2025Activity report​​Project-TeamAROMATH

3.1 High order​​​‌ geometric modeling

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​‌ Computer Aided Geometric Design​​ (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.

3.2 Robust​​ algebraic-geometric computation

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 a 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​​ progress has 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 localization 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.

4.1 Geometric modeling​​​‌ for Design and Manufacturing.​

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.

4.2 Geometric​‌ modeling for Numerical Simulation​​ and Optimization

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 $\le 2$).‌​‌

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.

5.1 Latest‌ software developments

6.1 An Effective Positivstellensatz​​​‌ over the Rational Numbers​ for Finite Semialgebraic Sets​‌

Participant: Lorenzo Baldi [Max-Planck-Institut​​ für Mathematik in den​​​‌ Naturwissenschaften, Germany], Térésa​ Krick [Universidad de Buenos​‌ Aires, Argentina], Bernard​​ Mourrain.

In 21​​​‌, we study the​ problem of representing multivariate​‌ polynomials with rational coefficients,​​ which are nonnegative and​​ strictly positive on finite​​​‌ semialgebraic sets, using rational‌ sums of squares.

We‌​‌ focus on the case​​ of finite semialgebraic sets​​​‌ S defined by equality‌ constraints, generating a zero-dimensional‌​‌ ideal I, and by​​ nonnegative sign constraints. First,​​​‌ we obtain existential results.‌ We prove that a‌​‌ strictly positive polynomial f​​ with coefficients in a​​​‌ subfield K of R‌ has a representation in‌​‌ terms of weighted Sums-of-Squares​​ with coefficients in this​​​‌ field, even if the‌ ideal I is not‌​‌ radical. We generalize this​​ result to the case​​​‌ where f is nonnegative‌ on S and $(‌‌f)+(I:f)‌=1$. We‌ deduce that nonnegative polynomials‌​‌ with coefficients in K​​ can be represented in​​​‌ terms of Sum-of-Squares of‌ polynomials with coefficients in‌​‌ K, when the ideal​​ is radical. Second, we​​​‌ obtain degree bounds for‌ such Sums-of-Squares representations, which‌​‌ depend linearly on the​​ regularity of the ideal​​​‌ and the degree of‌ the defining equations, when‌​‌ they form a graded​​ basis. Finally, we analyze​​​‌ the bit complexity of‌ the Sums-of-Squares representations for‌​‌ polynomials with coefficients in​​ Q, in the case​​​‌ where the ideal is‌ radical. The bitsize bounds‌​‌ are quadratic or cubic​​ in the Bezout bound,​​​‌ and linear in the‌ regularity, generalizing and improving‌​‌ previous results obtained for​​ special zero dimensional ideals.​​​‌ As an application in‌ the context of polynomial‌​‌ optimization, we retrieve and​​ improve results on the​​​‌ finite convergence and exactness‌ of the moment/Sums-of-Squares hierarchy.‌​‌

6.2 Singularity, approximation and​​ representation

Participant: Adam Parusiński​​​‌, Laurentiu Päunescu [Univ.‌ Sidney, Australia], Guillaume‌​‌ Rond [Univ. Aix-Marseille, France]​​.

The aim of​​​‌ the paper doi:10.4064/ap241218-2-6 is‌ to review how some‌​‌ approximation results in commutative​​ algebra are being used​​​‌ to construct equisingular deformations‌ of singularities. The first‌​‌ example of such an​​ approximation result appeared in​​​‌ A. Płoski’s PhD thesis.‌

A result of Teissier‌​‌ says that the cone​​ over one of classical​​​‌ polygon examples in the‌ real projective space gives,‌​‌ by complexification, a surface​​ singularity which is not​​​‌ Whitney equisingular to a‌ singularity defined over the‌​‌ field of rational numbers​​ $\mathbb{Q}$. In doi:10.4064/ap241208-25-4​​​‌, we correct the‌ example and give a‌​‌ complete proof of Teissier’s​​ result.

6.3 Separation of​​​‌ the orbits in representations‌ of SO${}_2$ and‌​‌ O${}_2$ over $\mathrm{ℝ}$ and $ℂ$

Participant: Martin​​​‌ Jalard.

