2023Activity reportProjectTeamGEOMERIX
RNSR: 202224337M Research center Inria Saclay Centre at Institut Polytechnique de Paris
 In partnership with:CNRS, Institut Polytechnique de Paris
 Team name: Geometrydriven Numerics
 In collaboration with:Laboratoire d'informatique de l'école polytechnique (LIX)
 Domain:Perception, Cognition and Interaction
 Theme:Interaction and visualization
Keywords
Computer Science and Digital Science
 A3.4.1. Supervised learning
 A3.4.2. Unsupervised learning
 A3.4.4. Optimization and learning
 A3.4.6. Neural networks
 A5.5. Computer graphics
 A5.5.1. Geometrical modeling
 A5.5.4. Animation
 A6.1.4. Multiscale modeling
 A6.1.5. Multiphysics modeling
 A6.2.5. Numerical Linear Algebra
 A6.2.6. Optimization
 A6.2.8. Computational geometry and meshes
 A6.5.1. Solid mechanics
 A6.5.2. Fluid mechanics
 A8.3. Geometry, Topology
 A8.7. Graph theory
 A8.12. Optimal transport
 A9.2. Machine learning
Other Research Topics and Application Domains
 B9.2.2. Cinema, Television
 B9.2.3. Video games
 B9.5.1. Computer science
 B9.5.2. Mathematics
 B9.5.3. Physics
 B9.5.5. Mechanics
 B9.5.6. Data science
1 Team members, visitors, external collaborators
Research Scientists
 Steve Oudot [Team leader, INRIA, Senior Researcher, HDR]
 Jiong Chen [INRIA, Starting Research Position, from Nov 2023]
 Mathieu Desbrun [INRIA, Advanced Research Position]
 Pooran Memari [CNRS, Researcher]
Faculty Member
 Maksims Ovsjanikovs [Ecole Polytechnique, Professor]
PostDoctoral Fellows
 Maysam Behmanesh [Ecole Polytechnique]
 Anouar Abdeldjaoued Ferfache [Inria, PostDoctoral Fellow, from Apr 2023]
 Roman Klokov [Ecole Polytechnique]
 Wei Li [Inria, PostDoctoral Fellow, until Jan 2023]
 Vincent Mallet [Ecole Polytechnique]
PhD Students
 Souhaib Attaiki [Ecole Polytechnique, from Oct 2023]
 Elisa Ballantoni [Telecom Paris, from Oct 2023]
 Francesco Bellantoni [Telecom Paris, from Oct 2023]
 Theo Braune [Ecole Polytechnique]
 Lucas Brifault [Dassault Systèmes]
 André Freitas [Telecom Paris, from Oct 2023]
 Souhail Hadgi [Ecole Polytechnique]
 Vadim Lebovici [ENS Paris, until Aug 2023]
 Jingyi Li [Ecole Polytechnique, from Nov 2023]
 Robin Magnet [Ecole Polytechnique]
 Nissim Maruani [3IA Côte d'Azur]
 Julie Mordacq [Ministère des Armées]
 Tim Scheller [Ecole Polytechnique, from Oct 2023]
 Ramana Sundararaman [Ecole Polytechnique]
 Jiayi Wei [Ecole Polytechnique, until Sep 2023]
 Jiayi Wei [Inria, from Oct 2023]
Interns and Apprentices
 Ethan André [ENS Paris, Intern, from May 2023 until Oct 2023]
 Diego Gomez [Ecole Polytechnique, from May 2023 until Jul 2023]
 Jingyi Li [Inria, Intern, from Mar 2023 until Sep 2023]
 Siddharth Setlur [Inria, Intern, until Jan 2023]
 Thomas Wimmer [FX Conseil, Intern, from Apr 2023 until Sep 2023]
Administrative Assistant
 Michael Barbosa [Inria]
Visiting Scientist
 Shreyas Samaga [Purdue University, from May 2023 until May 2023]
2 Overall objectives
Historical context. Geometry has been a unifying formalism for science: predictive models of the world around us have often been derived using geometric notions which formalize observable symmetries and experimental invariants. Tools such as differential geometry and tensor calculus quickly became invaluable in describing the complexity of natural phenomena and mechanical systems through concise equations, condensing local and global properties into simple relations between measurable quantities. Today, geometry (be it Euclidean or not) is at the core of many current physical theories: general relativity, electromagnetism (E&M), gauge theory, quantum mechanics, as well as solid and fluid mechanics, all have strong underlying structures that are best described and elucidated through geometric notions like differential forms, curvatures, vector bundles, connections, and covariant derivative. Geometry also creeps up in unexpected fields such as number theory and functional analysis, offering new insights and even breakthroughs, e.g., the use of algebraic geometry to address Fermat's last theorem.
Geometry in Digital Sciences. In sharp contrast, the role of geometry was mostly ignored at the inception of computer science. Yet, it has now become clear that digital sciences are imbued with an overwhelming amount of fundamentally geometric and topological concepts. Some are rather obvious, when dealing with the modeling of Euclidean shapes in computer graphics or the analysis of images in computer vision for instance; some are more subtle, such as the “manifold hypothesis” underlying a number of supervised or unsupervised learning techniques; and some are only nascent, such as the fields of Information Geometry (basically, the geometry used to understand probability distributions), Geometric Statistics (new statistical methodology for nonEuclidean entities), and Topological Data Analysis (where algebraic topology is used as a tool to enhance data analysis pipelines). In fact, even the discretization of physical theories needed to offer fast numerical simulation has brought geometry back to the forefront after it was understood that the loss of numerical fidelity in standard numerical methods is due to a fundamental failure to preserve geometric or topological structures of the underlying continuous models: partial differential equations (PDEs) modeling our physical world are typically encoding invariants and structures that are independent of the choice of coordinates used to express the equations and the tensors involved in them; but invariance to the choice of basis is often lost during discretization, as numerical approximations will in general not capture, let alone preserve, the key geometric structures that exist in the continuous case. Seeing these numerical issues through the lens of geometry is thus not just of academic interest: failure to maintain geometric invariants has serious consequences for the accuracy and stability of solutions.
Rationale. Given the unusual reach of geometry and its rich literature, there is an opportunity to assemble a team of experts in geometry and its vernacular, to help broadly impact digital science and technology. We thus propose the creation of a new projectteam whose core scientific mission is to use geometry as a bedrock for the development of numerical tools and algorithms: we wish to exploit the properties of infinitedimensional and finitedimensional spaces that are related with distance, shape, size, and relative position, and bringing them to bear on computational discretizations and algorithms for analysis, processing, and simulation. Adhering to geometric structures and invariants as a guiding principle for computations is a rich source of both theoretical and practical challenges, allowing to combine concepts and results from different areas of geometry broadly construed to produce new computational tools with solid mathematical foundations. While our team will be very focused in terms of the mathematical foundations and tools upon which it builds, it will also be very broad in terms of applications given the pervasiveness of geometry in sciences and technology. This makes for an unusual, yet powerful scientific setup that will facilitate interdisciplinary projects through the common use of geometric foundations and their specialized terminology. It will also allow us to contribute sporadically to pure and computational mathematics when appropriate in order to push our scientific mission forward.
Positioning. We see GeomeriX as first and foremost Inria Saclay’s graphics team, but with wider objectives afforded by the broad relevance of geometry. It is worth noting that graphics has evolved to the point where it often intersects with applied mathematics, machine learning, vision, and computational science in some of its efforts, and GeomeriX intends to continue this trend.
Objectives. Our projectteam's overall scientific objective is to contribute, through a geometric perspective, both foundational and practical methods for geometric data processing. In particular, we seek the development of predictive computational tools by drawing from the many facets of geometry and topology: whether it be discrete geometry, basic differential geometry or exterior calculus, symplectic geometry, persistent homology or sheaf theory, optimal transport, Riemannian or conformal geometry, these topics of geometry inform and guide both our discretizations and algorithmic designs towards computing. Note that we do not plan to merely adapt and exploit geometric concepts and understanding for numerical purposes, as our focus on digital data may even result in contributions to these mathematical fields, extending the current body of knowledge. While we intentionally leave the range of our mathematical foundations open so as not to restrict our potential teamwide explorations, we concentrate our research on four concrete themes, which we believe can be most significantly impacted by a geometric approach to developing new numerical tools:
 Euclidean shape processing: from computer graphics to geometry processing and vision, the analysis and manipulation of lowdimensional shapes (2D and 3D) is an important endeavor with applications covering a wide range of areas from entertainment and classical computeraided design, to reverse engineering and biomedical engineering. Our projectteam intends to lead efforts in this competitive field, with key contributions in shape matching, geometric analysis, and discrete calculus on meshes.
