RESEARCH WORKSHOP SUPPORTED BY THE EUROPEAN RESEARCH COUNCIL (ERC)

Topological data analysis meets symplectic topology

April 29 - May 3, 2018

Tel Aviv University, Israel

TITLES AND ABSTRACTS:


Alberto Abbondandolo: "On the question of existence of many closed magnetic geodesics"

Closed magnetic geodesics are critical points of a suitable functional - in the exact case - or zeroes of a suitable closed one-form - in the general case - on the space of free loops of arbitrary period. Minimax and Morse theoretical arguments imply often the existence of infinitely many of such critical points or zeroes. An important problem, in order to find lower bounds on the number of closed magnetic geodesics, is to understand whether a given critical point or zero corresponds to a primitive magnetic geodesic or to one of its k-fold covers. In this talk I will discuss some methods to address this problem and some of their consequences.


Yuliy Baryshnikov: "Statistics of jitter in Brownian paths"

Persistence diagrams for non-smooth function are increasingly used in applications (to material sciences, in particular) to characterize these functions. In this talk I'll discuss some recent results on the properties of the point process of zero-dimensional persistence diagrams for the baseline case of Brownian motions


Omer Bobrowski: "Topological Data Analysis and Probability"

In this talk we will review recent progress in the study of probabilistic problems motivated by models and questions in TDA. We will focus mainly on geometric complexes (e.g. the Cech and Vietoris-Rips complexes) generated by random samples in Euclidean spaces and Riemannian manifolds. The main objects of study are the homology and persistent homology of these complexes, and their marginal quantities. Recent progress includes the characterization of various phase transitions (e.g. “sudden” appearance/disappearance of cycles), analysis of limiting distributions (e.g. for persistent Betti numbers, Euler characteristic) , and extreme-value theoretic analysis (the study of outliers). Some of the potential applications of these results are the developing of null-models for statistical TDA, theoretical guarantees for topological inference problems, and noise filtering methods..


Dan Burghelea: "A computer friendly alternative to Morse-Novikov theory (for a closed one form)"

We extend the configurations \delta_r and \gamma_r, equivalently the closed, open, closed-open and open-closed bar codes from real- and angle-valued maps to topological closed one forms on a compact ANR. As a consequence one provides an extension of the classical Novikov complex associated to a closed smooth one form and a vector field the form is Lyapunov for, to a considerably larger class of situations. For these configurations we establish strong stability properties and Poincare duality properties when the underlying space is a closed manifold. Applications to Geometry, Dynamics and Data Analysis are envisioned. (A different approach towards such bar codes was proposed in Usher-Zhang's work).


Wojciech Chachólski: "What is persistence?"

What does it mean to understand shape? How can we measure it and make statistical conclusions about it? Do data sets have shapes and if so how to use their shape to extract information about the data? There are many possible answers to these questions.  Topological data analysis (TDA) aims at providing some of them using  homology.  In my presentation  I will describe a new approach to  TDA. I will illustrate  how our apporach  can be  used to give a machine intelligence to learn geometric shapes and how we can take advantage of this ability in data analysis.


Octav Cornea: "Triangulated metrics and Lagrangian Topology"

I will describe a procedure, related to the theory of persistence modules, that can be used to define metrics on the objects of certain triangulated categories.
This procedure will then be applied to the derived Fukaya category of Lagrangian submanifolds. The talk is based on joint work with Paul Biran and Egor Shelukhin.


Herbert Edelsbrunner: "The Multi-cover Persistence of Euclidean Balls"

Given a locally finite set X in R^d and a positive radius r, the k-fold cover of X consists of all points that have k or more points of X within distance r. The order-k Voronoi diagram decomposes the k-fold cover into convex regions, and we use the dual of this decomposition to compute homology and persistence in scale and in depth. The persistence in depth is interesting from a geometric as well as algorithmic viewpoint. The main tool in understanding its structure is a rhomboid tiling in R^{d+1} that combines the duals for all values of k into one. We mention a straightforward consequence, namely that the cells in the dual are generically not simplicial, unless k=1 or d=1,2. Joint work with Georg Osang. .


Maia Fraser: TBA

TBA.


Urs Frauenfelder: "The kei metric"

This is joint work with Lei Zhao. On the space of involutions one can define a product by conjugating one involution with another one. The algebraic structure one obtains in this way is referred by Takasaki as a kei. A kei is also known as an involutive quandle. Examples of keis are symmetric spaces however one can get as well infinite dimensional examples like the space of smooth involutions of a manifold or the space of (anti)symplectic involutions of a symplectic manifold. As in the theory of symmetric spaces one can associate to a kei its transvection group. For example in this way the Hamiltonian diffeomorphism group of a symplectic manifold can be interpreted as a transvection group of a kei. The kei structure gives rise to a biinvariant metric on its transvection group. This biinvariant metric is closely related to questions of reversibility. In the talk we will explore this.


