Our daily life environment is increasingly interacting with digital information. An important amount of this information is of geometric nature. It concerns the representation of our environment, the analysis and understanding of “real” phenomena, the control of physical mechanisms or processes. The interaction between physical and digital worlds is two-way. Sensors are producing digital data related to measurements or observations of our environment. Digital models are also used to “act” on the physical world. Objects that we use at home, at work, to travel, such as furniture, cars, planes, ... are nowadays produced by industrial processes which are based on digital representation of shapes. CAD-CAM (Computer Aided Design – Computer Aided Manufacturing) software is used to represent the geometry of these objects and to control the manufacturing processes which create them. The construction capabilities themselves are also expanding, with the development of 3D printers and the possibility to create daily-life objects “at home” from digital models.

The impact of geometry is also important in the analysis and understanding of phenomena. The 3D conformation of a molecule explains its biological interaction with other molecules. The profile of a wing determines its aeronautic behavior, while the shape of a bulbous bow can decrease significantly the wave resistance of a ship. Understanding such a behavior or analyzing a physical phenomenon can nowadays be achieved for many problems by numerical simulation. The precise representation of the geometry and the link between the geometric models and the numerical computation tools are closely related to the quality of these simulations. This also plays an important role in optimisation loops where the numerical simulation results are used to improve the “performance” of a model.

Geometry deals with structured and efficient representations of information and with methods to treat it. Its impact in animation, games and VAMR (Virtual, Augmented and Mixed Reality) is important. It also has a growing influence in e-trade where a consumer can evaluate, test and buy a product from its digital description. Geometric data produced for instance by 3D scanners and reconstructed models are nowadays used to memorize old works in cultural or industrial domains.

Geometry is involved in many domains (manufacturing, simulation, communication, virtual world...), raising many challenging questions related to the representations of shapes, to the analysis of their properties and to the computation with these models. The stakes are multiple: the accuracy in numerical engineering, in simulation, in optimization, the quality in design and manufacturing processes, the capacity of modeling and analysis of physical problems.

The accurate description of shapes is a long standing problem in mathematics, with an important impact in many domains, inducing strong interactions between geometry and computation. Developing precise geometric modeling techniques is a critical issue in CAD-CAM. Constructing accurate models, that can be exploited in geometric applications, from digital data produced by cameras, laser scanners, observations or simulations is also a major issue in geometry processing. A main challenge is to construct models that can capture the geometry of complex shapes, using few parameters while being precise.

Our first objective is to develop methods, which are able to describe accurately and in an efficient way, objects or phenomena of geometric nature, using algebraic representations.

The approach followed in CAGD, to describe complex geometry is based on parametric representations called NURBS (Non Uniform Rational B-Spline). The models are constructed by trimming and gluing together high order patches of algebraic surfaces. These models are built from the so-called B-Spline functions that encode a piecewise algebraic function with a prescribed regularity at knots. Although these models have many advantages and have become the standard for designing nowadays CAD models, they also have important drawbacks. Among them, the difficulty to locally refine a NURBS surface and also the topological rigidity of NURBS patches that imposes to use many such patches with trims for designing complex models, with the consequence of the appearing of cracks at the seams. To overcome these difficulties, an active area of research is to look for new blending functions for the representation of CAD models. Some examples are the so-called T-Splines, LR-Spline blending functions, or hierarchical splines, that have been recently devised in order to perform efficiently local refinement. An important problem is to analyze spline spaces associated to general subdivisions, which is of particular interest in higher order Finite Element Methods. Another challenge in geometric modeling is the efficient representation and/or reconstruction of complex objects, and the description of computational domains in numerical simulation. To construct models that can represent efficiently the geometry of complex shapes, we are interested in developing modeling methods, based on alternative constructions such as skeleton-based representations. The change of representation, in particular between parametric and implicit representations, is of particular interest in geometric computations and in its applications in CAGD.

We also plan to investigate adaptive hierarchical techniques, which can locally improve the approximation of a shape or a function. They shall be exploited to transform digital data produced by cameras, laser scanners, observations or simulations into accurate and structured algebraic models.

The precise and efficient representation of shapes also leads to the problem of extracting and exploiting characteristic properties of shapes such as symmetry, which is very frequent in geometry. Reflecting the symmetry of the intended shape in the representation appears as a natural requirement for visual quality, but also as a possible source of sparsity of the representation. Recognizing, encoding and exploiting symmetry requires new paradigms of representation and further algebraic developments. Algebraic foundations for the exploitation of symmetry in the context of non linear differential and polynomial equations are addressed. The intent is to bring this expertise with symmetry to the geometric models and computations developed by aromath.

In many problems, digital data are approximated and cannot just be
used as if they were exact. In the context of geometric modeling,
polynomial equations appear naturally, as a way to describe
constraints between the unknown variables of a problem. *An
important challenge is to take into account the input error in order
to develop robust methods for solving these algebraic constraints.*
Robustness means that a small perturbation of the input should produce
a controlled variation of the output, that is forward stability, when
the input-output map is regular. In non-regular cases,
robustness also means that the output is an exact solution, or the
most coherent solution, of a problem with input
data in a given neighborhood, that is backward stability.