 Simulation: traditional finiteelement treatments of various physical models have had tremendous success. Recently, a number of geometric integrators have upended the field, either through structurepreserving integration which offers improved statistical predictability by respecting the geometric properties of the exact flow of the differential equations, or through novel discretizations of the state space. We intend to continue introducing novel integration methods for increasingly complex multiphysics systems, as well as exploiting the use of learning methods to accelerate simulation.
 Dynamical systems: we intend to leverage the geometric nature of dynamical systems to investigate and promote highdimensional data analysis for dynamics: the study of dynamical systems from a limited number of observations of the state of a given system (for example, time series or a sparse set of trajectories) offers a unique opportunity to develop scalable computational tools to detect or characterize unusual features and coherent structures. Meanwhile, the study of dynamical systems from a combinatorial point of view opens up the possibility of characterizing their invariant sets and assessing their stability.
 Data science: finally, we are intent on exploring the underlying role of geometry in machine learning and statistical analysis. This role has been put forward in the recent years, with the emergence of approaches such as geometric deep learning or topological data analysis, whose aim is to leverage the underlying geometry or topology of the data to enhance the performance, robustness, or explainability of the methods used for their analysis. We will pursue investigations toward this goal, concentrating our efforts on topics related to explainable feature design, geometric feature learning, geometrydriven learning, and geometry for categorical and mixed data types.
Evidently, our research efforts may at times lie across multiple of these themes given our multidisciplinary objectives, and it is our hope that we will all eventually participate in the four themes.
3 Research program
Below we introduce the details of our four research themes, in four separate subsections. In each subsection, we first present the scientific focus and research objectives of the corresponding theme, then we detail the research topics we intend to address and how we plan to leverage topology and geometry for each one of them. For each theme, we list the most likely contributors, and organize the various subtopics within each theme from short to longterm goals, based on our current expectations and focus.
3.1 Geometry for Euclidean shape processing
Euclidean space is the default setting of classical geometry in two or three dimensions. Shapes in 3D space are of particular interest as they represent the typical objects we interact with. Geometry processing is an area of research focusing on these lowdimensional shapes in Euclidean space, with the goal to design algorithms, data structures, as well as analysis tools for their digital acquisition, reconstruction, analysis, manipulation, synthesis, classification, transmission, and animation. Digital shapes are typically discretized through either point clouds, triangle meshes, or polygonal meshes for surfaces, and through tetrahedron or polytopal meshes for volumes. Analyzing and manipulating these digital representations already involve fundamental difficulties in terms of efficiency, scalability, and robustness to arbitrary sampling, for which computational geometry and computer graphics have generated a number of key algorithms. Simple surface meshes in 3D also offer a simple context in which to define discrete notions of basic topological properties (quantities preserved through arbitrary stretching, such as Euler characteristic, genus, Betti numbers, etc) and relevant geometric properties (normal, curvatures, covariant derivatives, parallel transport, etc). Yet the digital counterpart of the lowdimensional case of Euclidean geometry is far from being settled or complete: it remains obviously relevant in a number of scientific fields on which we plan to focus. A few research directions of particular interest are described below.
Operatorbased methods for shape analysis
We plan to develop novel approaches for representing and manipulating geometric concepts as linear functional operators. Specifically we will focus on tools for shape matching, design and analysis of differential quantities such as vector fields or cross fields, shape deformation and shape comparison, where functional approaches have recently been shown to provide a natural and discretizationagnostic representation 110, 43, 44, 120. This “functional” point of view is classical in many scientific areas, including dynamical systems (where the pullback with respect to a map is closely related to the Koopman or composition operator, allowing the study ergodicity or mixing property of nonlinear maps through the spectral properties of a linear operator), differential geometry (where vector fields are often defined by their action on realvalued functions) and representation theory among others. However, it has only recently been adopted in geometry processing with tremendous and constantly growing potential in both axiomatic or even learningbased approaches 99, 88, 71. We will continue developing efficient and robust algorithms by considering shapes as functional spaces and by representing various geometric operations as linear operators acting on appropriate realvalued functions. In addition to the efficiency and robustness of methods obtained by considering this linear operator point of view of geometry processing and dynamical systems, another very significant advantage of these techniques is that they allow to express many different geometric operations in a common language. This means, for example, that it makes it easy to define the pushforward of a vector field with respect to a map by simply considering a composition of appropriate discrete operators. Despite the significant recent success of tools within this area, especially related to the functional map framework 111, there does not exist a unified coherent theoretical framework in which different geometric concepts can be represented and manipulated via their functional equivalents. Our main longterm goal therefore would be to establish a novel field within geometry processing by creating both a computational framework and a coherent theoretical formalism in which all of the different basic geometric operations can be expressed, and thus in which different concepts can “communicate” with one another. We believe that such a formalism and associated computational tools, already quite well developed, will not only greatly extend the scope of applicability of many existing geometry processing pipelines, but will also help expand this language to novel concepts, and ultimately help pave the way towards representationagnostic geometric data manipulation.
Discrete metrics and applications.
While threedimensional shapes are often encoded via their Euclidean embedding, numerous research efforts have focused on studying and discretizing their intrinsic metric. Regge calculus 118, an early approach to numerical relativity without coordinates, proposed the use of edge lengths to encode a piecewiseEuclidean metric per simplex, from which the Riemann curvature tensor can be easily computed to derive local areas or curvatures. This early work led to a series of alternative metric representations: tip angles, for instance, are known to encode the intrinsic geometry of a triangle mesh up to a scaling, while local measurements (an angle 119 or a length crossratio 95 per edge) later formed the basis of circle patterns 47, 93 as well as conformal representations 125; the discrete LaplaceBeltrami cotan formula 114 also determines the edge lengths of a triangle mesh (and thus its discrete metric) up to a global scaling 137. More recently, generalized notions of metrics were proposed; for instance, 85 presented a characterization of an augmented discrete metric resulting from the orthogonal primaldual structure of weighted triangulations. Common to many of these various metric characterizations is the existence of convex energies which allow to efficiently compute these metrics from various boundary conditions. We intend to investigate the discrete treatment of metric for lowdimensional manifolds as a counterpart to the discretization of antisymmetric tensors (differential forms), which is far less studied — and a discrete theory unifying symmetric and antisymmetric tensors remains elusive despite recent advances 84. Moreover, the metric of a surface is known in the continuous realm to induce Hodge stars and a canonical torsionfree LeviCivita connection (or parallel transport), but this picture is far less clear for discrete manifolds, even if the construction of arbitraryorder discrete Hodge stars and metric connections are well understood by now. A few research directions on generalized metrics seem particularly interesting due to their likelihood of resulting in novel algorithmic and computational frameworks:
 Metricdependent meshing: Given a set of metricbased operators, optimized mesh structures can be designed to offer optimal accuracy akin to Hodgestar mesh optimization for the augmented weighted metric proposed in 107. Another interesting research question is the existence and construction of intrinsic Delaunay triangulation, the most common discrete shape representation, with respect to a particular metric 48.
 Metricaware sampling: Metricdependent descriptors such as the pair correlation function are particularly efficient in characterizing statistical properties of point distributions for texture synthesis 72. Extending this framework to arbitrary nonflat domains through MultiDimensional Scaling (MDS) seem particularly promising.
 Shape characterization: Highly convoluted embeddings like the cortical surface of the brain and its functional connectivity graph are naturally hyperbolic in nature 53. However, investigating a link between cortical folding and the volumetric fiber bundle structure from a pure geometric viewpoint through a hyperbolic metric characterization has surprisingly not be done in brain analysis, despite striking visual similarities between brain folding and geometric realizations of the hyperbolic plane (see 130 and Taimiņa's crochet model). We are hoping that this intrinsic metric characterization can be investigated through recent discrete hyperbolic parametrization tools 80, which may also lead to other shape classification techniques in more general contexts.
 Piecewiselinear maps: We also wish to study the classification of the deformation of a triangle mesh through its induced metric change in the embedding space. Developing an approach to decompose such a diffeomorphic piecewiselinear map into canonical geometric transformations through either linear algebra or convex minimization could offer new discrete equivalences for conformal, equiareal, and curvaturepreserving maps between triangulations, with direct applications to mesh parameterization and more general processing of discrete meshes.
 Geodesic abstractions: curvenetwork representations 83 based on a few geodesics to describe a shape provide a compact encoding of surfaces. While it is increasingly useful for artistic depictions, we also want to study its relevance as a compact compression scheme from which the shape and its metric can be derived with controllable precision.

Metricdependent cage: Finally, we also want to understand how to define optimized metricdependent cages for intuitive & expressive deformation and animation of complex shapes 128, and how these cages can be understood as polygonal or polyhedral cells to locally simplify a simplicial complex.
Discrete differential and tensor calculus.
