Starting in the eighties, the emerging computational geometry community has put
a lot of effort into designing and analyzing algorithms for geometric problems.
The most commonly used framework was to study
the worst-case theoretical complexity of geometric problems
involving linear objects (points, lines, polyhedra...) in
Euclidean spaces.
This so-called
classical computational geometry has some known
limitations:

Even if these limitations have already got some attention from the community 53, a quick look at the flagship conference SoCG 1 proceedings shows that these topics still need a big effort.

It should be stressed that, in this document, the notion of certified algorithms is to be understood with respect to robustness issues. In other words, certification does not refer to programs that are proven correct with the help of mechanical proof assistants such as Coq, but to algorithms that are proven correct on paper even in the presence of degeneracies and computer-induced numerical rounding errors.

We address several of the above limitations:

$\u2022$ Non-linear computational geometry.
Curved objects are ubiquitous in the world we live in. However,
despite this ubiquity and decades of research in several
communities, curved objects
are far
from being robustly and efficiently manipulated by geometric algorithms. Our work on, for instance,
quadric intersections and certified drawing of plane curves has proven that
dramatic improvements can be accomplished when the right mathematics and
computer science concepts
are put into motion. In this direction, many problems
are fundamental
and solutions have potential industrial impact in Computer Aided
Design and Robotics for instance.
Intersecting NURBS (Non-uniform rational basis splines) and meshing
singular surfaces in a certified manner
are important examples of such problems.

$\u2022$ Non-Euclidean computational geometry.
Triangulations are central
geometric data structures in many areas of science and
engineering. Traditionally, their study has been limited to the
Euclidean setting. Needs for triangulations in non-Euclidean settings have emerged in many areas
dealing with objects whose sizes range from the
nuclear to the astrophysical scale, and both in academia and in industry.
It has become timely to extend the traditional focus on

$\u2022$ Probability in computational geometry.
The design of efficient algorithms is driven by the analysis of their
complexity. Traditionally, worst-case input and sometimes uniform distributions
are considered and many results in these settings have had a great influence on
the domain.
Nowadays, it is necessary to be more subtle and to prove new results in between these two extreme settings.
For instance, smoothed analysis, which was introduced for the simplex algorithm and which we applied successfully to
convex hulls, proves that
such promising alternatives exist.

$\u2022$ Discrete geometric structures. Many
geometric algorithms work, explicitly or implicitly, over discrete
structures such as graphs, hypergraphs, lattices that are
induced by the geometric input data. For example, convex hulls or
straight-line graph drawing are essentially based on orientation
predicates, and therefore operate on the so-called

As mentioned above, curved objects are ubiquitous in real world problems
and
in computer science and, despite this
fact, there are very few problems on curved objects that admit robust and efficient algorithmic
solutions without first discretizing the curved objects into meshes.
Meshing curved objects induces
a loss of accuracy which is sometimes not an issue but which can also be most problematic
depending on the application. In addition, discretization induces a combinatorial explosion which could
cause a loss in efficiency compared to a direct solution on the curved objects (as our work on
quadrics has demonstrated with flying colors
60, 61, 59, 63, 67).
But it is also crucial to know that even the
process of computing meshes that approximate curved objects is far from being resolved. As a matter
of fact there is no algorithm capable of computing in practice meshes with certified topology of
even rather simple singular 3D surfaces, due to the high constants in the
theoretical complexity and the difficulty of handling degenerate cases.
Part of the difficulty comes from the unintuitive fact that the structure of an algebraic
object can be quite
complicated, as depicted in the Whitney umbrella (see Figure 1), surface of the
equation

It is thus to be understood that producing practical robust and efficient algorithmic solutions to
geometric problems on curved objects is a challenge on all and even the most basic problems. The
basicness and fundamentality of two problems we mentioned above on the intersection of 3D quadrics
and on the drawing in a topologically certified way of plane algebraic curves show rather well that
the domain is still in its infancy. And it should be stressed that these two sets of results were
not anecdotal but flagship results produced during the lifetime of
the Vegas team (the team preceding Gamble).