Viktor Ginzburg: "Dynamics of Hamiltonian pseudo-rotations"

The main theme of this talk, based on a joint work with Basak Gurel, is the dynamics of Hamiltonian diffeomorphisms of projective spaces with the minimal possible number of periodic points (equal to n + 1 by Arnold’s conjecture), called Hamiltonian pseudo-rotations. This is an interesting and important class of Hamiltonian diffeomorphisms, and somewhat surprisingly one can say a lot about their dynamics, going beyond periodic orbits, by using Floer theoretical methods. In this talk, we discuss  the underlying machinery and some of the results focusing on Lagrangian Poincaré recurrence and, if time permits, the existence of invariant sets in arbitrarily small punctured neighborhoods of the fixed points, partially extending a theorem of Le Calvez and Yoccoz and Franks to higher dimensions.


Jean Gutt: "Equivariant symplectic capacities"

We study obstructions to symplectically embedding a cube (a polydisk with all factors equal) into another symplectic manifold with boundary of the same dimension. We find sharp obstructions in many cases, including all "convex toric domains" and "concave toric domains" in C^n. The proof uses analogues of the Ekeland-Hofer capacities, which are conjecturally equal to them, but which are defined using positive S^1-equivariant symplectic homology. This is joint work with Michael Hutchings.


Michael Hutchings: "Computing Reeb dynamics on 4d convex polytopes"

We describe algorithms for computing the periodic orbits and their properties for a combinatorial version of the Reeb vector field on the boundary of a convex polytope in four dimensions. The periods of these Reeb orbits enter into bar codes for contact homology. We present the results of computer experiments testing Viterbo’s conjecture and related conjectures. This is joint work with Julian Chaidez.



Claudia Landi: "The Reeb graph edit distance is universal"

In this talk I will consider those distances between Reeb graphs that are stable, meaning that functions which are similar in the supremum norm ought to have similar Reeb graphs. I will focus on an edit distance for Reeb graphs and prove that it is stable and universal, meaning that it provides an upper bound to any other stable distance. This is joint work with Ulrich Bauer and Facundo Mémoli.


Alexandru Oancea: "Poincaré duality for free loop spaces"

I will explain a framework in which one can make sense of Poincaré duality for free loop spaces. Such a symmetry between homological and cohomological quantities has long been observed in the context of the search for closed geodesics. This reports on joint work in progress with Kai Cieliebak and Nancy Hingston.



Steve Oudot: Decomposition of exact 2-d persistence modules

After a brief review of the current state of knowledge on the decomposition of multidimensional persistence modules and on the stability of these decompositions, I will focus on a specific class of 2-d persistence modules whose internal morphisms satisfy a certain exactness property. The main result I will present states that such modules decompose into thin summands whose supports are planar quadrants or bands. I will also mention some of the implications of this result, e.g. on the stability theory for zigzag modules and interlevel-sets persistence, or on the constructibility of sheaves of vector spaces over the open sets of the real line.


Amit Patel : "Persistent Local Systems"

In persistent homology, we start with a function and construct its sublevel set or its level set persistence module. We then study its barcode. In this talk, we consider constructible maps to manifolds. We generalize the persistence module to something we call the persistence stack of a map. The persistence stack is stable to all sufficiently small perturbations of the map. We also have a generalization of the barcode we call the étalage of a map. This is joint work with Robert MacPherson.


TBA: "Barcodes and  C^0 symplectic topology"

Hamiltonian homeomorphisms are those homeomorphisms of a symplectic manifold which can be written as uniform limits of Hamiltonian diffeomorphisms.  One difficulty in understanding Hamiltonian homeomorphisms (particularly in dimensions greater than two) is that we possess fewer tools for studying them.  For example, (filtered) Floer homology, which has been a very effective tool for studying Hamiltonian diffeomorphisms, is not well-defined for homeomorphisms.  We will show in this talk that using barcodes and persistence homology one can indirectly define (filtered) Floer homology for Hamiltonian homeomorphisms.  We will then apply our newly obtained tool to answer some questions about Hamiltonian homeomorphisms.  This talk is based on two joint projects with Buhovsky-Humiliére and Le Roux-Viterbo..


Egor Shelukhin: "Algebra of bars: bounds, deformations, and applications"

TBA.



Michael Usher: "Knotted symplectic embeddings and persistence barcodes"

The theory of persistence modules provides a useful language for various incarnations of filtered Floer homology in symplectic topology.  Equivariant symplectic homology is one such filtered Floer theory, and has been used in various ways to answer questions related to symplectic embeddings.  I will discuss how an effective obstruction to certain kinds of symplectic embeddings being isotopic to standard ones can be extracted from the persistence barcodes of equivariant symplectic homology.  This obstruction is based on studying a notion in persistence module theory that is related to an interleaving in the same way that an injection is related to a bijection.  Part of the talk will be based on joint work with Jean Gutt.







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