When working on lowdimensional spaces, the use of meshes (as opposed to just point clouds) pays dividends as it allows for the development of discrete versions of Exterior Calculus (see DEC 67 or FEEC 41), where $k$dimensional integrals can be directly evaluated in $k$cells, and differentiation can formally achieved through the boundary operator: the concept of chains and cochains from algebraic topology forms the basis of a discrete analog of Cartan's exterior calculus of differential forms, providing crucial numerical tools such as a discrete de Rham cohomology and a discrete HelmholtzHodge decomposition that precisely mimick their continuous counterparts. Moreover, finite elements of arbitrary order can be associated with these discrete forms through subdivision 82 to provide a powerful Isogeometric Analysis (IGA). Recent developments 100, 81 have offered also a discrete approach to tangent vector fields. While DEC encodes vector fields as 1forms, processing tangent vectors and, more generally, directional fields sampled at vertices of discrete surfaces requires the development of discrete (metric) connections 64, 100 (which can be seen as discrete equivalent to the Christoffel symbols) to handle the nonlinearity of nonflat domains. From these connections can be derived the usual continuous notions of covariant derivatives or Killing operator, and these discrete operators demonstrate the same intimate link between geometry and topology as exemplified by the hairy ball theorem (Hopf index theorem). While these operators apply equally well on discrete threemanifolds, much remains to do: properly defining the notion of curvature matrixvalued 2form or torsion vectorvalued 2form in 3D and checking that these definitions provide consistent Bianchi identities (i.e., there exists an exterior covariant derivative satisfying fundamental geometric and topological properties) is an exciting research direction. Not only will it allow to deal with the line singularities in hexahedral meshing robustly, but it will also provide a Bochner Laplacian (also called the vector Laplacian) in 3D devoid of the type of spurious modes that discrete Laplacians over flat domains can introduce if one does not enforce a proper discrete deRham complex. Such a tensor calculus for threemanifolds may allow us to explore possible applications in the context of general relativity in the longer term. Finally, the design of simplicial or cell meshes that guarantee accurate computations while approximating a given domain well remains an important endeavor for practical applications.
3.2 Geometry for simulation
Mathematical models of the evolution in time of mechanical systems generally involve systems of differential equations. Simulating a physical system consists in figuring out how to move the system forward in time from a set of initial conditions, allowing the computation of an actual trajectory through classical methods such as fourthorder RungeKutta or Newmark schemes. However, a geometric — instead of a traditional numericalanalytic — approach to the problem of time integration is particularly pertinent 86: the very essence of a mechanical system is indeed characterized by its symmetries and invariants (e.g., momenta), thus preserving these geometric notions into the discrete computational setting is of paramount importance if one wants discrete time integration to properly capture the underlying continuous motion. Considering mechanics from a variational point of view goes back to Euler, Lagrange and Hamilton 74, and Poincaré famously stated that geometry and physics are “indissociable”. The variational principle most important for continuous mechanics is due to Hamilton, and is often called Hamilton’s principle or the least action principle: it states that a dynamical system always finds an optimal course from one position to another. One consequence is that we can recast the traditional way of thinking about an object accelerating in response to applied forces, into a geometric viewpoint: the path followed by the object between two spacetime positions has optimal geometric properties, analogous to the notion of geodesics on curved surfaces. This point of view is equivalent to Newton’s laws in the context of classical mechanics, but is broad enough to encompass physical models ranging to E&M and quantum mechanics 103. While the idea of discretizing variational formulations of mechanics is standard for elliptic problems using Galerkin Finite Element methods for instance, only recently did it get used to derive variational timestepping algorithms for mechanical systems 104. These variational integrators have been shown to be remarkably versatile, powerful, and general for simulations of physical phenomena when compared to traditional numerical time stepping methods: the symplectic character of variational integrators guarantees good statistical predictability through accurate preservation of the geometric properties of the exact flow of the differential equations. We endeavor to continue contributing to this particular application of geometry and extend it further, as we foresee a number of interesting scientific developments and industrial applications.
Statespace discretization of statistical physics.
Kinetic equations are used to describe a variety of phenomena in various scientific fields, ranging from rarefied gas dynamics and plasma physics to biology and socioeconomics, and appear naturally when one considers a statistical description of a large particle system evolving in time. In incompressible fluid simulation, kinetic solvers based on the lattice Boltzmann method (LBM) have generated growing interest due to their use of the Boltzmann transport equation and to its unusual statespace discretization based on a computationallyefficient lattice 123: compared to macroscopic solvers directly integrating NavierStokes equations, LBM totally bypasses the difficult issue of discretizing advection to high order, and absence of global pressure solves makes for extremely efficient parallel implementations, which are now surpassing alternative discretizations 97. However, the numerical treatment of the collision operator of the Boltmann equation has not reached maturity; most surprising is the complete absence of geometric approaches to deal with Boltzmann equations. One should be able to formulate a variational approach to LBM based on Hamilton's principle to derive a systematic integrator with guaranteed accuracy and structurepreserving properties. Moreover, while dealing with isothermal and incompressible flows is a good starting point, the kinetic standpoint of fluid dynamics is not theoretically restricted to this case: far more complex physical systems, from compressible flow (with shocks), to thermal conductivity, to even acoustics for example, can be handled; but far less is known on how to handle these more involved cases computationally, because no systematic numerical approach to handle Boltzmann equations is known. Success in our geometric approach to LBM should offer a much better handle to deal with these difficult cases: between new Hermite regularization tools 49, 63 and the recent introduction of variational integrators for nonequilibrium thermodynamical systems mentioned above should provide the necessary theoretical foundations to establish a geometric solver for this generalized case.
Learningaided simulation.
Computational physics is experiencing a tectonic shift as datadriven approaches are quickly becoming mainstream. While we do not adhere to the idea being floated that numerical integration could be simply “learned” to improve current solvers, the fact is that many machine learning tools may have profound influence in practical applications using simulation. Long standing problems such as the design of perfectly matched layers (PML, an artificial absorbing layer for transport equations used to reduce the domain of simulation without suffering from reflected waves 61) or flux limiters in high resolution schemes 132 (to avoid the spurious oscillations (wiggles) that would otherwise occur due to shocks or sharp changes) could be found through training, and applied at very low numerical cost. We are curious to see if geometry can help design better architectures or approaches for this type of learningaided simulation, by helping with better loss functions (with soft constraints) or better architectures (to enforce hard constraints) that account for the importance of structure preservation. Learning the highly nonlinear and chaotic dynamics of fluids is also an interesting direction: we believe that one can infer predictive highfrequency details of a turbulent flow from a lowresolution simulation as it is an attractive alternative to nonlinear turbulence modeling, extending the computationallyexpensive ReynoldsAveraged NavierStokes (RANS 39), LargeEddy Simulation (LES 91), or DetachedEddy Simulation (DES 124) models used in CFD. Many other learning efforts in the domain of simulation are being explored, in particular towards the goal of allowing realtime design of shapes that satisfy some physical properties, such as lowest drag for improved aerodynamics or highest stiffness for a light cantilever.
Geometric integration of physical systems and multiphysics.
Although the use of geometric integrators for differential equations in computational physics has recently brought off many numerical improvements, the large body of knowledge in differential geometric mechanics remains vastly underutilized in discrete mechanics. Many mechanical systems require geometric objects such as diffeomorphisms, vector fields, or (principal) connections for which no structurepreserving discretization exists. Hydrodynamics, for instance, has well established and rich differential geometric foundations, but rare are the numerical methods that take advantage of this rich body of knowledge as yet. Yet, satisfying a form of “particle relabeling” symmetry 103 on a discrete level could directly enforce Kelvin’s circulation theorem, a momentum preservation as important as angular momentum preservation for rigid bodies. Relativity is another example, albeit much more involved, where structurepreserving numerics would strongly impact the scientific community: having discretizations automatically enforcing Bianchi’s identities would not only simplify the numerical procedures involved in gravitational theory (as spectral accuracy would no longer be required to avoid spurious modes), but could in fact result in conservation of energy and angular momentum. Moreover, multiphysics (coupled mechanical systems involving more than one simultaneously occurring physical field) can be consistently described through constrained variational principles: a simple, yet already interesting example is the case of the equations of motion for the garden hose, where rod dynamics coupled with fluid motion was only fully modeled (along with its nonlinear solutions of travelingwave type) a few years back 116 through such a geometric treatment. Now that a variational formulation of nonequilibrium thermodynamics extending Hamilton's principle to include irreversible processes has been proposed 78, we are particularly interested in advancing further the arsenal of computational methods for physical simulation.