29 provides‌ a minimal set of‌​‌ invariant polynomials separating the​​ orbits for representations of​​​‌ ${\mathrm{SO}}_2$ and ${\mathrm{O‌}}_2$ over $ℝ$ and‌​‌ $ℂ$. The idea​​ is to select only​​​‌ polynomials of support of‌ size 2 for ${\mathrm{SO‌‌}}_2$ and 4 for​​ ${\mathrm{O}}_2$ . Cardinalities​​​‌ in respectively $O(‌dim{(V)‌‌}^2)$ and $\mathrm{O}(dim{\left(\mathrm{V‌}\right)}^4)$ are‌ thus obtained. These cardinalities‌​‌ are much smaller than​​ for generating sets, which​​​‌ require polynomials of arbitrary‌ large supports. Yet a‌​‌ separating set is sufficient​​ for most of the​​​‌ applications. It appears also‌ that real separating sets‌​‌ are smaller than the​​​‌ complex ones, which helps​ significatively for applications over​‌ $ℝ$. The obtained​​ separating set are used​​​‌ to stratify the real​ representations by isotropy classes.​‌

6.4 Orbit separation and​​ stratification by isotropy classes​​​‌ of piezoelectricity tensors.

Participant:​ Evelyne Hubert, Martin​‌ Jalard.

In 27​​, we explore an​​​‌ innovative method to compute​ separating invariants in a​‌ real $\mathrm{G}$-variety $\mathrm{𝒱}$.

A refinement of​​​‌ Seshadri slice Lemma enables​ us to decompose $\mathrm{𝒱‌}$ into a union of​​ stable subsets $\mathrm{G}·‌{𝒵}_1\cup ...\mathrm{G}·{𝒵}_{\mathrm{r‌}}$. This reduces the​​ problem to separating the​​​‌ orbits in the slices​ ${𝒵}_i$ for their​‌ normalizers ${\mathrm{N}}_i<\mathrm{G}$. This sequencing​​​‌ allows also to identify​ efficiently the isotropy class​‌ of any point. After​​ the presentation of three​​​‌ types of Seshadri slices​ for representations of ${\mathrm{SO‌}}_3(ℝ)$, we apply the​​​‌ method to the space​ of piezoelectricity tensors. This​‌ provides a separating set​​ of low cardinality and​​​‌ a complete stratification of​ this space by isotropy​‌ classes.

6.5 Rationality of​​ the invariant field for​​​‌ a class of representations​ of the real orthogonal​‌ groups.

Participants: Evelyne Hubert​​, Martin Jalard.​​​‌

In 28, we​ give a criterion for​‌ the invariant field of​​ a representation of ${\mathrm{SO‌}}_n(ℝ)$ to be rational. We​‌ define a length ${\mathrm{\lambda }}_n$ on representations of​​​‌ ${\mathrm{SO}}_n\left(\mathrm{ℝ}\right)$, depending on​‌ their weights. This length​​ is at most $\frac{\mathrm{n‌}}{2}$ . If a​ representation of ${\mathrm{SO}}_{\mathrm{n‌}}\left(ℝ\right)$ or​​ ${\mathrm{O}}_n\left(\mathrm{ℝ‌}\right)$ on $𝒱$ contains​ the standard representation ${\mathrm{ℝ‌}}^n$ with multiplicity greater​​ than this length, its​​​‌ invariant field is rational.​ To prove it, we​‌ construct a sequence of​​ Seshadri slices, each reducing​​​‌ the problem to a​ repesentation of a subgroup​‌ of codimension one preserving​​ the inequality between the​​​‌ multiplicity of the standard​ representation and the length.​‌ The case of length​​ ${\lambda }_n\left(\mathrm{V‌}\right)=0$ corresponds​ to the natural action​‌ of ${\mathrm{SO}}_n(ℝ)$ on matrices​​​‌ ${\mathrm{M}}_{n,\mathrm{k}}\left(ℝ\right)$,​‌ for which we construct​​ a minimal basis.

6.6​​​‌ Bigraded Castelnuovo-Mumford regularity and​ Groebner bases.

Participant: Matias​‌ Bender [Inria, Tropical],​​ Laurent Busé, Carlès​​​‌ Checa [Univ. Copenhague],​ Elias Tsigaridas [Inria, Ouragan]​‌.