Our second long term objective is to develop methods to robustly and efficiently solve algebraic problems that occur in geometric modeling.

Robustness is a major issue in geometric modeling and algebraic computation. Classical methods in computer algebra, based on the paradigm of exact computation, cannot be applied directly in this context. They are not designed for stability against input perturbations. New investigations are needed to develop methods, which integrate this additional dimension of the problem. Several approaches are investigated to tackle these difficulties.

One relies on linearization of algebraic problems based on “elimination of variables” or projection into a space of smaller dimension. Resultant theory provides strong foundation for these methods, connecting the geometric properties of the solutions with explicit linear algebra on polynomial vector spaces, for families of polynomial systems (e.g., homogeneous, multi-homogeneous, sparse). Important progresses have been made in the last two decades to extend this theory to new families of problems with specific geometric properties. Additional advances have been achieved more recently to exploit the syzygies between the input equations. This approach provides matrix based representations, which are particularly powerful for approximate geometric computation on parametrized curves and surfaces. They are tuned to certain classes of problems and an important issue is to detect and analyze degeneracies and to adapt them to these cases.

A more adaptive approach involves linear algebra computation in a hierarchy of polynomial vector spaces. It produces a description of quotient algebra structures, from which the solutions of polynomial systems can be recovered. This family of methods includes Gröbner Basis, which provides general tools for solving polynomial equations. Border Basis is an alternative approach, offering numerically stable methods for solving polynomial equations with approximate coefficients. An important issue is to understand and control the numerical behavior of these methods as well as their complexity and to exploit the structure of the input system.

In order to compute “only” the (real) solutions of a polynomial system in a given domain, duality techniques can also be employed. They consist in analyzing and adding constraints on the space of linear forms which vanish on the polynomial equations. Combined with semi-definite programming techniques, they provide efficient methods to compute the real solutions of algebraic equations or to solve polynomial optimization problems. The main issues are the completness of the approach, their scalability with the degree and dimension and the certification of bounds.

Singular solutions of polynomial systems can be analyzed by computing differentials, which vanish at these points. This leads to efficient deflation techniques, which transform a singular solution of a given problem into a regular solution of the transformed problem. These local methods need to be combined with more global root localisation methods.

Subdivision methods are another type of methods which are interesting for robust geometric computation. They are based on exclusion tests which certify that no solution exists in a domain and inclusion tests, which certify the uniqueness of a solution in a domain. They have shown their strength in addressing many algebraic problems, such as isolating real roots of polynomial equations or computing the topology of algebraic curves and surfaces. The main issues in these approaches is to deal with singularities and degenerate solutions.

The main domain of applications that we consider for the methods we develop is Computer Aided Design and Manufacturing.

Computer-Aided Design (CAD) involves creating digital models defined by mathematical constructions, from geometric, functional or aesthetic considerations. Computer-aided manufacturing (CAM) uses the geometrical design data to control the tools and processes, which lead to the production of real objects from their numerical descriptions.

CAD-CAM systems provide tools for visualizing, understanding, manipulating, and editing virtual shapes. They are extensively used in many applications, including automotive, shipbuilding, aerospace industries, industrial and architectural design, prosthetics, and many more. They are also widely used to produce computer animation for special effects in movies, advertising and technical manuals, or for digital content creation. Their economic importance is enormous. Their importance in education is also growing, as they are more and more used in schools and educational purposes.

CAD-CAM has been a major driving force for research developments in geometric modeling, which leads to very large software, produced and sold by big companies, capable of assisting engineers in all the steps from design to manufacturing.

Nevertheless, many challenges still need to be addressed. Many problems remain open, related to the use of efficient shape representations, of geometric models specific to some application domains, such as in architecture, naval engineering, mechanical constructions, manufacturing ...Important questions on the robustness and the certification of geometric computation are not yet answered. The complexity of the models which are used nowadays also appeals for the development of new approaches. The manufacturing environment is also increasingly complex, with new type of machine tools including: turning, 5-axes machining and wire EDM (Electrical Discharge Machining), 3D printer. It cannot be properly used without computer assistance, which raises methodological and algorithmic questions. There is an increasing need to combine design and simulation, for analyzing the physical behavior of a model and for optimal design.

The field has deeply changed over the last decades, with the emergence of new geometric modeling tools built on dedicated packages, which are mixing different scientific areas to address specific applications. It is providing new opportunities to apply new geometric modeling methods, output from research activities.

A major bottleneck in the CAD-CAM developments is the lack of interoperability of modeling systems and simulation systems. This is strongly influenced by their development history, as they have been following different paths.

The geometric tools have evolved from supporting a limited number of tasks at separate stages in product development and manufacturing, to being essential in all phases from initial design through manufacturing.