3.3 Geometry for dynamical systems
Dynamical systems – whether physical, biological, chemical, or social – are ubiquitous in nature, and their study deals with the concept of change, rate of change, rate of rate of change, etc. Dynamical systems are often better elucidated and modeled through topology and geometry. Whether we consider a continuoustime dynamical system (flow) or discretetime dynamical system (map), the geometric theory of dynamical systems studies phase portraits: on the statespace manifold (a geometric model for the set of all possible states of the system), the global behavior of the dynamical system is determined by a cellular structure of basins enclosed by separatrices, each basin being dominated by a different specific behavior or fate. A system's trajectories on the statespace manifold determine velocity vectors by differentiation; conversely, velocity vectors determine trajectories by integration. Bifurcations can also be understood as geometric models for the controlled change of one system into another, while the rate of divergence of trajectories in phase space measures a system's stability. Given this overwhelming relevance of geometry in dynamical systems, we intend to dedicate some of our activities to develop geometrybased computational tools to study time series and dynamical systems: while classic dynamical systems theory has established solid foundations to study structures in steady and timeperiodic flows and maps, new tools are needed to analyze the complexity of time series or aperiodic largescale flows from sampled trajectories, and to automatically generate a simplified skeleton of the overall dynamics of a system from input data. We discuss a few directions we are interested in further impacting next.
Time series.
Geometric methods play an important part in the study of time series. Of particular interest are timedelay embeddings, which are generically able to capture the underlying state space and dynamics from which the time series data have been acquired, by the Takens embedding theorem 127. Such embeddings transform discrete time series into point clouds in Euclidean space, so that the underlying geometry of the point cloud reflects the geometry of the phase space the data originate from. By doing so, questions related to the seasonality or anomalous behavior of the time series are naturally reformulated into questions about the geometry or topology of their embeddings 113. Beside this approach, other more direct methods apply geometric or topological tools in the original physical or frequency domain, which, despite its simplicity, has proven to be relevant in various contexts 66, 70. A common thread to all these developments is their restriction to numerical time series, including (but not restricted to) data for which geometry plays an obvious role—e.g. inertial or gyroscopic sensor data. With potential medical applications in mind, one of our main longterm goals will be to adapt and extend these approaches to handle categorical data, in connection to the item in the Geometry for data science theme. We also plan to find principled methods to tuning the various parameters involved in the techniques, e.g. the window size in timedelay embeddings: we will seek to optimize or learn these parameters automatically, in connection to the item Geometrydriven learning in the Geometry for data science theme. We will also seek to make these parameters adaptive, e.g. using timevarying window sizes in timedelay embeddings of irregular time series, in order to obtain more accurate data representations and improved learning performance.
Coherent structures.
Another interesting area in need of new numerical methods concerns coherent structures, i.e., persisting features of a flow over long periods that tend to favor or inhibit material transport between distinct flow regions. While there is no universally agreedupon definition for coherent structures (there exist ergodicitybased 52, observerbased 105, and probabilistic 76 approaches to their definition), most variants and associated computational methods assume a fine knowledge of the Eulerian velocity field in space and time to deduce a good approximation of the flow. However, flows are often known only as a set of sparse particle trajectories in time (an example is the trajectory of buoys in the ocean). Such a sparse sampling of the dynamical system does not lend itself well to a geometric analysis of transport, so topological methods have recently been proposed to extract structures from a sparse set of trajectories by measuring their entanglement 129, 40, 136 based on the theory of braid groups, a classical area of topology. Coherent regions can then be defined as containing particles that possibly mix with other particles within the region itself but do not mix with particles outside the region; the set of trajectories arising from the particles within a coherent region forms a coherent bundle. Even if the use of braid groups offers sound foundations and numerical tools for the definition of coherent structures in 2D, there has been only limited efforts in developing practical and scalable computational tools for the efficient analysis of flow structures in 3D, offering a clear opportunity for us to try new geometric insights.
Invariant sets.
Much of the theory of dynamical systems revolves around the existence and structure of invariant sets, which by definition are subsets of the state space that are invariant under the action of the dynamics. Invariant sets come in many different forms (stationary solutions, periodic orbits, connecting orbits, chaotic invariant sets, etc), and their structure can be very complicated and can undergo drastic changes under perturbations of the system, thus making their study difficult. This is all the more true in practical applications, where the systems are only known through space and/or time discretizations. Conley index theory 62 overcomes these issues by restricting the focus to invariant sets that admit an isolating neighborhood, and by introducing a topological invariant—the Conley index—that characterizes whether such isolated invariant sets are attracting, repelling, or saddlelike. It is defined as the homotopy type of a pair of compact subsets of the neighborhood, and it is proven to be independent of the choice of neighborhood—thus characterizing the invariant set itself. We are interested in the study of invariant sets in the discrete space and continuous time setting, where the space is typically described by a simplicial complex and the dynamics by a combinatorial vector (or multivector) field. Building upon Forman's seminal work in combinatorial dynamical systems 73, recent advances 45, 98 have shown that isolated invariant sets and their Conley indices can be properly defined even in this setting, and that they can be related to the dynamics of some upper semicontinuous acyclic multivalued map defined on the geometric realization of the simplicial complex; in simpler terms, not only can Conley index theory be adapted to the combinatorial setting, but it also connects to its classical analog in the underlying space. Two important questions for applications arise from this line of work: (1) how to compute the invariant sets and their Conley indices (including choosing relevant isolating neighboroods) efficiently? (2) how do they behave under perturbations of the input vector field or simplicial complex? These questions have just started to be addressed 68, 69, mostly through the lens of singleparameter topological persistence theory, developed in the context of topological data analysis. We intend to push this direction further, notably using multiparameter persistence theory to cope with some of the key difficulties such as the choice of isolating neighborhoods.
3.4 Geometry for data science
The last decade has seen the advent of machine learning (ML), and in particular deep learning (DL), in a large variety of fields, including some directly connected to geometry. For instance, DLbased approaches have become increasingly popular in geometry processing 117 due to their ability to outperform stateoftheart, domainspecific methods by leveraging the everincreasing amounts of available labeled data. On the downside, DL approaches suffer from a general lack of explainability. Moreover, their performances can be disappointing on small data due to their large numbers of parameters; this is especially true with endtoend learning pipelines, which tend to require humongous amounts of training data to learn the right data representation. Finally, DL is by essence tied to Euclidean data representations, and as such it requires intermediate transforms in order to be applicable to nonEuclidean data types such as graphs or probability measures. Because of these limitations, we are seeing a rise of geometric and topological methods for data science in general, and for ML and DL in particular, whose aim is to help address the aforementioned challenges as well as others. For instance, geometric deep learning 50 tries to generalize deep neural models to nonEuclidean domains. This includes for instance using information geometry to apply deep neural models in probability spaces. Topological data analysis (TDA) 108 is another popular approach to enhance ML and DL methods. It contributes to data science in at least three different ways: first, by providing data mining tools that can help users uncover hidden structures in data; second, by providing generic descriptors for geometric data that can be turned into features for ML and DL with provable stability properties; third, by integrating itself deeply into existing ML methods or DL architectures to enhance their performances or to analyze their behavior 58, 101. Other contributions of geometry to data science at large include: the use of Forman’s Ricci curvature and its corresponding Ricci flow in networks, to understand the networks' properties and growth 133; the application of the HodgeHemholtz decomposition to statistical ranking problems with sparse response data, with theoretical connections to both PageRank and LASSO 90; the use of Reeb graphs or MorseSmale complexes in statistical inference 59 as well as in data visualization 131. These important developments reinforce our argument that geometry and topology have their role to play in the elaboration of the nextgeneration data analysis tools. We plan to focus on a few research directions related to these developments, which are of particular interest in our view.
Deep learning for largescale 3D geometric data analysis.
We first propose to develop efficient algorithms and mathematical tools for analyzing large geometric data collections using Deep Learning techniques. This includes 3D shapes represented as triangle or quad meshes, volumetric data, point clouds possibly embedded in highdimensions, and graphs representing geometric (e.g. proximity) data. Our project is motivated by the fact that large annotated collections of geometric models have recently become available 57, 135, and that machine learning algorithms applied to such collections have shown promising initial results, both for data analysis as well as synthesis. We believe that these results can be significantly extended by building on recent advances in geometry processing, optimization and learning. Our ultimate goal is to design novel deep learning techniques capable both of handling geometric data directly and of combining and integrating different data sources into a unified analysis pipeline. A key challenge in this project is the fact that geometric data can come in a myriad different representations, such as point clouds and meshes among others, with variable sampling and discretization. Furthermore, geometric shapes can undergo both rigid and nonrigid deformations. Unfortunately, most existing deep learning approaches focus only on a particular type of representations and deformation classes (e.g., considering purely extrinsic or purely intrinsic methods). Instead we propose to place special focus on designing learning techniques capable of handing diverse multimodal data sources undergoing arbitrary deformations, in a coherent theoretical and practical framework. Moreover we propose to develop novel powerful interactive tools for analysis and annotation, to help harness user input, and also provide better mechanisms for exploration of variability in the data 120, 112.