In 22,​​ we study the relation​​​‌ between the bigraded Castelnuovo-Mumford​ regularity of a bihomogeneous​‌ ideal $I$ in the​​ coordinate ring of the​​​‌ product of two projective​ spaces and the bidegrees​‌ of a Groebner basis​​ of $I$ with respect​​​‌ to the degree reverse​ lexicographical monomial order in​‌ generic coordinates. For the​​ single-graded case, Bayer and​​​‌ Stillman unraveled all aspects​ of this relationship forty​‌ years ago and these​​ results led to complexity​​​‌ estimates for computations with​ Groebner bases. We build​‌ on this work to​​ introduce a bounding region​​​‌ of the bidegrees of​ minimal generators of bihomogeneous​‌ Groebner bases for $\mathrm{I}$. We also use​​ this region to certify​​​‌ the presence of some‌ minimal generators close to‌​‌ its boundary. Finally, we​​ show that, up to​​​‌ a certain shift, this‌ region is related to‌​‌ the bigraded Castelnuovo-Mumford regularity​​ of $I$.

6.7​​​‌ Solving bihomogeneous polynomial systems‌ with a zero-dimensional projection.‌​‌

Participants: Matias Bender [Inria,​​ Tropical], Laurent Busé​​​‌, Carlès Checa [Univ.‌ Copenhague], Elias Tsigaridas‌​‌ [Inria, Ouragan].

In​​ 32, we study​​​‌ bihomogeneous systems defining, non-zero‌ dimensional, biprojective varieties for‌​‌ which the projection onto​​ the first group of​​​‌ variables results in a‌ finite set of points.‌​‌ To compute (with) the​​ 0-dimensional projection and the​​​‌ corresponding quotient ring, we‌ introduce linear maps that‌​‌ greatly extend the classical​​ multiplication maps for zero-dimensional​​​‌ systems, but are not‌ those associated to the‌​‌ elimination ideal; we also​​ call them multiplication maps.​​​‌ We construct them using‌ linear algebra on the‌​‌ restriction of the ideal​​ to a carefully chosen​​​‌ bidegree or, if available,‌ from an arbitrary Gröbner‌​‌ bases. The multiplication maps​​ allow us to compute​​​‌ the elimination ideal of‌ the projection, by generalizing‌​‌ FGLM algorithm to bihomogenous,​​ non-zero dimensional, varieties. We​​​‌ also study their properties,‌ like their minimal polynomials‌​‌ and the multiplicities of​​ their eigenvalues, and show​​​‌ that we can use‌ the eigenvalues to compute‌​‌ numerical approximations of the​​ zero-dimensional projection. Finally, we​​​‌ establish a single exponential‌ complexity bound for computing‌​‌ multiplication maps and Gröbner​​ bases, that we express​​​‌ in terms of the‌ bidegrees of the generators‌​‌ of the corresponding bihomogeneous​​ ideal. This work is​​​‌ a collaboration with Matias‌ Bender, Carlès Checa and‌​‌ Elias Tsigaridas.

6.8 Construction​​ of birational trilinear volumes​​​‌ via tensor rank criteria.‌

Participant: Laurent Busé,‌​‌ Pablo Mazon [Univ. CUNEF,​​ Madrid].

In 23​​​‌, in collaboration with‌ Pablo Mazon (former PhD‌​‌ student of the team),​​ we provide effective methods​​​‌ to construct and manipulate‌ trilinear birational maps $\mathrm{\varphi ‌‌}:{\left({\mathbb{P}}^{\mathrm{1}}\right)}^3\to {\mathrm{\mathbb{P}‌}}^3$ by establishing a‌ novel connection between birationality‌​‌ and tensor rank. These​​ yield four families of​​​‌ nonlinear birational transformations between‌ 3D spaces that can‌​‌ be operated with enough​​ flexibility for applications in​​​‌ computer-aided geometric design. More‌ precisely, we describe the‌​‌ geometric constraints on the​​ defining control points of​​​‌ the map that are‌ necessary for birationality, and‌​‌ present constructions for such​​ configurations. For adequately constrained​​​‌ control points, we prove‌ that birationality is achieved‌​‌ if and only if​​ a certain $2\times ‌2\times 2$ tensor‌ has rank one. As‌​‌ a corollary, we prove​​ that the locus of​​​‌ weights that ensure birationality‌ is ${\mathbb{P}}^1\times ‌‌{\mathbb{P}}^1\times {\mathrm{\mathbb{P}}}^1$. Additionally, we​​​‌ provide formulas for the‌ inverse ${\varphi }^{-\mathrm{1‌‌}}$ as well as the​​ explicit defining equations of​​​‌ the irreducible components of‌ the base loci. Finally,‌​‌ we introduce a notion​​ of “distance to birationality”​​​‌ for trilinear rational maps,‌ and explain how to‌​‌ continuously deform birational maps.​​

