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.[-]

In my talk, I will discuss an application of the theory of motives to transcendence theory, concentrating on the formal aspects. The Period Conjecture predicts that all relations between period numbers are induced by properties of the category of motives. It is a theorem for motives of points and curves, but wide open in general. The Period Conjecture also implies fullness of the Hodge-de Rham realization on Nori motives. If time permits, I will discuss how this generalizes (conjecturally) to triangulated motives and thus to motivic cohomology.[-]

Quadratic enumerative geometry extends classical enumerative geometry. In this enriched setting, the answers to enumerative questions are classes of quadratic forms and live in the Grothendieck-Witt ring GW(k) of quadratic forms. In the talk, we will compute some quadratic enumerative invariants (this can be done, for example, using Marc Levine's localization methods), for example, the quadratic count of lines on a smooth cubic surface.
We will then study the geometric significance of this count: Each line on a smooth cubic surface contributes an element of GW(k) to the total quadratic count. We recall a geometric interpretation of this contribution by Kass-Wickelgren, which is intrinsic to the line and generalizes Segre's classification of real lines on a smooth cubic surface. Finally, we explain how to generalize this to lines of hypersurfaces of degree 2n − 1 in Pn+1. The latter is a joint work with Felipe Espreafico and Stephen McKean.[-]

We report on the development of localization methods useful for quadratic enumerative invariants, replacing the classical Gm-action with an action by the normalizer of the diagonal torus in SL2.
We discuss applications to quadratic counts of twisted cubics in hypersurfaces and complete intersections (joint with Sabrina Pauli) as well as work by Anneloes Vierever, and our joint work with Viergever on quadratic DT invariants for Hilbert schemes of points on P3 and on (P1)3.[-]

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.[-]

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[-]

We consider clustering problems that are fundamental when dealing with trajectory and time series data. The Fréchet distance provides a natural way to measure similarity of curves under continuous reparametrizations. Applied to trajectories and time series, it has proven to be very versatile as it allows local non-linear deformations in time and space. Subtrajectory clustering is a variant of the trajectory clustering problem, where the start and endpoints of trajectory patterns within the collected trajectory data are not known in advance. We study this problem in the form of a set cover problem for a given polygonal curve: find the smallest number k of representative curves such that any point on the input curve is contained in a subcurve that has Fréchet distance at most a given r to a representative curve.[-]

This talk will focus on the fluctuations of a linear spectral statistic around its mean for $P\left(W_N, D_N\right)$ where $P$ is a polynomial, $W_N$ a Wigner matrix and $D_N$ a deterministic diagonal matrix. I will first consider the case when $P\left(W_N,D_N\right)=W_N+D_N$, based on a joint work with M. Février (U. Paris-Saclay). In the general case of $P$ a selfadjoint noncommutative polynomial, I will present results for the special case of the Stieltjes transform, based on a joint work with S. Belinschi (CNRS, U. Toulouse), M. Capitaine (CNRS,U. Toulouse) and M. Février (U. Paris-Saclay).[-]