Explainable geometric and topological features for data.
Another of our goals is to design geometric and topological features that can capture richer content from the data, while maintaining the robustness and stability properties that the existing features enjoy. If we can make our features rich enough so that they characterize the input data (or their underlying geometric structures, assuming such structures exist) completely, then we will be able to leverage them in the context of explainable AI, to compute preimages with guarantees on the corresponding interpretations. In cases where our features cannot completely describe the data, we will study the geometry of the fibers of the feature extraction step, in order to quantify the discrepancy that may appear between different interpretations of the same feature. We envision two complementary approaches for this:
 The first approach relies on feature aggregation. In the context of TDA for instance, one may consider using multiple filtrations (or filter functions on a fixed simplicial complex), computing their corresponding topological descriptors, then aggregating these descriptors together to form a feature vector.
 The second approach relies on more elaborate geometric and topological tools to design the features. The idea is to encode the joint effect of multiple geometric and topological constructions on the data, in a more integrated way than just by aggregating the corresponding features. By encoding more complex effects, we hope to extract a richer content using smaller constructions.
Research on the first approach in TDA started with 65, 79, who proved that, in the special case where the data are sampled from some subanalytic compact sets in Euclidean space ${\mathbb{R}}^{n}$, the compact sets themselves are fully described by the aggregated features obtained by orthogonal projections onto lines. This follows from a more fundamental result on the invertibility of the Radon transforms of constructible functions 122, to which the above aggregated features belong. This initial result has sparked a thriving new direction of research, exploring larger and larger classes of compact sets 89, 102, 109. Many important questions arise from this line of work, some of which have been partially addressed, including: what kind of stability or robustness properties do these aggregated features enjoy? Can the size of the collection of filter functions used be reduced, to become finite and (more importantly) independent of the compact set under consideration? Can the aggregated features be computed efficiently? Can nonEuclidean compact sets, such as manifolds or length spaces, be considered as well, with similar guarantees?
The second approach is related to the development of multiparameter persistence 54, which is undeniably the most widely open and longstanding research topic in TDA today. The core challenge is to define computationally tractable algebraic invariants that can capture as much of the joint structure of multiple topological constructions as possible. The notorious difficulty of this question comes from the fact that the algebraic objects underlying multiparameter topological constructions are significantly more complicated than the ones underlying singleparameter constructions. The question also connects to notoriously hard problems in other areas of pure mathematics, such as the classification of isomorphism classes of indecomposable poset representations in quiver representation theory for instance. It can benefit from these connections, as mathematical tools that have been developed for those problems can be imported into the TDA literature—several promising such imports have been made in the recent past, including from representation theory 46 and from sheaf theory 92. In turn, mathematical and algorithmic advances made in multiparameter persistence may benefit these other areas of mathematics as well. This is clearly a highrisk and longterm research topic, but if successful, it may eventually have an enormous impact on TDA and related areas.
Geometric feature learning.
Geometry and topology have played a key role in the design of feature extraction pipelines for certain types of data. The numerous existing geometric features for geometry processing (shape contexts 75, differential and integral invariants 115, heat or wave kernel signatures 42, 126, etc.) are a sign of the importance of this topic for the computer graphics community. Meanwhile, the TDA community has developed generic feature extraction pipelines, based on combinatorial constructions and their algebraic invariants, which have proven to be useful in a variety of application domains 108. All these approaches are, however, handcrafted, with hyperparameters being tuned via manual, grid, or random search. Our goal is to make these approaches transition from a paradigm of feature engineering to that of feature learning, in order to set up endtoend learning pipelines for improved performances and adaptability. Two complementary directions are considered:
 designing piecewisesmooth variants of the existing pipelines, with a fine control over the underlying stratification. This will make it possible to apply variational optimization methods, typically stochastic (sub)gradient descent, and to optimize the gradient sampling steps for improved convergence rates.
 designing novel pipelines based on a combination of geometric/topological tools and deep learning, in order to get the best out of both worlds.
Research in the first direction is still in its infancy. Promising theoretical advances were made recently, towards understanding the piecewise differentiability of the basic topological persistence operator in full generality 96, as well as towards optimizing its parameters using classical stochastic gradient descent 55. Can the knowledge gained in these studies about the underlying stratification of the operator be leveraged to optimize the gradient sampling step and thus improve the convergence rates? Can these results be extended to more advanced pipelines, such as the one for Mapper or for zigzags and multiparameter persistence?
The idea behind the second direction is to integrate topological or geometric layers into neural network architectures such as autoencoders or GANs for feature extraction — the challenge being to determine how to do it in the appropriate way, so that we can make the most of this combination. This question connects to the research topic described further down in this section.
Geometrydriven learning.
Most of the contributions of geometry and topology to machine learning until recently have been to the design of preprocessing steps (e.g. feature extraction) to enhance the performances of the learning pipeline. There is now a thriving effort of the community toward integrating geometric and/or topological computations deeper into the core of the pipeline. This includes for instance: ToMATo 58, which integrates a TDAbased feedback loop into density based algorithms to improve their stability and robustness; topological regularizers 60, 87, which add topologybased regularization terms to the loss in supervised statistical learning; topological layers 56, 77, 94, which are meant to be incorporated into neural networks. Meanwhile, geometry and topology have been used to analyze the behavior of neural networks 121, 51. This exciting line of work is just emerging, and our intent is to push this direction further, in particular to address the following important questions:
 How can we generalize the use of topological layers in neural networks? This question is connected to the differentiability of the TDA pipeline, addressed in the research topic Geometric feature learning. Inded, generalizing the current (nascent) framework for differential calculus and optimization with the TDA pipeline will be key to designing both generic and effective topological layers. Another more practical aspect of the question is to evaluate the contribution of topological layers as initial or intermediate layers, depending on the neural network architecture that they are combined with and on the data they are applied to.
 The same question arises for topological regularizers, with similar theoretical and practical challenges.
 The development of richer families of geometric and topological descriptors, undertaken in the item Richer geometric and topological features for data, will eventually lead to the question of generalizing the current differentiable framework to these new descriptors, in order to make them as widely applicable as the current descriptors, and also to the practical question of determining how to best combine them with existing loss functions, regularizers, or neural network architectures.
 The aforementioned contributions and research directions concern mostly supervised learning. Can we contribute as well to unsupervised learning problems, including clustering (as ToMATo does already for densitybased clustering), dimensionality reduction, or unsupervised feature learning? This question connects also to the research topic Geometric feature learning described previously. One direction we may explore is the design of geometric or topological layers to be inserted in unsupervised neural network architectures such as autoencoders or GANs.
 Finally, as TDA is concerned primarily with topology, an obvious (yet still wide open) question to ask is whether it can contribute to the current effort towards generating neural network architectures automatically.
Geometry for categorical and mixed data types.
Categorical data types are notoriously hard to deal with in the context of ML and AI. Indeed, most of the existing ML toolbox has been designed specifically to work with numerical variables, usually sitting in some vector or metric space. By contrast, spaces of categorical data do not naturally come equipped with a linear structure nor a metric. More importantly, these spaces are discrete by nature, so choices of metrics or (dis)similarity measures can be scarce, with limited effects on the learning efficiency. To make things worse, categorical variables are often mixed with numerical variables, and choosing a proper weighting for them is a challenge in its own right. Meanwhile, categorical variables play an important part in many applications: for instance, in precision medicine, where the monitoring of patients relies on collected longitudinal data that include not only numerical variables such as temperature or blood pressure, but also categorical variables such as illness antecedents or symptoms lists. Thus, handling categorical and mixed data types represents an important challenge today. Unfortunately, with very few exceptions 134, it has been mostly overlooked so far in the development of topological methods for ML and AI, so our goal will be to help fix this situation. The standard approach for handling categorical variables is to define a proper vector representation, then to apply—either offtheshelf or with minor adaptations—an analysis method designed for numerical variables to the new data representation. A prototypical instance of this approach is Multiple Correspondance Analysis for dimensionality reduction 38, which applies classical PCA to the onehot encoding matrix of the input data. A variant of the approach replaces the vector representation by a suitable metric or (dis)similarity measure on the initial categorical variables or on some transformed version of those. For instance, in clustering, one can define a metric on the input data, e.g. Jaccard or Hamming distance, then apply a hierarchical bottomup clustering algorithm such as singlelinkage to the resulting distance matrix. This variant seems quite appropriate for geometric or topological methods, since the latter typically work with metric or (dis)similarity spaces. The challenge is to determine with which metrics or (dis)similarity measures, and on which data types, geometric or topological methods will be provably better.