6.9 An algebraic framework​​​‌ for geometrically continuous splines‌

Participants: Angelos Mantzaflaris,‌​‌ Bernard Mourrain, Nelly​​​‌ Villamizar [Swansea University, UK]​, Beihui Yuan [Swansea​‌ University, UK].

Geometrically​​ continuous splines are piecewise​​​‌ polynomial functions defined on​ a collection of patches​‌ which are stitched together​​ through transition maps. They​​​‌ are called G${}^{\mathrm{r}}$-splines if, after composition​‌ with the transition maps,​​ they are continuously differentiable​​​‌ functions to order $\mathrm{r}$ on each pair of​‌ patches with stitched boundaries.​​ This type of splines​​​‌ has been used to​ represent smooth shapes with​‌ complex topology for which​​ (parametric) spline functions on​​​‌ fixed partitions are not​ sufficient. In 31,​‌ we develop new algebraic​​ tools to analyze G​​​‌${}^r$-spline spaces. We​ define G${}^r$-domains​‌ and transition maps using​​ an algebraic approach, and​​​‌ establish an algebraic criterion​ to determine whether a​‌ piecewise function is G​​${}^r$-continuous on the​​​‌ given domain. In the​ proposed framework, we construct​‌ a chain complex whose​​ top homology is isomorphic​​​‌ to the G${}^{\mathrm{r}}$-spline space. This complex​‌ generalizes Billera-Schenck-Stillman homological complex​​ used to study parametric​​​‌ splines. Additionally, we show​ how previous constructions of​‌ G${}^r$-splines fit​​ into this new algebraic​​​‌ framework, and present an​ algorithm to construct a​‌ basis for G${}^{\mathrm{r}}$-spline spaces. We illustrate​​​‌ how our algebraic approach​ works with concrete examples,​‌ and prove a dimension​​ formula for the G​​​‌${}^r$-spline space in​ terms of invariants to​‌ the chain complex. In​​ some special cases, explicit​​​‌ dimension formulas in terms​ of the degree of​‌ splines are also given.​​

6.10 Geometric tools in​​​‌ structural bioinformatics

Participant: Ioannis​ Emiris.

In a​‌ couple of papers, with​​ two distinct groups of​​​‌ co-authors, we employ various​ tools from computational geometry​‌ to model the 3d​​ shape of molecules and​​​‌ their spatial interactions. In​ 30, we focus​‌ on the 4 human​​ Argonaute (AGO) proteins, critical​​​‌ in RNA interference and​ gene regulation. We investigated​‌ the underexplored structural relationships​​ of these paralogs through​​​‌ microsecond-scale molecular dynamics simulations.​ Our findings reveal that​‌ AGO proteins adopt similar,​​ yet unsynchronized, open-close states.​​​‌ We observed similar and​ unique local conformations, interdomain​‌ distances and intramolecular interactions.​​ By combining simulation data​​​‌ with large datasets of​ experimental structures and AlphaFold's​‌ predictions, we identified proteins​​ with genomic and proteomic​​​‌ similarities. In 24 we​ study the plant root​‌ microbiome, vital in plant​​ health, nutrient uptake, and​​​‌ environmental resilience. We present​ the MetagRoot database that​‌ integrates metagenomic, metatranscriptomic, and​​ reference genome-derived protein data​​​‌ to characterize 71,091 enriched​ protein families. These families​‌ are annotated with multiple​​ sequence alignments, hidden Markov​​​‌ models, taxonomic and functional​ classifications, and predicted 3D​‌ structures using AlphaFold2.