There are many problems in this theme that are expected to have high long-term impacts. Intersecting NURBS (Non-uniform rational basis splines) in a certified way is an important problem in computer-aided design and manufacturing. As hinted above, meshing objects in a certified way is important when topology matters. The 2D case, that is essentially drawing plane curves with the correct topology, is a fundamental problem with far-reaching applications in research or R&D. Notice that on such elementary problems it is often difficult to predict the reach of the applications; as an example, we were astonished by the scope of the applications of our software on 3D quadric intersection 2 which was used by researchers in, for instance, photochemistry, computer vision, statistics and mathematics.

Triangulations, in particular Delaunay triangulations, in the
Euclidean spaceet al.41). Some members of Gamble have been contributing to these algorithmic advances
(see, e.g. 45, 77, 58, 44); they have also
contributed robust and efficient triangulation packages through the
state-of-the-art Computational Geometry Algorithms Library
Cgal whose impact extends far
beyond computational geometry.
Application fields include particle physics, fluid dynamics, shape
matching, image processing, geometry processing, computer graphics,
computer vision, shape reconstruction, mesh generation, virtual
worlds, geophysics, and medical
imaging. 3

It is fair to say that little has been done on non-Euclidean spaces,
in spite of the large number of questions raised by application
domains. Needs for simulations or modeling in a variety of
domains 4
ranging from the infinitely small (nuclear matter, nano-structures,
biological data) to the infinitely large (astrophysics) have led us to
consider 3D periodic Delaunay triangulations, which can be seen
as Delaunay triangulations in the 3D flat torus, quotient of
Cgal package 49 is the only publicly available software
that computes Delaunay triangulations of
a 3D flat torus, in the special case where the domain is cubic. This case, although restrictive, is already useful. 5
We have also generalized this algorithm
to the case of general

Interestingly, even for the simple case of triangulations on the sphere, the software
packages that are
currently
available are far from offering satisfactory solutions in terms of
robustness and efficiency 52.

Moreover, while our solution for computing triangulations in
hyperbolic spaces can be considered as ultimate 42, the case
of hyperbolic manifolds has hardly been explored. Hyperbolic manifolds are
quotients of a hyperbolic space by some group of hyperbolic
isometries. Their triangulations can be seen as hyperbolic
periodic triangulations. Periodic hyperbolic triangulations and
meshes appear for instance in geometric modeling
73, neuromathematics 54, or physics
74. Even the case of the Bolza surface (a surface
of genus 2, whose fundamental domain is the regular octagon in the
hyperbolic plane) shows mathematical
difficulties 43, 66.

In most computational geometry papers, algorithms are analyzed in the worst-case setting. This often yields too pessimistic complexities that arise only in pathological situations that are unlikely to occur in practice. On the other hand, probabilistic geometry provides analyses with great precision 71, 72, 48, but using hypotheses with much more randomness than in most realistic situations. We are developing new algorithmic designs improving state-of-the-art performance in random settings that are not overly simplified and that can thus reflect many realistic situations.

Sixteen years ago, smooth analysis was introduced by Spielman and Teng analyzing the simplex algorithm by averaging on some noise on the data 76 (and they won the Gödel prize). In essence, this analysis smoothes the complexity around worst-case situations, thus avoiding pathological scenarios but without considering unrealistic randomness. In that sense, this method makes a bridge between full randomness and worst case situations by tuning the noise intensity. The analysis of computational geometry algorithms within this framework is still embryonic. To illustrate the difficulty of the problem, we started working in 2009 on the smooth analysis of the size of the convex hull of a point set, arguably the simplest computational geometry data structure; then, only one very rough result from 2004 existed 55 and we only obtained in 2015 breakthrough results, but still not definitive 57, 56, 62.

Another example of a problem of different flavor concerns Delaunay triangulations, which are rather ubiquitous in computational geometry. When Delaunay triangulations are computed for reconstructing meshes from point clouds coming from 3D scanners, the worst-case scenario is, again, too pessimistic and the full randomness hypothesis is clearly not adapted. Some results exist for “good samplings of generic surfaces” 40 but the big result that everybody wishes for is an analysis for random samples (without the extra assumptions hidden in the “good” sampling) of possibly non-generic surfaces.

