numbers and discuss certain situations in which we can explicitly compute contact surgery numbers.
This talk is based on joint work with John Etnyre and Sinem Onaran and on joint work with Rima Chatterjee.
Michael Usher
Title:
Interlevel persistence in Floer theory
Abstract:
The usual filtered Floer homology groups are formal analogues of the homologies of the sublevel sets of a Morse function on a manifold. In the Morse setting, by instead considering interlevel sets (preimages of general intervals) one obtains an algebraic structure that is classified by a barcode that refines the usual sublevel persistence barcode. I will describe an algebraic formalism that allows one to adapt this to Floer-theoretic settings. In the case of Hamiltonian Floer theory on closed manifolds, this gives rise to a pairing between distinguished spectral invariants of a Hamiltonian flow and of its inverse that satisfies stability and duality theorems. A version on Liouville manifolds is related to the canonical map from symplectic cohomology to symplectic homology.