6.11​​ O-GEST: Overground gait events​​​‌ detector using B-Spline-based geometric​ models for marker-based and​‌ markerless analysis.

Participant: Laurent​​ Busé, Mehran Hatamzadeh​​​‌ [Univ. Gustave Eiffel, Lyon]​, Katia Turcot [Univ.​‌ Laval, Canada], Raphaël​​ Zory [Univ. Côte d'Azur]​​​‌.

Accurate gait events​ detection is imperative for​‌ reliable assessment of normal​​ and pathological gaits. However,​​​‌ this detection becomes challenging​ in the absence of​‌ force plates. In 26​​, we introduce two​​ geometric models integrated into​​​‌ an automatic algorithm (O-GEST)‌ for overground gait events‌​‌ detection using kinematic data.​​ O-GEST employs B-Spline-based geometric​​​‌ models to represent the‌ horizontal trajectory of foot‌​‌ landmarks. It leverages gait-dependent​​ thresholds, along with optimal​​​‌ coefficients to detect events‌ and compute spatiotemporal parameters‌​‌ on healthy and pathological​​ gaits. To validate the​​​‌ proposed algorithm, timing differences‌ in the detected events‌​‌ using the force plates​​ and O-GEST were calculated​​​‌ and also compared between‌ different methods on the‌​‌ gait data of 390​​ subjects. This dataset includes​​​‌ 200 healthy subjects, 100‌ subjects with unilateral hip‌​‌ osteoarthritis, 50 stroke survivors,​​ 26 individuals diagnosed with​​​‌ Parkinson’s disease, and 14‌ children with cerebral palsy.‌​‌ The validation results show​​ that O-GEST detects gait​​​‌ events with an overall‌ accuracy of 13.5 ms‌​‌ for foot-strike and 12.6​​ ms for foot-off. It​​​‌ also demonstrates significantly more‌ accurate results than the‌​‌ common deep learning-based and​​ kinematic-based methods. O-GEST offers​​​‌ several advantages, including its‌ applicability for events detection‌​‌ across various pathologies, capability​​ to handle noisy trajectories,​​​‌ and usability in the‌ absence of certain foot‌​‌ landmarks. Development of such​​ algorithms could lead to​​​‌ enhanced accuracy and reliability‌ of gait analysis in‌​‌ force-plate-less environments, especially in​​ markerless gait analysis setups.​​​‌ This work is a‌ collaboration with Mehran Hatamzadeh‌​‌ (former PhD student of​​ the team), Katia Turcot​​​‌ and Raphael Zory.

6.12‌ Efficient alternating and joint‌​‌ distance minimization methods for​​ adaptive spline surface fitting.​​​‌

Participants: Mantzaflaris Angelos,‌ Giannelli Carlotta [Univ. Florence,‌​‌ Italy], Imperatore Sofia​​ [Univ. Florence, Italy],​​​‌ Mokris Mokriš [MTU Aero‌ Engines].

In 25‌​‌ we propose a new​​ paradigm for scattered data​​​‌ fitting with adaptive spline‌ constructions based on the‌​‌ key interplay between parameterization​​ and adaptivity. Specifically, we​​​‌ introduce two novel adaptive‌ fitting schemes that combine‌​‌ moving parameterizations with adaptive​​ spline refinement, for highly​​​‌ accurate CAD models reconstruction‌ from real-world scattered point‌​‌ clouds. The first scheme​​ alternates surface fitting and​​​‌ data parameter optimization. The‌ second scheme jointly optimizes‌​‌ the parameters and the​​ surface control points. To​​​‌ combine the proposed fitting‌ methods with adaptive spline‌​‌ constructions, we present a​​ key treatment of boundary​​​‌ points. Industrial examples show‌ that updating the parameterization,‌​‌ within an adaptive spline​​ approximation framework, significantly reduces​​​‌ the number of degrees‌ of freedom needed for‌​‌ a certain accuracy, especially​​ if spline adaptivity is​​​‌ driven by suitably graded‌ hierarchical meshes. The numerical‌​‌ experiments employ Truncated Hierarchical​​ B-splines, thus exploiting the​​​‌ existing CAD integration within‌ the considered industrial setting,‌​‌ nevertheless, any adaptive spline​​ construction can be chosen.​​​‌

7.1‌​‌ Bilateral grants with industry​​

$•$ G1 Splines in​​​‌ TopSolid

Participants: Angelos Mantzaflaris‌, Michelangelo Marsala [Post-Doctorate,‌​‌ Univ. Firenze], Bernard​​ Mourrain.