Trade-offs between full randomness and worst case may also appear in other forms such as dependent distributions, or random distributions conditioned to be in some special configurations. Simulating these kinds of geometric distributions is currently out of reach for more than a few hundred points 65 although it has practical applications in physics or networks.

Our work on discrete geometric structures develops in several directions, each one probing a different type of structure. Although these objects appear unrelated at first sight, they can be tackled by the same set of probabilistic and topological tools.

A first research topic is the study of Order types. Order types
are combinatorial encodings of finite (planar) point sets, recording
for each triple of points the orientation (clockwise or
counterclockwise) of the triangle they form. This already determines
properties such as convex hulls or half-space depths, and the
behaviour of algorithms based on orientation predicates. These
properties for all (infinitely many)

A second research topic is the study of Embedded graphs and
simplicial complexes. Many topological structures can be
effectively discretized, for instance combinatorial maps record
homotopy classes of embedded graphs and simplicial complexes represent
a large class of topological spaces. This raises many structural and
algorithmic questions on these discrete structures; for example, given
a closed walk in an embedded graph, can we find a cycle of the graph
homotopic to that walk? (The complexity status of that problem is
unknown.) Going in the other direction, some purely discrete
structures can be given an associated topological space that reveals
some of their properties (e.g. the Nerve theorem for
intersection patterns). An open problem is for instance to obtain
fractional Helly theorems for set system of bounded topological
complexity.

Another research topic is that of Sparse inclusion-exclusion formulas.
For any family of sets

where

Many domains of science can benefit from the results developed
by Gamble.
Curves and surfaces are ubiquitous in all sciences to
understand and interpret raw data as well as experimental results.
Still, the non-linear problems we address are rather basic and
fundamental, and it is often difficult to predict the impact of
solutions in that area.
The short-term industrial impact is likely to be small because, on basic
problems, industries have used ad hoc solutions for decades and have thus got
used to it.

The example of our work on quadric intersection is typical: even though we were fully convinced that intersecting 3D quadrics is such an elementary/fundamental problem that it ought to be useful, we were the first to be astonished by the scope of the applications of our software 6 (which was the first and still is the only one —to our knowledge— to compute robustly and efficiently the intersection of 3D quadrics) which has been used by researchers in, for instance, photochemistry, computer vision, statistics, and mathematics. Our work on certified drawing of plane (algebraic) curves falls in the same category. It seems obvious that it is widely useful to be able to draw curves correctly (recall also that part of the problem is to determine where to look in the plane) but it is quite hard to come up with specific examples of fields where this is relevant. A contrario, we know that certified meshing is critical in mechanical-design applications in robotics, which is a non-obvious application field. There, the singularities of a manipulator often have degrees higher than 10 and meshing the singular locus in a certified way is currently out of reach. As a result, researchers in robotics can only build physical prototypes for validating, or not, the approximate solutions given by non-certified numerical algorithms.

The fact that several of our
pieces of software for computing non-Euclidean triangulations
had already been requested by users
long before they become public in Cgal is a good sign for their wide future
impact. This will not come as a
surprise, since most of the questions that we have been studying
followed from
discussions with
researchers outside computer science and pure mathematics.
Such researchers are either
users of our algorithms and software, or we meet them in
workshops.
Let us only
mention a few names here. Rien van de
Weijgaert 64, 78 (astrophysicist,
Groningen, NL) and Michael Schindler 75
(theoretical physicist, ENSPCI, CNRS, France) used our
software for 3D periodic weighted triangulations.
Stephen Hyde and Vanessa Robins (applied mathematics and
physics at Australian National University) used our package for 3D
periodic meshing. Olivier Faugeras (neuromathematics, INRIA Sophia
Antipolis) had come to us and mentioned his needs for good meshes of the Bolza
surface 54 before we started to study them.
Such contacts are very important both to get feedback about our
research and to help us choose problems that are relevant for
applications. These problems are at the same time challenging
from the mathematical and algorithmic points of view. Note that our research and our
software are generic, i.e., we are studying fundamental geometric
questions, which do not depend on any specific application. This
recipe has made the sucess of the Cgal library.