Current Finite Element Analysis (FEA) technology was already well
established 40 years ago, when CAD-systems just started to
appear, and its success stems from using approximations of both the
geometry and the analysis model with low order finite elements (most
often of degree

There has been no requirement between CAD and numerical simulation, based on Finite Element Analysis, leading to incompatible mathematical representations in CAD and FEA. This incompatibility makes interoperability of CAD/CAM and FEA very challenging. In the general case today this challenge is addressed by expensive and time-consuming human intervention and software developments.

Improving this interaction by using adequate geometric and functional descriptions should boost the interaction between numerical analysis and geometric modeling, with important implications in shape optimization. In particular, it could provide a better feedback of numerical simulations on the geometric model in a design optimization loop, which incorporates iterative analysis steps.

The situation is evolving. In the past decade, a new paradigm has emerged to replace the traditional Finite Elements by B-Spline basis element of any polynomial degree, thus in principle enabling exact representation of all shapes that can be modeled in CAD. It has been demonstrated that the so-called isogeometric analysis approach can be far more accurate than traditional FEA.

It opens new perspectives for the interoperability between geometric modeling and numerical simulation. The development of numerical methods of high order using a precise description of the shapes raises questions on piecewise polynomial elements, on the description of computational domains and of their interfaces, on the construction of good function spaces to approximate physical solutions. All these problems involve geometric considerations and are closely related to the theory of splines and to the geometric methods we are investigating. We plan to apply our work to the development of new interactions between geometric modeling and numerical solvers.

This is a joint work with Simon Telen and Marc Van Barel, Department of Computer Science - K.U.Leuven.

This is a joint work with Gabriel Sticlaru (Ovidius University of Constanta).

The construction of optimal resultant formulae for polynomial systems is one of the main areas of research in computational algebraic geometry. However, most of the constructions are restricted to formulae for unmixed polynomial systems, that is, systems of polynomials which all have the same support. Such a condition is restrictive, since mixed systems of equations arise frequently in many problems. Nevertheless, resultant formulae for mixed polynomial systems is a very challenging problem. In , we introduce a square, Koszul-type, matrix, the determinant of which is the resultant of an arbitrary (mixed) bivariate tensor-product polynomial system. The formula generalizes the classical Sylvester matrix of two univariate polynomials, since it expresses a map of degree one, that is, the elements of the corresponding matrix are up to sign the coefficients of the input polynomials. Interestingly, the matrix expresses a primal-dual multiplication map, that is, the tensor product of a univariate multiplication map with a map expressing derivation in a dual space. In addition we prove an impossibility result which states that for tensor-product systems with more than two (affine) variables there are no universal degree-one formulae, unless the system is unmixed. Last but not least, we present applications of the new construction in the efficient computation of discriminants and mixed discriminants.

This is joint work with Elias Tsigaridas (Ouragan, Inria).

This is a joint work with E. Tsigaridas (Ouragan).

This is a joint work with Cédric Josz and Jean-Bernard Lasserre (Équipe Méthodes et Algorithmes en Commande, LAAS).

This is a joint work with Juan Elias and Roser Homs (Dep. de Matematiques i Informatica, Universitat de Barcelona).

This is a joint work with Enrique Arrondo (UCM - Universidad Complutense de Madrid, Spain), Alessandra Bernardi (Department of Mathematics, University of Trento, Italy) Pedro Macias Marques (Departamento de Matemática da Universidade de Évora, Spain).

This is a joint work with E. Tsigaridas (Ouragan), and J. Legersky (JK University, Linz, Austria).

The software was developed during Katsamaki's internship in 2018 at Sophia-Antipolis under the supervision of Bernard Mourrain. The problem has been proposed by our industrial collaborator ANSYS Hellas. The paper is based on Katsamaki's MSc thesis.

This is joint work with Vassilis Margonis (NKUA), and is based on his MSc thesis.

This is a joint work with Ludovic Calès (JRC - European Commission - Joint Research Centre, Ispra).

This is a joint work with Mingyang Zhao, Xiaohong Jia (KLMM - Key Laboratory of Mathematics Mechanization, Beijing, China), Changhe Tu (Shandong University, China), Wenping Wang (Computer Graphics Group, Department of Computer Science, Hong Kong, China).

Convolution surfaces with 1D skeletons have been limited to close-to-circular normal sections. The new formalism and method
presented in
allows for ellipsoidal normal sections. Anisotropy is prescribed on G

This is a joint work with Cédric Zanni (MFX Inria NGE).

The need to compute the intersections between a line and a high-order curve or surface arises in a large number of finite element applications. Such intersection problems are easy to formulate but hard to solve robustly. In , we introduce a non-iterative method for computing intersections by solving a matrix singular value decomposition (SVD) and an eigenvalue problem. That is, all intersection points and their parametric coordinates are determined in one-shot using only standard linear algebra techniques available in most software libraries. As a result, the introduced technique is far more robust than the widely used Newton-Raphson iteration or its variants. The maximum size of the considered matrices depends on the polynomial degree

This is joint work with Xiao Xiao and Fehmi Cirak (Cambridge, UK).