A more refined version of the approach learns the new data representation instead of engineering it, which is particularly relevant when endtoend learning pipelines are sought for. The methods are usually taylored to a specific data type, for instance word2vec 106 computes word embeddings for text data using a twolayer neural network. Our developments in the research topic Geometrydriven learning will make it possible to combine TDA layers with such networks, and thus to benefit from the most recent advances on representation learning for these data types. The challenge will be to understand when and how to make the most of this combination.
4 Application domains
Our work aims at a wide range of applications covering 3D shape analysis and processing, simulation, and data science in general. While we typically focus on contributions that are of a fundamental, mathematical and algorithmic nature, we seek collaborations with academics and industrial from applied fields, who can use our tools on practical and concrete problems. Here are a few examples of collaborations:
 In the context of 3D geometry processing, we collaborate with Dassault Systèmes for a) the PhD of Lucas Brifault on the design of novel geometric representations for shapes through measure theory and b) the PhD of Mariem Mezghanni on the design of physical simulation layers for 3D modeling.
 In the context of personalized medicine, we collaborate with statisticians and medical doctors to incorporate our geometric and topological features into learning pipelines to design better dynamic treatment regimens (AEx PreMediT).
 In a collaboration with the French Ministry of Defense, we seek to develop tools to analyze multimodal time series data in order to predict the appearance of GLOCs among fighter jet pilots in training or in operation (PhD of Julie Mordacq).
Beside these few illustrative examples, GeomeriX also maintains regular collaborations with Sanofi, EDF, Danone R&D, Immersion Tools, as well as with several key players in the worldwide tech industry, including Ansys, Adobe Research, Disney/Pixar, NVidia.
5 Highlights of the year
5.1 Awards
 The paper “Functional Maps: A Flexible Representation of Maps Between Shapes” coauthored by Maks Ovsjanikov and colleagues, has received the SIGGRAPH 2023 TestofTime award.
 Maks Ovsjanikov has received an ERC Consolidator Grant in 2023.
 The paper “Implicit fairing of irregular meshes using diffusion and curvature flow” coauthored by Mathieu Desbrun and colleagues has been included in the ACM list of “Seminal Graphics Papers: Pushing the Boundaries”.
5.2 Distinctions
 S. Oudot was a CAS fellow at the Norwegian Academy of Science and Letters for the academic year 20222023.
 Maks Ovsjanikov became a fellow of ELLIS, the European Laboratory for Learning and Intelligent Systems, bringing together top AI researchers in Europe.
 Mathieu Desbrun received an INRIA Chair on Geometry and AI.
6 New results
We list our new results for each of the four themes that our team is articulated around.
6.1 Geometry for Euclidean shape processing
6.1.1 Patternshop: Editing Point Patterns by Image Manipulation
In collaboration with Xingchang Huang, HansPeter Seidel and Gurprit Singh (MPI Saarbrucken) and Tobias Ritschel (UCL).
Point patterns are characterized by their density and correlation. While spatial variation of density is wellunderstood, analysis and synthesis of spatiallyvarying correlation is an open challenge. No tools are available to intuitively edit such point patterns, primarily due to the lack of a compact representation for spatially varying correlation. In this work 13, we propose a lowdimensional perceptual embedding for point correlations. This embedding can map point patterns to common threechannel raster images, enabling manipulation with offtheshelf image editing software. To synthesize back point patterns, we propose a novel edgeaware objective that carefully handles sharp variations in density and correlation. The resulting framework allows intuitive and backwardcompatible manipulation of point patterns, such as recoloring, relighting to even texture synthesis that have not been available to 2D point pattern design before. Effectiveness of our approach is tested in several user experiments. Our proposed framework was patented under the reference M35655EP. The corresponding code is however publicly available at https://github.com/xchhuang/patternshop.
6.1.2 Robust Pointset Denoising of PiecewiseSmooth Surfaces through Line Processes
In collaboration with Damien Rohmer (LIX).
Denoising is a common, yet critical operation in geometry processing aiming at recovering highfidelity models of piecewise smooth objects from noisecorrupted pointsets. Despite a sizable literature on the topic, there is a dearth of approaches capable of processing very noisy and outlierridden input pointsets for which no normal estimates and no assumptions on the underlying geometric features or noise type are provided. In this paper 21, we propose a new robuststatistics approach to denoising pointsets based on line processes to offer robustness to noise and outliers while preserving sharp features possibly present in the data. While the use of robust statistics in denoising is hardly new, most approaches rely on prescribed filtering using dataindependent blending expressions based on the spatial and normal closeness of samples. Instead, our approach deduces a geometric denoising strategy through robust and regularized tangent plane fitting of the initial pointset, obtained numerically via alternating minimizations for efficiency and reliability. Key to our variational approach is the use of line processes to identify inliers vs. outliers, as well as the presence of sharp features. We demonstrate that our method can denoise sampled piecewisesmooth surfaces for levels of noise and outliers at which previous works fall short.
6.1.3 FeatureSized Sampling for Vector Line Art
In collaboration with Stefan Ohrhallinger (TU Wien) and Amal D. Parakkat (Telecom Patis).
In this work 27, by introducing a firstofitskind quantifiable sampling algorithm based on feature size, we present a fresh perspective on the practical aspects of planar curve sampling. Following the footsteps of $\u03f5$sampling, which was originally proposed in the context of curve reconstruction to offer provable topological guarantees (Crust algorithm) under quantifiable bounds, we propose an arbitrarily precise $\u03f5$sampling algorithm for sampling smooth planar curves (with a prior bound on the minimum feature size of the curve). This paper not only introduces the first such algorithm which provides usercontrol and quantifiable precision but also highlights the importance of such a sampling process under two key contexts: 1) To conduct a first study comparing theoretical sampling conditions with practical sampling requirements for reconstruction guarantees that can further be used for analysing the upper bounds of $\u03f5$ for various reconstruction algorithms with or without proofs, 2) As a featureaware sampling of vector line art that can be used for applications such as coloring and meshing.
6.1.4 BioSketch: A new medium for interactive storytelling, illustrated by the phenomenon of infection
In collaboration with Pauline Olivier, Renaud Chabrier, MariePaule Cani (LIX), JeanLuc Coll (INSERM, CNRS, Institute for Advanced Biosciences, Grenoble).
In the field of biology, digital illustrations play a crucial role in conveying complex phenomena, allowing for idealized shapes and motion, in contrast to data visualization. In the absence of suitable media, scientists often rely on oversimplified 2D figures or have to call in professional artists to create better illustrations, which can be limiting. In this work 28 we introduce BioSketch, a novel progressive sketching system designed to ease the creation of animated illustrations, as exemplified here in the context of the infection phenomenon. Our solution relies on a new progressive sketching paradigm that seamlessly combines 3D modeling and patternbased shape distribution to create background volume and temporal animation control. The elements created can be assembled into a complex scenario, enabling narrative design experiments for educational applications in biology. Our results and first feedback from experts in illustration and biology demonstrate the potential of BioSketch to assist communication on the infection phenomenon, helping to bridge the gap between expert and nonexpert audiences.
6.1.5 Somigliana Coordinates
In collaboration with Fernando de Goes (Pixar).
In this work 24, we present a novel cage deformer based on elasticityderived matrixvalued coordinates. In order to bypass the typical shearing artifacts and lack of volume control of existing cage deformers, we promote a more elastic behavior of the cage deformation by deriving our coordinates from the Somigliana identity, a boundary integral formulation based on the fundamental solution of linear elasticity. Given an initial cage and its deformed pose, the deformation of the cage interior is deduced from these Somigliana coordinates via a corotational scheme, resulting in a matrixweighted combination of both vertex positions and face normals of the cage. Our deformer thus generalizes Green coordinates, while producing physicallyplausible spatial deformations that are invariant under similarity transformations and with interactive bulging control. We demonstrate the efficiency and versatility of our method through a series of examples in 2D and 3D.
6.2 Geometry for simulation
6.2.1 Building a Virtual WeaklyCompressible Wind Tunnel Testing Facility
In collaboration with Chaoyang Lyu, Kai Bai, Yiheng Wu, and Xiaopei Liu (all from Shanghaitech University, China), and Changxi Zheng (Columbia University, USA).