Probabilistic models for geometric data are widely used to model various situations ranging from cell phone distribution to quantum mechanics. The impact of our work on probabilistic distributions is twofold. On the one hand, our studies of properties of geometric objects built on such distributions will yield a better understanding of the above phenomena and has potential impact in many scientific domains. On the other hand, our work on simulations of probabilistic distributions will be used by other teams, more maths oriented, to study these distributions.

Xavier Goaoc received the best paper award
for his paper at the 36th International Symposium on Computational
Geometry coauthored with Emo Welzl (ETH Zurich)

Monique Teillaud received the best paper award at the
28th European Symposium on Algorithms (Track Engineering and
Applications) for her paper
Generalizing CGAL Periodic Delaunay Triangulations
coauthored with Georg Osang (IST Austria) and Mael Rouxel-Labbé (Geometry
Factory)

Monique Teillaud gave a keynote talk at the 36th European Workshop on
Computational Geometry:
Triangulations in CGAL - To non-Euclidean spaces and beyond! 26.

The major event of the year is of course the corona virus crisis. All ongoing works were slowed down by working remotely most of the time.

This work, published in the journal of Mathematics in Computer Science,
gives the first algorithm for finding a set of natural

In collaboration with R. Imbach and C. Yap (Courant Institute of
Mathematical Sciences, New York University, USA).

Isolating the singularities of a plane curve is the first step towards computing its topology. For this, numerical methods are efficient but not certified in general. We are interested in developing certified numerical algorithms for isolating the singularities. In order to do so, we restrict our attention to the special case of plane curves that are the projections of smooth curves in higher dimensions. This type of curves appears naturally in robotics applications and scientific visualization. In this setting, we show that the singularities can be encoded by a regular square system whose solutions can be isolated with certified numerical methods. Our analysis is conditioned by assumptions that we prove to be generic using transversality theory, and we also provide a semi-algorithm to check their validity 38.

We consider geometric triangulations of surfaces, i.e., triangulations whose edges can be realized by disjoint geodesic segments. We prove that the flip graph of geometric triangulations with fixed vertices of a flat torus or a closed hyperbolic surface is connected. We give upper bounds on the number of edge flips that are necessary to transform any geometric triangulation on such a surface into a Delaunay triangulation 3.

In collaboration with Jean-Marc Schlenker (University of Luxembourg)

Even though Delaunay originally introduced his famous triangulations in the case of infinite point sets with translational periodicity, a software that computes such triangulations in the general case is not yet available, to the best of our knowledge. Combining and generalizing previous work, we present a practical algorithm for computing such triangulations. The algorithm has been implemented and experiments show that its performance is as good as the one of the CGAL package, which is restricted to cubic periodicity 21.

In collaboration with Georg Osang (IST Austria) and Mael Rouxel-Labbé (Geometry Factory).

Let us consider a regular octagon in the Poincaré disk that is centered at the origin. The hyperbolic isometries identifying its opposite edges pairwise generate a group. The Bolza surface can be seen as the quotient of the hyperbolic plane under the action this group. We consider generalized Bolza surfaces

In collaboration with Matthijs Ebbens and Gert Vegter (Bernoulli Institute for Mathematics and Computer Science and Artificial Intelligence, University of Groningen).

Given a finite point cloud

Let

In collaboration with Michaël Rao and Stéphan Thomassé (LIP)

We present a technique for the enumeration of all isotopically distinct ways of tiling, with disks, a hyperbolic surface of finite genus, possibly nonorientable and with punctures and boundary. This provides a generalization of the enumeration of Delaney-Dress combinatorial tiling theory on the basis of isotopic tiling theory. To accomplish this, we derive representations of the mapping class group of the orbifold associated to the symmetry group of the tiling under consideration as a set of algebraic operations on certain generators of the symmetry group. We derive explicit descriptions of certain subgroups of mapping class groups and of tilings as embedded graphs on orbifolds. We further use this explicit description to present an algorithm that we illustrate by producing an array of examples of isotopically distinct tilings of the hyperbolic plane with symmetries generated by rotations that are commensurate with the prominent Primitive, Diamond and Gyroid triply-periodic minimal surfaces, outlining how the approach yields an unambiguous enumeration. We also present the corresponding 3-periodic graphs on these surfaces 24.