This is a joint work with Agostino Martinelli and Alexander Oliva (CHROMA, Inria Grenoble).

In isogeometric analysis (IGA) the domain of interest is usually represented by B-spline or NURBS patches, as they are present in standard CAD models. Complex domains can often be represented as a union of simple overlapping subdomains, parameterized by (tensor-product) spline patches. Numerical simulation on such overlapping multi-patch domains is a serious challenge in IGA. To obtain non-overlapping subdomains one would usually reparameterize the domain or trim some of the patches. Alternatively, one may use methods that can handle overlapping subdomains. In we propose a non-iterative, robust and efficient method defined directly on overlapping multi-patch domains. Consequently, the problem is divided into several sub-problems, which are coupled in an appropriate way. The resulting system can be solved directly in a single step. We compare the proposed method with iterative Schwarz domain decomposition approaches and observe that our method reduces the computational cost significantly, especially when handling subdomains with small overlaps. Summing up, our method significantly simplifies the domain parameterization problem, since we can represent any domain of interest as a union of overlapping patches without the need to introduce trimming curves/surfaces. The performance of the proposed method is demonstrated by several numerical experiments for the Poisson problem and linear elasticity in two and three dimensions.

This is a joint work with S. Kargaran, B. Jüttler, S. Kleiss and T. Takacs. (RICAM - Johann Radon Institute for Computational and Applied Mathematics and Institute of Applied Geometry, Linz, Austria)

This is joint work with B. Jüttler and F. Scholz. (Institute of Applied Geometry, Linz, Austria).

Many interface formulations, e.g. based on asymptotic thin interphase models or material surface theories, involve higher-order differential operators and discontinuous solution fields. In we are taking first steps towards a variationally consistent discretization framework that naturally accommodates these two challenges by synergistically combining recent developments in isogeometric analysis and cut-cell finite element methods. Its basis is the mixed variational formulation of the elastic interface problem that provides access to jumps in displacements and stresses for incorporating general interface conditions. Upon discretization with smooth splines, derivatives of arbitrary order can be consistently evaluated, while cut-cell meshes enable discontinuous solutions at potentially complex interfaces. We demonstrate via numerical tests for three specific nontrivial interfaces (two regimes of the Benveniste–Miloh classification of thin layers and the Gurtin–Murdoch material surface model) that our framework is geometrically flexible and provides optimal higher-order accuracy in the bulk and at the interface.

This is joint work with Zhilin Han, Changzheng Cheng, (HFUT - Hefei University of Technology, China), Chien-Ting Wu, S. Stoter, S. Mogilevskaya, and D. Schillinger (Department of Civil, Environmental and Geo-Engineering, University of Minnesota, USA).

Self-supporting surfaces are widely used in contemporary architecture, but their design remains a challenging problem. This paper aims to provide a heuristic strategy for the design of complex self-supporting surfaces. In our method, presented in non-uniform rational B-spline (NURBS) surfaces are used to describe the smooth geometry of the self-supporting surface. The equilibrium state of the surface is derived with membrane shell theory and Airy stresses within the surfaces are used as tunable variables for the proposed heuristic design strategy. The corresponding self-supporting shapes to the given stress states are calculated by the nonlinear isogeometric analysis (IGA) method. Our validation using analytic catenary surfaces shows that the proposed method finds the correct self-supporting shape with a convergence rate one order higher than the degree of the applied NURBS basis function. Tests on boundary conditions show that the boundary's influence propagates along the main stress directions in the surface. Various self-supporting masonry structures, including models with complex topology, are constructed using the presented method. Compared with existing methods such as thrust network analysis and dynamic relaxation, the proposed method benefits from the advantages of NURBS-based IGA, featuring smooth geometric description, good adaption to complex shapes and increased efficiency of computation.

This is joint work with Yang Xia, Ping Hu (Dalian University of Technology, China), Bert Jüttler (Institute of Applied Geometry, Linz, Austria), Hao Pan (Microsoft Research Asia, China), Wenping Wang (CSE - Department of Computer Science and Engineering, HKUST, Honk Kong, China).

This work was done jointly with F. Scholz and I. Toulopoulos (RICAM - Johann Radon Institute for Computational and Applied Mathematics, Linz, Austria).

** $\u2022$ NURBSFIX: Repairing the topology of a NURBS model in view of its approximation.**
We have a research contract with the industrial partner GeometryFactory, in collaboration with the project-team Titane (Pierre Alliez). The post-doc of Xiao Xiao is funded by this research contract together with a PEPS from the labex AMIES.