The transfer​​​‌ concerns the integration of‌ new techniques for constructing‌​‌ G${}^1$ splines from​​ quadrangular meshes into the​​​‌ products of the company‌ TopSolid. The technology‌​‌ enables the real-time generation​​ of smooth spline surfaces​​​‌ or surfaces with specific‌ sharp edges from a‌​‌ quadrangular control mesh. It​​​‌ provides new features in​ TopSolid software that enable​‌ the simple and intuitive​​ creation of free forms​​​‌ with complex geometries. It​ takes into account the​‌ treatment of sharp and​​ evanescent edges. Based on​​​‌ a Julia prototype, a​ C++ package G1ACC was​‌ produced, intensive and integrated​​ in TopSolid software. The​​​‌ technique used, relying on​ the computation of dedicated​‌ masks for spline surface​​ control points, is based​​​‌ on the thesis work​ of Mr. Marsala [tel-04459277]​‌.

8.1 International research​​​‌ visitors

8.2 European initiatives

8.3 National‌ initiatives

Participants: Angelos Mantzaflaris‌​‌, Bernard Mourrain,​​ Régis Duvigneau [ACUMES].​​​‌

• Title:
  ANR PRCI: “RFF-Splines:‌ High-order Isogeometric simulation with‌​‌ geometrically continuous functions”
• Duration:​​
  2025–2028, administrative start October​​​‌ 2025
• Coordinator:
  Angelos Mantzaflaris‌
• Partners:
  University of Linz,‌​‌ Austria
• Summary:

  Our project​​ aims at the development​​​‌ of a novel framework‌ for high-order discretization of‌​‌ partial differential equations on​​ general domains. Smoothness requirements​​​‌ and superior approximation power‌ are paramount for efficient‌​‌ simulations. We focus on​​ the paradigm of isogeometric​​​‌ analysis that uses spline‌ functions for design and‌​‌ analysis on real-world (both​​ man-made and occurring in​​​‌ nature) geometries. General domains‌ pose challenges due to‌​‌ their topology, in particular​​ in the vicinity of​​​‌ so-called extraordinary vertices, which‌ are essentially artifacts created‌​‌ by the discretization (i.e.,​​ meshing) procedures. We propose​​​‌ a framework of geometrically‌ continuous splines, called Refinable‌​‌ FreeForm (RFF-) Splines, to​​ enable numerical schemes for​​​‌ topologically unrestricted design and‌ analysis.

  The novelty of‌​‌ the construction is based​​ on three main points:​​​‌ First, we will establish‌ theoretical guarantees for approximation‌​‌ power. These will​​ be derived based on​​​‌ the property of local‌ polynomial reproduction, using an‌​‌ adapted construction for spline​​ projectors on domain manifolds.​​​‌ Second, we will focus‌ on the efficient construction‌​‌ of the basis functions,​​ notably concerning the evaluation​​​‌ and the matrix assembly‌ for simulation via numerical‌​‌ integration. Third, we will​​ emphasize adaptivity and approximate​​​‌ evaluation through local refinement‌. We will employ‌​‌ the truncation mechanism to​​ reduce the support of​​​‌ the basis functions (thereby‌ increasing sparsity) and to‌​‌ preserve the partition of​​ unity property, again with​​​‌ theoretical guarantees regarding the‌ approximation power.