In collaboration with Myfanwy Evans (University of Potsdam)

In this paper, we develop the mathematical tools needed to explore isotopy classes of tilings on hyperbolic surfaces of finite genus, possibly non-orientable, with boundary, and punctured. More specifically, we generalize results on Delaney-Dress combinatorial tiling theory using an extension of mapping class groups to orbifolds, in turn using this to study tilings of covering spaces of orbifolds. Moreover, we study finite subgroups of these mapping class groups. Our results can be used to extend the Delaney-Dress combinatorial encoding of a tiling to yield a finite symbol encoding the complexity of an isotopy class of tilings. The results of this paper provide the basis for a complete and un-ambiguous enumeration of isotopically distinct tilings of hyperbolic surfaces 18.

In collaboration with Myfanwy Evans (University of Potsdam).

Based on the mathematical theory of isotopic tilings on hyperbolic surfaces and mapping class groups, we present the, to the best of our knowledge, first algorithms for the enumeration of isotopy classes of tilings by compact disks with a given symmetry group on hyperbolic surfaces, which is moreover combinatorial in nature. This enumeration is relevant for crystallography and material science. Using the theory of automatic groups, we give some results on the computational tractability of the presented algorithm. We also extend data structures for combinatorial classes of tilings to isotopy classes and give an implementation of the proposed algorithm for certain classes of tilings and illustrate the enumeration with examples 37.

We present a method to enumerate tile-transitive crystallographic tilings of the Euclidean and hyperbolic planes by unbounded ribbon tiles up to equivariant equivalence. The hyperbolic case is relevant to self-assembly of branched polymers. This is achieved by combining and extending known methods for enumerating crystallographic disk-like tilings. We obtain a natural way of describing all possible stabiliser subgroups of tile-transitive tilings using a topological viewpoint of the tile edges as a graph embedded in an orbifold, and a group theoretical one derived from the structure of fundamental domains for discrete groups of planar isometries 36.

In collaboration with Vanessa Robins (Australian National University).

We study online routing algorithms on the

In collaboration with Prosenjit Bose (University Carleton) and
Jean-Lou De Carufel (University of Ottawa)

We examine how the measure and the number of vertices of the convex
hull of a random sample of

In collaboration with
Imre Barany (Rényi Institute of Mathematics)
Matthieu Fradelizi (Laboratoire d'Analyse et de Mathématiques Appliquées)
Alfredo Hubard (Laboratoire d'Informatique Gaspard-Monge)
Günter Rote (Institut für Informatik, Berlin)

We analyze several optimal transportation problems between determinantal point processes. We show how to estimate some of the distances between distributions of DPP they induce. We then apply these results to evaluate the accuracy of a new and fast DPP simulation algorithm. We can now simulate in a reasonable amount of time more than ten thousand points 31 (see also Section 6).

In collaboration with Laurent Decreusefond (Laboratoire Traitement et Communication de l'Information)

We establish the following two main results on order types of points in general position in the plane (realizable simple planar order types, realizable uniform acyclic oriented matroids of rank 3):

Result (a) generalizes to arbitrary dimension d for labeled order types with the average number of extreme points

In collaboration with
Emo Welzl (ETH Zürich)

This chapter (in French) offers an introduction to combinatorial convexity, its algorithmic applications and its extensions in topological combinatorics 28.

In most layered additive manufacturing processes, a tool solidifies or deposits material while following pre-planned trajectories to form solid beads. Many interesting problems arise in this context, among which one concerns the planning of trajectories for filling a planar shape as densely as possible. This is the problem we tackle in the present paper. Recent works have shown that allowing the bead width to vary along the trajectories helps increase the filling density. We present a novel technique that, given a deposition width range, constructs a set of closed beads whose width varies within the prescribed range and fills the input shape. The technique outperforms the state of the art in important metrics: filling density (while still guaranteeing the absence of bead overlap) and smoothness of trajectories. We give a detailed geometric description of our algorithm, explore its behavior on example inputs and provide a statistical comparison with the state of the art. We show that it is possible to obtain high quality fabricated layers on commodity FDM printers 16.