Because of their flexibility and accuracy, NURBS (Non-Uniform Rational Basis Spline) models have become a standard in the modeling community for generating and representing complex shapes. They are made of several surface patches and a collection of curves that are used for trimming. As a direct consequence of software quirks, designer errors, and representation flaws, these NURBS models have inconsistencies that introduce small gaps and overlaps between surface patches. They are mainly located on the singularity graph of a NURBS model, near the trimming curves, especially near singularities such as sharp edges or corners. Building a correct approximation of a NURBS model in the presence of inconsistencies is a challenging problem. Most of the current approaches are based on the repairing of the geometry of the surface patches. This requires an interactive process which is difficult to control and rarely completely successful. In this project, we develop another approach which consists in repairing the topology of the singularity graph within a tolerance volume. This tolerance volume will be considered as a protected region that will not receive any query of geometric computations. Based on that, three types of approximations will be treated: triangular isotropic surface meshing of NURBS models, volume approximation of multi-domains delimited by NURBS surfaces, and NURBS models approximation within a given tolerance volume.

Program: Marie Skłodowska-Curie ETN

Project acronym: ARCADES

Project title: Algebraic Representations in Computer-Aided Design for complEx Shapes

Duration: January 2016 - December 2019

Coordinator: I.Z. Emiris (NKUA, Athens, Greece, and ATHENA Research Innovation Center)

Scientist-in-charge at Inria: L. Busé

Other partners: U. Barcelona (Spain), Inria Sophia Antipolis (France), J. Kepler University, Linz (Austria), SINTEF Institute, Oslo (Norway), U. Strathclyde, Glascow (UK), Technische U. Wien (Austria), Evolute GmBH, Vienna (Austria).

Webpage: http://

Abstract: ARCADES aims at disrupting the traditional paradigm in Computer-Aided Design (CAD) by exploiting cutting-edge research in mathematics and algorithm design. Geometry is now a critical tool in a large number of key applications; somewhat surprisingly, however, several approaches of the CAD industry are outdated, and 3D geometry processing is becoming increasingly the weak link. This is alarming in sectors where CAD faces new challenges arising from fast point acquisition, big data, and mobile computing, but also in robotics, simulation, animation, fabrication and manufacturing, where CAD strives to address crucial societal and market needs. The challenge taken up by ARCADES is to invert the trend of CAD industry lagging behind mathematical breakthroughs and to build the next generation of CAD software based on strong foundations from algebraic geometry, differential geometry, scientific computing, and algorithm design. Our game-changing methods lead to real-time modelers for architectural geometry and visualisation, to isogeometric and design-through-analysis software for shape optimisation, and marine design and hydrodynamics, and to tools for motion design, robot kinematics, path planning, and control of machining tools.

Program: Marie Skłodowska-Curie ITN

Project acronym: POEMA

Project title: Polynomial Optimization, Efficiency through Moments and Algebra

Duration: January 2019 - December 2022 (48 months)

Coordinator: B. Mourrain (Aromath, Inria Sophia Antipolis)

Other partners: LAAS - CNRS, Toulouse (France), Sorbonne Université, Paris (France), Centrum Wiskunde & Informatica, Amsterdam (The Netherlands), Stichting Katholieke Universiteit Brabant, Tilburd (The Netherlands), Universität Konstanz (Germany), Università degli Studi di Firenze (Italy), University of Birmingham (United Kingdom), Friedrich Alexander University Erlangen-Nuremberg (Germany), Universitet I Tromsø (Norway), ARTELYS SAS, Paris (France).

Webpage: http://

Abstract: Non-linear optimization problems are present in many real-life applications and in scientific areas such as operations research, control engineering, physics, information processing, economy, biology, etc. However, efficient computational procedures, that can provide the guaranteed global optimum, are lacking for them. The project will develop new polynomial optimization methods, combining moment relaxation procedures with computational algebraic tools to address this type of problems. Recent advances in mathematical programming have shown that the polynomial optimization problems can be approximated by sequences of Semi-Definite Programming problems. This approach provides a powerful way to compute global solutions of non-linear optimization problems and to guarantee the quality of computational results. On the other hand, advanced algebraic algorithms to compute all the solutions of polynomial systems, with efficient implementations for exact and approximate solutions, were developed in the past twenty years. The network combines the expertise of active European teams working in these two domains to address important challenges in polynomial optimization and to show the impact of this research on practical applications.

POEMA aims to train scientists at the interplay of algebra, geometry and computer science for polynomial optimization problems and to foster scientific and technological advances, stimulating interdisciplinary and intersectoriality knowledge exchange between algebraists, geometers, computer scientists and industrial actors facing real-life optimization problems.

Program: Marie Skłodowska-Curie ETN

Project acronym: GRAPES

Project title: Learning, Processing and Optimising Shapes

Duration: December 2019 - November 2023

Coordinator: I.Z. Emiris (NKUA, Athens, and ATHENA Research Center, Greece)

Scientist-in-charge at Inria: L. Busé

Other partners: U. Barcelona (Spain), Inria Sophia-Antipolis (France), J. Kepler University, Linz (Austria), SINTEF Institute, Oslo (Norway), U. Strathclyde, Glascow (UK), RWTH Aachen (Germany), U. Svizzera Italiana (Switzerland), U. Tor Vergata (Italy), Vilnius U. (Lithuania), GeometryFactory SARL (France).