Virtual wind tunnel testing is a key ingredient in the engineering design process for the automotive and aeronautical industries as well as for urban planning: through visualization and analysis of the simulation data, it helps optimize lift and drag coefficients, increase peak speed, detect high pressure zones, and reduce wind noise at low cost prior to manufacturing. In this paper 18, we develop an efficient and accurate virtual wind tunnel system based on recent contributions from both computer graphics and computational fluid dynamics in highperformance kinetic solvers. Running on one or multiple GPUs, our massivelyparallel lattice Boltzmann model meets industry standards for accuracy and consistency while exceeding current mainstream industrial solutions in terms of efficiency Ð especially for unsteady turbulent flow simulation at very high Reynolds number (on the order of ${10}^{7}$) – due to key contributions in improved collision modeling and boundary treatment, automatic construction of multiresolution grids for complex models, as well as performance optimization. We demonstrate the efficacy and reliability of our virtual wind tunnel testing facility through comparisons of our results to multiple benchmark tests, showing an increase in both accuracy and efficiency compared to stateoftheart industrial solutions. We also illustrate the fine turbulence structures that our system can capture, indicating the relevance of our solver for both VFX and industrial product design.
6.2.2 FluidSolid Coupling in Kinetic TwoPhase Flow Simulation
Reallife flows exhibit complex and visually appealing behaviors such as bubbling, splashing, glugging and wetting that simulation techniques in graphics have attempted to capture for years. While early approaches were not capable of reproducing multiphase flow phenomena due to their excessive numerical viscosity and low accuracy, kinetic solvers based on the lattice Boltzmann method have recently demonstrated the ability to simulate waterair interaction at high Reynolds numbers in a massivelyparallel fashion. However, robust and accurate handling of fluidsolid coupling has remained elusive: be it for CG or CFD solvers, as soon as the motion of immersed objects is too fast or too sudden, pressures near boundaries and interfacial forces exhibit spurious oscillations leading to blowups. Built upon a phasefield and velocitydistribution based latticeBoltzmann solver for multiphase flows, this paper 16 spells out a series of numerical improvements in momentum exchange, interfacial forces, and twoway coupling to drastically reduce these typical artifacts, thus significantly expanding the types of fluidsolid coupling that we can efficiently simulate. We highlight the numerical benefits of our solver through various challenging simulation results, including comparisons to previous work and real footage.
6.2.3 HighOrder MomentEncoded Kinetic Simulation of Turbulent Flows
In collaboration with Tongtong Wang, Zherong Pang, Xifeng Gao, and Kui Wu (Tencent Lightspeed Studios, China).
Kinetic solvers for incompressible fluid simulation were designed to run efficiently on massively parallel architectures such as GPUs. While these lattice Boltzmann solvers have recently proven much faster and more accurate than the macroscopic NavierStokesbased solvers traditionally used in graphics, it systematically comes at the price of a very large memory requirement: a mesoscopic discretization of statistical mechanics requires over an order of magnitude more variables per grid node than most fluid solvers in graphics. In order to open up kinetic simulation to gaming and simulation software packages on commodity hardware, we propose a HighOrder MomentEncoded LatticeBoltzmannMethod solver which we coined HOMELBM, requiring only the storage of a few moments per grid node 17, with little to no loss of accuracy in the typical simulation scenarios encountered in graphics. We show that our lightweight and lightspeed fluid solver requires three times less memory and runs ten times faster than stateoftheart kinetic solvers, for a nearlyidentical visual output.
6.3 Geometry for data science
6.3.1 Stable Vectorization of Multiparameter Persistent Homology using Signed Barcodes as Measures
In collaboration with David Loiseaux (Inria, Datashape team), Luis Scoccola (Northeastern University), Magnus Botnan (Vrije Universiteit Amsterdam), and Mathieu Carrière (Inria, Datashape team).
Persistent homology (PH) provides topological descriptors for geometric data, such as weighted graphs, which are interpretable, stable to perturbations, and invariant under, e.g., relabeling. Most applications of PH focus on the oneparameter case – where the descriptors summarize the changes in topology of data as it is filtered by a single quantity of interest – and there is now a wide array of methods enabling the use of oneparameter PH descriptors in data science, which rely on the stable vectorization of these descriptors as elements of a Hilbert space. Although the multiparameter PH (MPH) of data that is filtered by several quantities of interest encodes much richer information than its oneparameter counterpart, the scarceness of stability results for MPH descriptors has so far limited the available options for the stable vectorization of MPH. In this work 25 we aim to bring together the best of both worlds by showing how the interpretation of signed barcodes – a recent family of MPH descriptors – as signed measures leads to natural extensions of vectorization strategies from one parameter to multiple parameters. The resulting feature vectors are easy to define and to compute, and provably stable. While, as a proof of concept, we focus on simple choices of signed barcodes and vectorizations, we already see notable performance improvements when comparing our feature vectors to stateoftheart topologybased methods on various types of data.
6.3.2 Local characterizations for decomposability of 2parameter persistence modules
In collaboration with Magnus Botnan (Vrije Universiteit Amsterdam).
In this work 12 we investigate the existence of sufficient local conditions under which poset representations decompose as direct sums of indecomposables from a given class. In our work, the indexing poset is the product of two totally ordered sets, corresponding to the setting of 2parameter persistence in topological data analysis. Our indecomposables of interest belong to the socalled interval modules, which by definition are indicator representations of intervals in the poset. While the whole class of interval modules does not admit such a local characterization, we show that the subclass of rectangle modules does admit one and that it is, in some precise sense, the largest subclass to do so.
6.3.3 A Gradient Sampling Algorithm for Stratified Maps with Applications to Topological Data Analysis
In collaboration with Jacob Leygonie (University of Oxford), Mathieu Carrière (Inria, Datashape team), and Théo Lacombe (Université de Marne la Vallée).
In this work 15 we introduce a novel gradient descent algorithm refining the wellknown Gradient Sampling algorithm on the class of stratifiably smooth objective functions, which are defined as locally Lipschitz functions that are smooth on some regular pieces—called the strata—of the ambient Euclidean space. On this class of functions, our algorithm achieves a sublinear convergence rate. We then apply our method to objective functions based on the (extended) persistent homology map computed over lowerstar filters, which is a central tool of Topological Data Analysis. For this, we propose an efficient exploration of the corresponding stratification by using the Cayley graph of the permutation group. Finally, we provide benchmarks and novel topological optimization problems that demonstrate the utility and applicability of our framework.
6.3.4 Shape Nonrigid Kinematics (SNK): A ZeroShot Method for NonRigid Shape Matching via Unsupervised Functional Map Regularized Reconstruction
In this work 32, we present Shape Nonrigid Kinematics (SNK), a novel zeroshot method for nonrigid shape matching that eliminates the need for extensive training or ground truth data. SNK operates on a single pair of shapes, and employs a reconstructionbased strategy using an encoderdecoder architecture, which deforms the source shape to closely match the target shape. During the process, an unsupervised functional map is predicted and converted into a pointtopoint map, serving as a supervisory mechanism for the reconstruction. To aid in training, we have designed a new decoder architecture that generates smooth, realistic deformations. SNK demonstrates competitive results on traditional benchmarks, simplifying the shape matching process without compromising accuracy.
6.3.5 ZeroShot 3D Shape Correspondence
In collaboration with Ahmed Abdelreheem, Abdelrahman Eldesokey, and Peter Wonka, from King Abdullah University of Science and Technology (KAUST), Saudi Arabia.
In this work 29 we propose a novel zeroshot approach to computing correspondences between 3D shapes. Existing approaches mainly focus on isometric and nearisometric shape pairs (e.g., human vs. human), but less attention has been given to strongly nonisometric and interclass shape matching (e.g., human vs. cow). To this end, we introduce a fully automatic method that exploits the exceptional reasoning capabilities of recent foundation models in language and vision to tackle difficult shape correspondence problems. Our approach comprises multiple stages. First, we classify the 3D shapes in a zeroshot manner by feeding rendered shape views to a languagevision model (e.g., BLIP2) to generate a list of class proposals per shape. These proposals are unified into a single class per shape by employing the reasoning capabilities of ChatGPT. Second, we attempt to segment the two shapes in a zeroshot manner, but in contrast to the cosegmentation problem, we do not require a mutual set of semantic regions. Instead, we propose to exploit the incontext learning capabilities of ChatGPT to generate two different sets of semantic regions for each shape and a semantic mapping between them. This enables our approach to match strongly nonisometric shapes with significant differences in geometric structure. Finally, we employ the generated semantic mapping to produce coarse correspondences that can further be refined by the functional maps framework to produce dense pointtopoint maps. Our approach, despite its simplicity, produces highly plausible results in a zeroshot manner, especially between strongly nonisometric shapes.
6.3.6 VoroMesh: Learning Watertight Surface Meshes with Voronoi Diagrams
In collaboration with Pierre Alliez (Inria).