In collaboration with
Samuel Hornus, Jonàs Martinez, Sylvain Lefebvre (project-team
MFX
), Marc Glisse (project-team
DataShape
), and Tim Kuipers (Ultimaker).

A cover for a family

In collaboration with
Otfried Cheong (KAIST) and Marc Glisse (project-team
DataShape
).

An effective way to reduce clutter in a graph drawing that has (many) crossings is to group edges that travel in parallel into bundles. Each edge can participate in many such bundles. Any crossing in this bundled graph occurs between two bundles, i.e., as a bundled crossing. We consider the problem of bundled crossing minimization: A graph is given and the goal is to find a bundled drawing with at most

In collaboration with
Thomas van Dijk, Myroslav Kryven, Alexander Wolff (Universität Würzburg), Steven Chaplick (Maastricht and Würzburg universities), and Alexander Ravsky (Pidstryhach Institute for Applied Problems of Mechanics and Mathematics).

Company: Waterloo Maple Inc.

Duration: 2 years

Participants: Gamble and Ouragan Inria teams

Abstract:
A two-years licence and cooperation agreement was signed on April 1st, 2018
between Waterloo Maple Inc., Ontario, Canada (represented by Laurent Bernardin,
its Executive Vice President Products and Solutions) and Inria.
On the Inria side, this contract involves the teams Gamble and Ouragan (Paris), and it is coordinated by Fabrice
Rouillier (Ouragan).

F. Rouillier and Gamble are the developers of the Isotop software for the computation
of topology of curves. The transfer of a version of Isotop to Waterloo Maple Inc. should be done on the long run.

Company: GeometryFactory

Duration: permanent

Participants: Inria and GeometryFactory

Abstract:
Cgal packages developed in Gamble are commercialized by GeometryFactory.

Project title: Structures on Surfaces

Duration: 4 years + Covid'19 extension

Starting Date: April 1st, 2018

Coordinator: Monique Teillaud

Participants:

SoS is co-funded by ANR (ANR-17-CE40-0033) and FNR (INTER/ANR/16/11554412/SoS) as a PRCI (Projet de Recherche Collaborative Internationale).

The central theme of this project is the study of geometric and combinatorial structures related to surfaces and their moduli. Even though they work on common themes, there is a real gap between communities working in geometric topology and computational geometry and SoS aims to create a long-lasting bridge between them. Beyond a common interest, techniques from both ends are relevant and the potential gain in perspective from long-term collaborations is truly thrilling.

In particular, SoS aims to extend the scope of computational geometry, a field at the interface between mathematics and computer science that develops algorithms for geometric problems, to a variety of unexplored contexts. During the last two decades, research in computational geometry has gained wide impact through CGAL, the Computational Geometry Algorithms Library. In parallel, the needs for non-Euclidean geometries are arising, e.g., in geometric modeling, neuromathematics, or physics. Our goal is to develop computational geometry for some of these non-Euclidean spaces and make these developments readily available for users in academy and industry.

To reach this aim, SoS will follow an interdisciplinary approach, gathering researchers whose expertise cover a large range of mathematics, algorithms and software. A mathematical study of the objects considered will be performed, together with the design of algorithms when applicable. Algorithms will be analyzed both in theory and in practice after prototype implementations, which will be improved whenever it makes sense to target longer-term integration into CGAL.

Our main objects of study will be Delaunay triangulations and circle patterns on surfaces, polyhedral geometry, and systems of disjoint curves and graphs on surfaces.

Project website: members.loria.fr/Monique.Teillaud/collab/SoS/.

The Covid'19 crisis has forbidden travelling since mid-March.

Otfried Cheong visited Gamble for one week in March.

Monique Teillaud visited Gert Vegter's group, Bernoulli Institute for Mathematics, Computer Science and Artificial Intelligence, University of Groningen (two weeks in February).

Project title: Analyse et Simulation Probabilistes d'Algorithmes Géométriques

Duration: 4 years + Covid'19 extension

Starting date: January 1st, 2018

Coordinator: Olivier Devillers

Participants:

Abstract: The ASPAG projet is funded by ANR under number ANR-17-CE40-0017 .

