In a seminal paper Henry Whitehead introduced four elementary operators, collapses and expansions (the inverse of a collapse), perforations and fillings (the inverse of a perforation), which correspond to an homotopy equivalence between two simplicial complexes. In this talk, we consider some transformations which are obtained by the means of these four operators. The presentation is composed of two parts. We begin the first part by introducing a certain axiomatic approach for combinatorial topology, which is settled in the framework of completions. Completions are inductive properties which may be expressed in a declarative way and may be combined. Then, we present a transformation that is based solely on collapses and expansions. This transformation involves homotopic pairs, it may be seen as a refinement of simple homotopy, which takes as input a single object. A homotopic pair is a couple of objects (X, Y ) such that X is included in Y and (X, Y ) may be transformed to a trivial couple by collapses and expansions that keep X inside Y . Our main result states that the collection of all homotopic pairs may be fully described by four completions which correspond to four global properties. After, we consider a transformation that is based on collapses, expansions, perforations, and fillings. This transformation involves contractible pairs, which are extensions of homotopic pairs. Again we show that the collection of all contractible pairs may be fully described by four completions which correspond to four global properties. Three of these completions are the same as the ones describing homotopic pairs. In the second part of the presentation, we introduce the notion of a Morse sequence, which provides a very simple approach to discrete Morse theory. A Morse sequence is obtained by considering only expansions and fillings of a simplicial complex, or, in a dual manner, by considering only collapses and perforations. A Morse sequence may be seen as an alternative way to represent the gradient vector field of an arbitrary discrete Morse function. We introduce reference maps, which are maps that associate a set of critical simplexes to each simplex appearing in a Morse sequence. By considering the boundary of each critical simplex, we obtain a chain complex from these maps, which corresponds precisely to the Morse complex. Then, we define extension maps. We show that, when restricted to homology, an extension map is the inverse of a reference map. Also we show that these two maps allow us to recover directly the isomorphism theorem between the homology of an object and the homology of its Morse complex[-]

Engineering analysis should match an underlying designed shape and not restrict the quality of the shape. I.e. one would like finite elements matching the geometric space optimized for generically good shape. Since the 1980s, classic tensor-product splines have been used both to define good shape geometry and analysis functions (finite elements) on the geometry. Polyhedral-net splines (PnS) generalize tensor-product splines by allowing additional control net patterns required for free-form surfaces: isotropic patterns, such as n quads surrounding a vertex, an n-gon surrounded by quads, polar configurations where many triangles join, and preferred direction patterns, that adjust parameter line density, such as T-junctions. PnS2 generalize C1 bi-2 splines, generate C1 surfaces and can be output bi-3 Bezier pieces. There are two instances of PnS2 in the public domain: a Blender add-on and a ToMS distribution with output in several formats. PnS3 generalize C2 bi-3 splines for high-end design. PnS generalize the use of higher-order isoparametric approach from tensor-product splines. A web interface offers solving elliptic PDEs on PnS2 surfaces and using PnS2 finite elements.[-]

Cellular A1-homology is a new homology theory for smooth algebraic varieties over a perfect field, which is often entirely computable and is expected to give the correct motivic analogue of Poincaré duality for smooth manifolds in classical topology. I will introduce cellular A1-homology, describe the precise conjectures about cellular A1-homology of smooth projective varieties and discuss how they can be verified for smooth projective rational surfaces. The talk is based on joint work with Fabien Morel.[-]

Studying the (closure of the) (semi-)conjugacy class of a given group action on a 1-manifold is interesting from many points of view. Depending on the manifold and/or the differentiability involved, one is faced with problems concerning small denominators, growth of groups / orbits, distortion elements, bounded cohomology, group orderability, etc. In this minicourse we will explore several general results on this topic such as the $C^1$ smoothing via (semi-)conjugacies of small group actions and obstructions in class $C^2$ and higher. We will also explore some of the ideas involved in the proof of the connectedness of the space of $\mathbb{Z}^d$ actions by diffeomorphisms of $C^{1+ac}$ regularity (obtained in collaboration with H. Eynard-Bontemps).[-]

Taut co-orientable foliations are associated with non-trivial elements of Heegard-Floer homology, hence, if a 3-manifold admits a taut, co-oriented foliation, it is not an L-space (Kronheimer-Mrowka-Ozsváth-Szabó). Conjecturally (Boyer-Gordon-Watson, Juhász), the converse is also true for irreducible manifolds. Thus far, the evidence from Dehn surgery on knots in S3 is consistent with this conjecture. We consider the L-space Knot Conjecture: if a knot has no reducible or L-space surgeries, then it is persistently foliar, meaning that for each boundary slope there is a taut, co-oriented foliation meeting the boundary of the knot complement in curves of that slope. For rational slopes, these foliations may be capped off by disks to obtain a taut, co-oriented foliation in every manifold obtained by Dehn surgery on that knot. I will describe an approach, applicable in a variety of settings, to constructing families of foliations realizing all boundary slopes. Recalling the work of Ghiggini, Hedden, Ni, Ozsváth-Szabó (and more recently, Juhász and Baldwin-Sivek) revealed that Dehn surgery on a knot in S3 can yield an L-space only if the knot is fibered and strongly quasipositive, we note that this approach seems to apply more easily when the knot is far from being fibered. As applications of this approach, we find that among the alternating and Montesinos knots, all those without reducible or L-space surgeries are persistently foliar. In addition, we find that any connected sum of alternating knots, Montesinos knots, or fibered knots is persistently foliar. Furthermore, any composite knot with a persistently foliar summand is easily shown to be persistently foliar. This work is joint with Charles Delman.[-]

I will explain how to combine tools of local tropical geometry and logarithmic geometry in order to study the structure of Milnor fibers of smoothings of isolated complex singularities, up to homeomorphisms. I will partly follow the paper “The Milnor fiber conjecture of Neumann and Wahl, and an overview of its proof”, written in collaboration with Marıa Angelica Cueto and Dmitry Stepanov.This course replaces a course on the same topic that should have been delivered by Angelica Cueto.[-]

I will explain how to combine tools of local tropical geometry and logarithmic geometry in order to study the structure of Milnor fibers of smoothings of isolated complex singularities, up to homeomorphisms. I will partly follow the paper “The Milnor fiber conjecture of Neumann and Wahl, and an overview of its proof”, written in collaboration with Marıa Angelica Cueto and Dmitry Stepanov.This course replaces a course on the same topic that should have been delivered by Angelica Cueto.[-]

Left-handed flows are 3-dimensional flows which have a particular topological property, namely that every pair of periodic orbits is negatively linked. This property (introduced by Ghys in 2007) implies the existence of as many Bikrhoff sections as possible, and therefore allows to reduce the flow to a suspension in many different ways. It then becomes natural to look for examples. A construction of Birkhoff (1917) suggests that geodesic flows are good candidates. In this conference we determine on which hyperbolic orbifolds is the geodesic flow left-handed: the answer is that yes if the surface is a sphere with three cone points, and no otherwise.
dynamical system - geodesic flow - knot - periodic orbit - global section - linking number - fibered knot[-]

Hilbert's Fifth Problem asks whether every topological group which is a manifold is in fact a (smooth!) Lie group; this was solved in the affirmative by Gleason and Montgomery-Zippin. A stronger conjecture is that a locally compact topological group which acts faithfully on a manifold must be a Lie group. This is the Hilbert--Smith Conjecture, which in full generality is still wide open. It is known, however (as a corollary to the work of Gleason and Montgomery-Zippin) that it suffices to rule out the case of the additive group of p-adic integers acting faithfully on a manifold. I will present a solution in dimension three.[-]