  The starting‌​‌ point of our work​​ is provided by a​​​‌ now classic work of‌ Prautzsch in 1997, which‌​‌ focused on the fundamental​​ property of polynomial reproduction.​​​‌ Obviously, this property is‌ essential for the derivation‌​‌ of theoretical guarantees for​​ the approximation performance. The​​​‌ project goes all the‌ way from the design‌​‌ of the construction and​​ the theory development to​​​‌ the algorithmic aspects and‌ the efficient implementation in‌​‌ C++, as well as​​ experimental evaluation in demanding​​​‌ applications that involve high‌ order partial differential equations.‌​‌

9.1 Promoting​​​‌ scientific activities

9.2​​​‌ Teaching - Supervision -​ Juries - Educational and​‌ pedagogical outreach

9.3‌​‌ Popularization

10.1 Major publications‌​‌

• 1 articleL.Lorenzo​​ Baldi and B.Bernard​​​‌ Mourrain. On the‌ Effective Putinar’s Positivstellensatz and‌​‌ Moment Approximation.Mathematical​​ Programming, Series ASeptember​​​‌ 2022HALDOI
• 2‌ articleE.Evangelos Bartzos‌​‌, I. Z.Ioannis​​ Z. Emiris and J.​​​‌Josef Schicho. On‌ the multihomogeneous Bézout bound‌​‌ on the number of​​ embeddings of minimally rigid​​​‌ graphs.Applicable Algebra‌ in Engineering, Communication and‌​‌ Computing315-62020​​, 325-357HAL
• 3​​​‌ articleE.Evangelos Bartzos‌, I. Z.Ioannis‌​‌ Z. Emiris and R.​​Raimundas Vidunas. New​​​‌ upper bounds for the‌ number of embeddings of‌​‌ minimally rigid graphs.​​Discrete and Computational Geometry​​​‌6832022,‌ 796HALDOI
• 4‌​‌ articleL.Laurent Busé​​, Y.Yairon Cid-Ruiz​​​‌ and C.Carlos D'Andrea‌. Degree and birationality‌​‌ of multi-graded rational maps​​.Proceedings of the​​​‌ London Mathematical Society121‌42020, 743-787‌​‌HALDOI
• 5 article​​L.Laurent Busé and​​​‌ A.Anna Karasoulou.‌ Resultant of an equivariant‌​‌ polynomial system with respect​​ to the symmetric group​​​‌.Journal of Symbolic‌ Computation762016,‌​‌ 142-157HALDOI
• 6​​ articleL.Ludovic Calès​​​‌, A.Apostolos Chalkis‌, I. Z.Ioannis‌​‌ Z. Emiris and V.​​Vissarion Fisikopoulos. Practical​​​‌ volume approximation of high-dimensional‌ convex bodies, applied to‌​‌ modeling portfolio dependencies and​​ financial crises.Computational​​​‌ Geometry109February 2023‌, 101916HALDOI‌​‌
• 7 articleA.Apostolos​​ Chalkis, I. Z.​​​‌Ioannis Z. Emiris and‌ V.Vissarion Fisikopoulos.‌​‌ A Practical Algorithm for​​ Volume Estimation based on​​​‌ Billiard Trajectories and Simulated‌ Annealing.ACM Journal‌​‌ of Experimental Algorithmics28​​May 2023, 1-34​​​‌HALDOI
• 8 article‌A.Apostolos Chalkis,‌​‌ I. Z.Ioannis Z.​​ Emiris, V.Vissarion​​​‌ Fisikopoulos, E.Elias‌ Tsigaridas and H.Haris‌​‌ Zafeiropoulos. Geometric algorithms​​ for sampling the flux​​​‌ space of metabolic networks‌.Journal of Computational‌​‌ Geometry1412023​​HALDOI
• 9 article​​​‌E.Emmanouil Christoforou,‌ H.Hari Leontiadou,‌​‌ F.Frank Noé,​​ J.Jannis Samios,​​​‌ I. Z.Ioannis Z.‌ Emiris and Z.Zoe‌​‌ Cournia. Investigating the​​ Bioactive Conformation of Angiotensin​​​‌ II Using Markov State‌ Modeling Revisited with Web-Scale‌​‌ Clustering.Journal of​​​‌ Chemical Theory and Computation​189September 2022​‌, 5636-5648HALDOI​​