Webpage: http://

Abstract: GRAPES aims at advancing the state of the art in Mathematics, Computer-Aided Design, and Machine Learning in order to promote game changing approaches for generating, optimising, and learning 3D shapes, along with a multisectoral training for young researchers. Recent advances in the above domains have solved numerous tasks concerning multimedia and 2D data. However, automation of 3D geometry processing and analysis lags severely behind, despite their importance in science, technology and everyday life, and the well-understood underlying mathematical principles. GRAPES spans the spectrum from Computational Mathematics, Numerical Analysis, and Algorithm Design, up to Geometric Modelling, Shape Optimisation, and Deep Learning. This allows the 15 PhD candidates to follow either a theoretical or an applied track and to gain knowledge from both research and innovation through a nexus of intersectoral secondments and Network-wide workshops. Horizontally, our results lead to open-source, prototype implementations, software integrated into commercial libraries as well as open benchmark datasets. These are indispensable for dissemination and training but also to promote innovation and technology transfer. Innovation relies on the active participation of SMEs, either as a beneficiary hosting an ESR or as associate partners hosting secondments. Concrete applications include simulation and fabrication, hydrodynamics and marine design, manufacturing and 3D printing, retrieval and mining, reconstruction and visualisation, urban planning and autonomous driving.

Program: PHC Alliance

Project title: High-order methods for computational design and data-driven engineering

Duration: 01/2020–12/2021

Coordinator: Angelos Mantzaflaris

Other partners: Swansea University, UK

Abstract: The aim of this project is to develop a mathematical framework for the integration of geometric modeling and simulation using spline-based finite elements of high degree of smoothness. High-order methods are known to provide a robust and efficient methodology to tackle complex challenges in multi-physics simulations, shape optimization, and the analysis of large-scale datasets arising in data-driven engineering and design. However, the analysis and design of high-order methods is a daunting task requiring a concurrent effort from diverse fields such as applied algebraic geometry, approximation theory and splines, topological data analysis, and computational mathematics. Our strategic vision is to create a research team combining a uniquely broad research expertise in these areas by establishing a link between the team AROMATH at Inria Sophia-Antipolis and Swansea University.

Program: NSFC

Project title: “Research on theory and method of time-varying parameterization for dynamic isogeometric analysis”,

Duration: 2018-2021.

Collaboration project with Gang Xu, Hangzhou Dianzi University, China.

Gang Xu, Hangzhou Dianzi University, China, visited AROMATH team (9 - 20 Oct.) to work on Isogeometric Analysis and Geometric Modeling.

Ibrahim Adamou, Univ. Dan Dicko Dankoulodo de Maradi, Niger, visited B. Mourrain (28 Oct. - 21 Dec.) to work on medial axes of curve arcs.

Martin Jalard (L3, Ecole normale supérieure de Rennes)
for his *introduction to research* internship
explored during 6 weeks (May 13th to June 21st)
the application of Norton's lemma to
the computation of isotypic decompositions.

Evelyne Hubert was awarded a Simons fellowship within the program *Geometry, compatibility and structure preservation in computational differential equations*, from July to December 2019, at the Isaac Newton Institute in Cambridge (UK).

For the month of April, Evelyne Hubert was a guest professor a the University of the Arctic for *Pure Mathematics in Norway*.

Angelos Mantzaflaris visited in April the Computational Foundry, Swansea University, UK in the frame of the College of Science International Visitor Scheme.

Laurent Busé organized a CIMPA school at Joao Pessoa, Brazil, November 4-13, on the topic "Syzygies: from theory to applications". Six courses delivered by international experts of this topic were scheduled during this school that hosted about 40 international PhD students and young researchers. For more details, see http://

Laurent Busé co-organized with C. D’Andrea and Y. Cid Ruiz a mini-symposium on «Syzygies and Applications to Geometry » at the SIAM Conference on Applied Algebraic Geometry, Bern, Switzerland, July 9-13, 2019.

Angelos Mantzaflaris and Bernard Mourrain co-organized with E. Tsigaridas a mini-symposium on "Symbolic-numeric methods for non-linear equations: Algorithms and applications" at the SIAM Conference on Applied Algebraic Geometry, Bern, Switzerland, July 9-13, 2019.

Laurent Busé was a member of the international program committee of the 2019 Symposium on Physical and Solid Modeling (SPM), Vancouver, Canada, June 17-19. He was also a member of the scientific committee fro the conference "Ideals, Varieties, Applications" celebrating the influence of David Cox, Amherst, USA, June 10-14.

Angelos Mantzaflaris served in the PC of the 2019 Solid and Physical Modeling (SPM), https://

Laurent Busé reviewed for *Symposium on Physical and Solid Modeling conference*, *SIGGRAPH conference*, *ISSAC conference* and *MEGA conference*.

Ioannis Emiris reviewed for
the *Symposium of Computational Geometry*,
*ICALP: Intern. Conf. Automata, Logic & Programming* and
*MEGA: Méthodes Effectives en Géométrie Algébrique*.

Angelos Mantzaflaris reviewed for the * Symposium on Physical and Solid Modeling*.