The analysis and processing of geometric data has become routine in a variety of human activities ranging from computer-aided design in manufacturing to the tracking of animal trajectories in ecology or geographic information systems in GPS navigation devices. Geometric algorithms and probabilistic geometric models are crucial to the treatment of all this geometric data, yet the current available knowledge is in various ways much too limited: many models are far from matching real data, and the analyses are not always relevant in practical contexts. One of the reasons for this state of affairs is that the breadth of expertise required is spread among different scientific communities (computational geometry, analysis of algorithms and stochastic geometry) that historically had very little interaction. The Aspag project brings together experts of these communities to address the problem of geometric data. We will more specifically work on the following three interdependent directions.

(1) Dependent point sets: One of the main issues of most models is the core assumption that the data points are independent and follow the same underlying distribution. Although this may be relevant in some contexts, the independence assumption is too strong for many applications.

(2) Simulation of geometric structures: The phenomena studied in (1) involve intricate random geometric structures subject to new models or constraints. A natural first step would be to build up our understanding and identify plausible conjectures through simulation. Perhaps surprisingly, the tools for an effective simulation of such complex geometric systems still need to be developed.

(3) Understanding geometric algorithms: the analysis of algorithms is an essential step in assessing the strengths and weaknesses of algorithmic principles, and is crucial to guide the choices made when designing a complex data processing pipeline. Any analysis must strike a balance between realism and tractability; the current analyses of many geometric algorithms are notoriously unrealistic. Aside from the purely scientific objectives, one of the main goals of Aspag is to bring the communities closer in the long term. As a consequence, the funding of the project is crucial to ensure that the members of the consortium will be able to interact on a very regular basis, a necessary condition for significant progress on the above challenges.

Project website: members.loria.fr/Olivier.Devillers/aspag/.

Project title: MIN-MAX

Duration: 4 years

Starting date: 2019

Coordinator: Stéphane Sabourau (Université Paris-Est Créteil)

Participants:

Abstract: The MinMax projet is funded by ANR under number ANR-19-CE40-0014

This collaborative research project aims to bring together researchers from various areas – namely, geometry and topology, minimal surface theory and geometric analysis, and computational geometry and algorithms – to work on a precise theme around min-max constructions and waist estimates.

Project website: bezout.univ-paris-est.fr/anr-project-minmax-accepted/

All members of the team are regular reviewers for the conferences of our field, namely Symposium on Computational Geometry (SoCG), European Symposium on Algorithms (ESA), Symposium on Discrete Algorithms (SODA), International Symposium on Symbolic and Algebraic Computation (ISSAC), etc.

Monique Teillaud is a managing editor of JoCG, Journal of Computational Geometry. She was a member of the editorial board of IJCGA, International Journal of Computational Geometry and Applications, until December 31st, 2020.

All members of the team are regular reviewers for the journals of our field, namely Discrete and Computational Geometry (DCG), Journal of Computational Geometry (JoCG), International Journal on Computational Geometry and Applications (IJCGA), Journal on Symbolic Computations (JSC), SIAM Journal on Computing (SICOMP), Mathematics in Computer Science (MCS), etc.

Marc Pouget and Monique Teillaud are members of the CGAL editorial board.

Monique Teillaud was chairing the Steering Committee of Computational Geometry until June 2020.

M. Teillaud is maintaining the Computational Geometry Web Pages www.computational-geometry.org/, hosted by Inria Nancy - Grand Est. This site offers general interest information for the computational geometry community, in particular the Web proceedings of the Video Review of Computational Geometry, part of the Annual/International Symposium on Computational Geometry (SoCG).

Members of Gamble are occasionally reviewing proposals or applications for foreign research agencies (e.g., FWF, NWO, NSERC, ISF, etc.) or foreign universities.
We are not giving more details here to preserve anonymity.

Team members are involved in various committees managing the scientific life of the lab or at a national level.

G. Moroz is a member of the Mathematics Olympiads committee of the Nancy-Metz academy.

L. Dupont participated to "Day of Science", in Oct. 2020 at "Mines of Neuves-Maisons" : Virtual Reality presentation.