• 10 articleI. Z.​​​‌Ioannis Z. Emiris and​ I.Ioannis Psarros.​‌ Products of Euclidean Metrics,​​ Applied to Proximity Problems​​​‌ among Curves.ACM​ Transactions on Spatial Algorithms​‌ and Systems64​​August 2020, 1-20​​​‌HALDOI
• 11 article​A. J.Alvaro Javier​‌ Fuentes Suárez and E.​​Evelyne Hubert. Scaffolding​​​‌ skeletons using spherical Voronoi​ diagrams: feasibility, regularity and​‌ symmetry.Computer-Aided Design​​102May 2018,​​​‌ 83 - 93HAL​DOI
• 12 articleA.​‌Alessandro Giust, B.​​Bert Jüttler and A.​​​‌Angelos Mantzaflaris. Local​ (T)HB-spline projectors via restricted​‌ hierarchical spline fitting.​​Computer Aided Geometric Design​​​‌80June 2020,​ 101865HALDOI
• 13​‌ articleP.Paul Görlach​​, E.Evelyne Hubert​​​‌ and T.Théo Papadopoulo​. Rational invariants of​‌ even ternary forms under​​ the orthogonal group.​​​‌Foundations of Computational Mathematics​192019, 1315-1361​‌HALDOI
• 14 article​​E.Evelyne Hubert and​​​‌ E.Erick Rodriguez Bazan​. Algorithms for fundamental​‌ invariants and equivariants: (of​​ finite groups).Mathematics​​​‌ of Computation91337​2022, 2459-2488HAL​‌DOI
• 15 articleZ.​​Zbigniew Jelonek and A.​​​‌André Galligo. Elimination​ ideals and Bezout relations​‌.Journal of Algebra​​5622020, 621-626​​​‌HALDOI
• 16 inproceedings​L.Loukas Kavouras,​‌ K.Konstantinos Tsopelas,​​ G.Giorgos Giannopoulos,​​​‌ D.Dimitris Sacharidis,​ E.Eleni Psaroudaki,​‌ N.Nikolaos Theologitis,​​ D.Dimitrios Rontogiannis,​​​‌ D.Dimitris Fotakis and​ I. Z.Ioannis Z.​‌ Emiris. Fairness Aware​​ Counterfactuals for Subgroups.​​​‌NeurIPS 2023 - 37th​ Conference on Neural Information​‌ Processing SystemsProceedings Thirty-seventh​​ Conference on Neural Information​​​‌ Processing SystemsNew-Orleans, Lousiane,​ United StatesJune 2023​‌HAL
• 17 articleA.​​Angelos Mantzaflaris, B.​​​‌Bert Jüttler, B.​Boris Khoromskij and U.​‌Ulrich Langer. Low​​ Rank Tensor Methods in​​​‌ Galerkin-based Isogeometric Analysis.​Computer Methods in Applied​‌ Mechanics and Engineering316​​April 2017, 1062-1085​​​‌HALDOI
• 18 article​B.Bernard Mourrain.​‌ Polynomial-Exponential Decomposition from Moments​​.Foundations of Computational​​​‌ Mathematics186December​ 2018, 1435--1492HAL​‌DOI
• 19 articleS.​​Simon Telen, B.​​​‌Bernard Mourrain and M.​Marc Van Barel.​‌ Solving Polynomial Systems via​​ a Stabilized Representation of​​​‌ Quotient Algebras.SIAM​ Journal on Matrix Analysis​‌ and Applications393​​October 2018, 1421--1447​​​‌HALDOI
• 20 inproceedings​K.Konstantinos Tertikas,​‌ D.Despoina Paschalidou,​​ B.Boxiao Pan,​​​‌ J. J.Jeong Joon​ Park, M. A.​‌Mikaela Angelina Uy,​​ I. Z.Ioannis Z.​​​‌ Emiris, Y.Yannis​ Avrithis and L.Leonidas​‌ Guibas. PartNeRF: Generating​​ Part-Aware Editable 3D Shapes​​​‌ without 3D Supervision.​CVPR 2023 - IEEE/CVF​‌ Conference on Computer Vision​​ and Pattern RecognitionProceedings​​​‌ of the 2023 IEEE/CVF​ Conference on Computer Vision​‌ and Pattern RecognitionVancouver,​​ CanadaIEEEMarch 2023​​​‌, 4466-4478HALDOI​

10.2 Publications of the​‌ year