Bernard Mourrain reviewed for *Symposium on Physical and Solid Modeling conference*, *ISSAC conference* and *MEGA conference*.

Ioannis Emiris is editorial board member of *J. Symbolic Computation (Elsevier)* and *Mathematics for Computer Science (Springer)*.

Evelyne Hubert is on the editorial board of Foundation of Computational Mathematics (since 2017) and the Journal of Symbolic Computation (since 2007).

Bernard Mourrain is associate editor of the Journal of Symbolic Computation (since 2007) and of the SIAM Journal on Applied Algebra and Geometry (since 2016).

Laurent Busé reviewed for *Graphical Models journal*, *Advances in Applied Mathematics journal*, *SIAM SIAGA journal*, *International Mathematics Research Notices*, *Journal of Algebra*, *Applicable Algebra in Engineering, Communication and Computing journal*, *Computer-Aided Design journal*, *Computer Aided Geometric Design
journal*, *Journal of Computer Science and Technology*, *AMS Mathematics of Computations journal*, *Israel Journal of Mathematics* and *Journal of Symbolic Computation*.

Evelyne Hubert reviewed for Journal für die reine und angewandte Mathematik, Proceedings of the American Mathematical Society, Advances in Applied Mathematics, Foundation of Computational Mathematics, SIAM Journal on Applied Algebra and Geometry, Journal of Symbolic Computation, and Computer Aided Geometric Design.

Bernard Mourrain reviewed for
*Annales Henri Lebesgue*,
*Collectanea Mathematica journal*
*Computer-Aided Design journal*,
*Computer Aided Geometric Design journal*,
*Journal of Global Optimization*,
*Journal of Symbolic Computation*,
and *Mathematics of Computations journal*,

Angelos Mantzaflaris reviewed for Computer Methods in Applied Mechanics and Engineering, Computer Aided Geometric Design, Computers & Mathematics with Applications, and Journal of Symbolic Computation.

Ahmed Blidia was an invited speaker in the minisymposium *Multivariate spline approximation and algebraic geometry*
at SIAM Applied Algebraic Geometry meeting, Bern, Switzerland in June 2019.

Laurent Busé was invited to give two lectures at the "27th National School on Algebra: Graded modules over polynomial rings with applications to free divisors", Bucarest, Romania, May 19-25; he was invited to give a plenary talk at the conference "Ideals, Varieties, Applications", in honor of David Cox, Amherst, USA, June 10-14; he gave an invited course during the school "TIME2019: Curves and Surfaces, a History of Shapes", Levico Terme, Italy, Septembre 2-6.

Ioannis Emiris was an invited speaker at KAUST, Visual computing center, Saudi Arabia in February 2019, and SIAM Applied Algebraic Geometry meeting, Bern, Switzerland in June 2019.

Alvaro Fuentes Suarez gave a seminar talk in the MFX team at Inria NGE in February 2019.

Evelyne Hubert presented a series of lectures at the national conference
*Equations Fonctionnelles et Interactions* (Anglet).
She was invited to give talks at the
* ${A}^{3}$ - Arctic Applied Algebra*
conference (Tromsø, Norway, April 2019)
and the conference

Angelos Mantzaflaris gave an invited talk at the 4th Workshop of the ERC project CHANGE, Centro Congressi dell'Annunziata, Sestri Levante, Italy (November 2019) and at the Schloss Dagstuhl – Leibniz Center for Informatics seminar on Interactive Design and Simulation, Germany (December 2019). He also delivered a mini-symposium presentation at the annual conference on Isogeometric Analysis (IGA 2019, Munich, Germany, September 2019).

Both Angelos Mantzaflaris and Bernard Mourrain were invited at the meeting on ”Isogeometric Splines: Theory and Applications“ of the Banff International Research Station for Mathematical Innovation and Discovery (BIRS, February 2019), at the Oberwolfach Mini-Workshop on Mathematical Foundations of Isogeometric Analysis (July 2019), and at the Algebraic Spline Geometry Meeting in Swansea, UK (August 2019).

Bernard Mourrain was invited
to give a talk at the conference * ${A}^{3}$ - Arctic Applied Algebra*
(Tromsø, Norway, April 1-4);
an invited speaker of the conference MEGA (Madrid, Spain, June),
to talk at the minisymposium

He was also invited to give two courses on splines at the University of Montpellier, (October 23-24, 2019).

Erick Rodriguez Bazan was an invited speaker in the minisymposium
*Symmetry in algorithmic questions of real algebraic geometry *
at SIAM Applied Algebraic Geometry meeting, Bern, Switzerland in June 2019.

Fatmanur Yildirim was an invited speaker in the minisymposium
*Syzygies and applications to geometry*
at SIAM Applied Algebraic Geometry meeting, Bern, Switzerland in June 2019.

Ioannis Emiris is member of the Scientific Board of Hellenic Foundation for Research & Innovation (http://

Bernard Mourain has been elected vice chair of the SIAM Algebraic Geometry group.

Evelyne Hubert was on the hiring committee for Junior Research Scientists at Inria NGE.

Bernard Mourrain is member of the BCEP (Bureau du Comité des Equipes Projet) of the center Inria- Sophia Antipolis.

Evelyne Hubert was a member of Inria Evaluation committee (until June 2019).

Laurent Busé is a member of the admnistrative and scientific committee of the labex AMIES. He is also member of the CDT at Inria Sophia-Antipolis and the CPRH of the Mathematics Laboratory Jean-Alexandre Dieudonné of the University of Nice.

Licence : Ioannis Emiris, Discrete Math, 52 h (L1), NKU Athens

Licence : Ioannis Emiris, Software development, 26 h (L3), NKU Athens

Master : Ioannis Emiris, Geometric data science, 52 h (M2), NKU Athens

Master : Ioannis Emiris, Structural bioinformatics, 39 h (M2), NKU Athens

Master : Laurent Busé, Geometric Modeling, 18h (M2), engineer school of the University of Nice Sophia-Antipolis (EPU).

Undergraduate: Angelos Mantzaflaris ”Fondements mathématiques 2 (L1 - TD Partie analyse), University of Côte d'Azur, spring semester 2019.

PhD in progress: Lorenzo Baldi, Structure of moment problems and applications to polynomial optimization. POEMA Marie Skłodowska-Curie ITN, started in October 2019, supervised by Bernard Mourrain.

PhD in progress: Ahmed Blidia, New geometric models for the design and computation of complex shapes. ARCADES Marie Skłodowska-Curie ITN, started in September 2016, supervised by Bernard Mourrain.

PhD in progress: Rima Khouja, Tensor decomposition, best approximations, algorithms and applications. Cotutelle Univ. Liban, started in November 2018, cosupervised by Houssam Khalil and Bernard Mourrain.

PhD: Ioannis Psarros, Proximity problems for high-dimensional data. Greek scholarships foundation. Defended in Jun. 2019, NKUA, supervised by Ioannis Emiris.

PhD in progress: Evangelos Bartzos, Algebraic elimination and Distance graphs. ARCADES Marie Skłodowska-Curie ITN, started in June 2016, NKUA, supervised by Ioannis Emiris.

PhD in progress: Clément Laroche, Algebraic representations of geometric objects. ARCADES Marie Skłodowska-Curie ITN, started in Nov. 2016, NKUA, supervised by Ioannis Emiris.

PhD in progress: Apostolos Chalkis, Sampling in high-dimensional convex regions, Google Summer of Code, started in June 2018, NKUA, supervised by Ioannis Emiris.

PhD in progress: Emmanouil Christoforou, Geometric approximation algorithms for clustering, Structural Bioinformatics national infrastructure "Inspire", started in Jan. 2018, NKUA, supervised by Ioannis Emiris.

PhD in progress: Tobias Metzlaff. Multivariate orthogonal polynomials and applications to global optimization. POEMA Marie Skłodowska-Curie ITN, started in December 2019, supervised by Evelyne Hubert.

PhD in progress: Erick Rodriguez Bazan, Symmetry in Algebraic Computations. Inria. Started in November 2017, supervised by Evelyne Hubert.

PhD : Alvaro Fuentes Suarez defended September 2019 . Modeling shapes with skeletons: scaffolds & anisotropic convolution. ARCADES Marie Skłodowska-Curie ITN, started October 2016 and supervised by Evelyne Hubert.

PhD in progress: Thomas Laporte, Towards a 4D model of the respiratory system. Fellowship from ED SFA/UCA. Started on October 2019, co-supervised by Benjamin Mouroy (UCA) and Angelos Mantzaflaris.

PhD in progress: Riccardo Di Dio, Building a diagnosis tool to detect broncho-constrictions, BoostUrCAreer Marie Skłodowska-Curie COFUND fellowship. Started on November 2019, co-supervised by Benjamin Mouroy (UCA) and Angelos Mantzaflaris.

Bernard Mourrain was a member of the PhD committee of Matias R. Bender *Algorithms for sparse polynomial systems : Gröbner basis and resultants*, Sorbonne Université, Paris, June 3rd;
a reviewer and member of the committee of the HDR of Frédéric Holweck entitled *On the projective geometry of entanglement and contextuality*, University Bourgogne Franche-Comté, Belfort, France, September 11th.

Evelyne Hubert was a reviewer for the Habilitation thesis of Georg Regensbuger, Johannes Kepler University (Austria): *Algebraic and algorithmic Approached to Analysis: Integro-differential equations, positive steady states, and wavelets*

Laurent Busé was a reviewer and member of the committee the PhD thesis of Navid Nemati, *Syzygies: Algebra, Combinatorics and
Geometry*, Sorbonne Université, Paris, May 28;
he was also a
member of the PhD committee of Matias R. Bender, *Algorithms for sparse polynomial systems : Gröbner basis and resultants*, Sorbonne Université, Paris, June 3rd.

Ioannis Emiris was an invited speaker at “From Open Access to Science", Athens, May 2019. and “30 years celebration of ATHENA Research Center", Athens, November 2019.