In this work 26, we present VoroMesh, a novel and differentiable Voronoibased representation of watertight 3D shape surfaces. From a set of 3D points (called generators) and their associated occupancy, we define our boundary representation through the Voronoi diagram of the generators as the subset of Voronoi faces whose two associated (equidistant) generators are of opposite occupancy: the resulting polygon mesh forms a watertight approximation of the target shape’s boundary. To learn the position of the generators, we propose a novel loss function, dubbed VoroLoss, that minimizes the distance from groundtruth surface samples to the closest faces of the Voronoi diagram which does not require an explicit construction of the entire Voronoi diagram. A direct optimization of the Voroloss to obtain generators on the Thingi32 dataset demonstrates the geometric efficiency of our representation compared to axiomatic meshing algorithms and recent learningbased mesh representations. We further use VoroMesh in a learningbased mesh prediction task from input SDF grids on the ABC dataset, and show comparable performance to stateoftheart methods while guaranteeing closed output surfaces free of selfintersections.
6.3.7 SATR: ZeroShot Semantic Segmentation of 3D Shapes
In collaboration with Ahmed Abdelreheem, Ivan Skorokhodov, and Peter Wonka, from from King Abdullah University of Science and Technology (KAUST), Saudi Arabia.
In this work 30, we explore the task of zeroshot semantic segmentation of 3D shapes by using largescale offtheshelf 2D image recognition models. Surprisingly, we find that modern zeroshot 2D object detectors are better suited for this task than contemporary text/image similarity predictors or even zeroshot 2D segmentation networks. Our key finding is that it is possible to extract accurate 3D segmentation maps from multiview bounding box predictions by using the topological properties of the underlying surface. For this, we develop the Segmentation Assignment with Topological Reweighting (SATR) algorithm and evaluate it on ShapeNetPart and our proposed FAUST benchmarks. SATR achieves stateoftheart performance and outperforms a baseline algorithm by 1.3% and 4% average mIoU on the FAUST coarse and finegrained benchmarks, respectively, and by 5.2% average mIoU on the ShapeNetPart benchmark.
6.3.8 Spatially and Spectrally Consistent Deep Functional Maps
Joint work with Mingze Sun (Tsinghua Shenzhen International Graduate School, China) , Shiwei Mao (Tsinghua Shenzhen International Graduate School, China) , Puhua Jiang (Tsinghua Shenzhen International Graduate School, China, Peng Cheng Laboratory, China), Ruqi Huang (Tsinghua Shenzhen International Graduate School, China).
In this paper 34, we investigate the utility of cycle consistency in Deep Functional Maps. We first justify that under certain conditions, the learned maps, when represented in the spectral domain, are already cycle consistent. Furthermore, we identify the discrepancy that spectrally consistent maps are not necessarily spatially, or pointwise, consistent. In light of this, we present a novel design of unsupervised Deep Functional Maps, which effectively enforces the harmony of learned maps under the spectral and the pointwise representation. By taking advantage of cycle consistency, our framework produces stateoftheart results in mapping shapes even under significant distortions. Beyond that, by independently estimating maps in both spectral and spatial domains, our method naturally alleviates overfitting in network training, yielding superior generalization performance and accuracy within an array of challenging tests for both nearisometric and nonisometric datasets.
6.3.9 TIDE: Time Derivative Diffusion for Deep Learning on Graphs
In collaboration with Maximilian Krahn (Aalto University, Finland).
In this paper 33, we present a novel method based on time derivative graph diffusion (TIDE) to overcome the structural limitations of the messagepassing framework in graph neural networks. Our approach allows for optimizing the spatial extent of diffusion across various tasks and network channels, thus enabling medium and longdistance communication efficiently. Furthermore, we show that our architecture design also enables local messagepassing and thus inherits from the capabilities of local messagepassing approaches. We show that on both widely used graph benchmarks and synthetic mesh and graph datasets, the proposed framework outperforms stateoftheart methods by a significant margin.
6.3.10 ReVISOR: ResUNets with visibility and intensity for structured outlier removal
In collaboration with Guillaume Terrasse and Guillaume Thibault (EDF R&D).
In this paper 14, we make several contributions to address the problem of reflectioninduced outlier detection. First, to overcome the scarcity of annotated data, we introduce a new dataset tailored for this task. Second, to capture nonlocal dependencies, we study and demonstrate, for the first time, the utility of deep learning based semantic segmentation architectures for reflectioninduced outlier detection. By doing so, we bring together the fields of shape denoising/repair and semantic segmentation. Third, we demonstrate that additional nonlocal cues in the form of laser intensity and a computed visibility signal help boost the performance considerably. We denote our pipeline as ResUNets with Visibility and Intensity for Structured Outlier Removal, or ReVISOR, and demonstrate its superior performance against existing baselines on realworld data.
6.3.11 Assessing craniofacial growth and form without landmarks: A new automatic approach based on spectral methods
In collaboration with Kevin Bloch (Institut Necker EnfantsMalades), Maxime Taverne (Institut Necker EnfantsMalades) , Simone Melzi (University of MilanoBicocca), Maya Geoffroy (Institut Necker EnfantsMalades), Roman Khonsari (Institut Necker EnfantsMalades)
In this paper 19 we present a novel method for the morphometric analysis of series of 3D shapes, and demonstrate its relevance for the detection and quantification of two craniofacial anomalies: trigonocephaly and metopic ridges, using CTscans of young children. Our approach is fully automatic, and does not rely on manual landmark placement and annotations. Our approach furthermore allows to differentiate shape classes, enabling successful differential diagnosis between trigonocephaly and metopic ridges, two related conditions characterized by triangular foreheads. These results were obtained using recent developments in automatic nonrigid 3D shape correspondence methods and specifically spectral approaches based on the functional map framework. Our method can capture local changes in geometric structure, in contrast to methods based, for instance, on global shape descriptors. As such, our approach allows to perform automatic shape classification and provides visual feedback on shape regions associated with different classes of deformations. The flexibility and generality of our approach paves the way for the application of spectral methods in quantitative medicine.
6.3.12 Affection: Learning Affective Explanations for RealWorld Visual Data
In collaboration with Panos Achlioptas (Snap Inc.), Leonidas Guibas (Stanford University), Sergey Tulyakov (Snap Inc.)
In this work 31, we explore the space of emotional reactions induced by realworld images. For this, we first introduce a largescale dataset that contains both categorical emotional reactions and freeform textual explanations for 85,007 publicly available images, analyzed by 6,283 annotators who were asked to indicate and explain how and why they felt when observing a particular image, with a total of 526,749 responses. Although emotional reactions are subjective and sensitive to context (personal mood, social status, past experiences)we show that there is significant common ground to capture emotional responses with a large support in the subject population. In light of this observation, we ask the following questions: i) Can we develop neural networks that provide plausible affective responses to realworld visual data explained with language? ii) Can we steer such methods towards producing explanations with varying degrees of pragmatic language, justifying different emotional reactions by grounding them in the visual stimulus? Finally, iii) How to evaluate the performance of such methods for this novel task? In this work, we take the first steps in addressing all of these questions, paving the way for more humancentric and emotionallyaware image analysis systems.
6.3.13 Generalizable Local Feature Pretraining for Deformable Shape Analysis
In this paper 22, we analyze the link between feature locality and transferability in tasks involving deformable 3D objects, while also comparing different backbones and losses for local feature pretraining. We observe that with proper training, learned features can be useful in such tasks, but, crucially, only with an appropriate choice of the receptive field size. We then propose a differentiable method for optimizing the receptive field within 3D transfer learning. Jointly, this leads to the first learnable features that can successfully generalize to unseen classes of 3D shapes such as humans and animals. Our extensive experiments show that this approach leads to stateoftheart results on several downstream tasks such as segmentation, shape correspondence, and classification.
6.3.14 Understanding and Improving Features Learned in Deep Functional Maps
In this paper 23, we show that under some mild conditions, the features learned within deep functional map approaches can be used as pointwise descriptors and thus are directly comparable across different shapes, even without the necessity of solving for a functional map at test time. Furthermore, informed by our analysis, we propose effective modifications to the standard deep functional map pipeline, which promote structural properties of learned features, significantly improving the matching results. Finally, we demonstrate that previously unsuccessful attempts at using extrinsic architectures for deep functional map feature extraction can be remedied via simple architectural changes, which encourage the theoretical properties suggested by our analysis. We thus bridge the gap between intrinsic and extrinsic surfacebased learning, suggesting the necessary and sufficient conditions for successful shape matching.
6.3.15 Scalable and Efficient Functional Map Computations on Dense Meshes
In this paper 20, we propose a new scalable version of the functional map pipeline that allows to efficiently compute correspondences between potentially very dense meshes. Unlike existing approaches that process dense meshes by relying on ad‐hoc mesh simplification, we establish an integrated end‐to‐end pipeline with theoretical approximation analysis. In particular, our method overcomes the computational burden of both computing the basis, as well the functional and pointwise correspondence computation by approximating the functional spaces and the functional map itself. Errors in the approximations are controlled by theoretical upper bounds assessing the range of applicability of our pipeline. With this construction in hand, we propose a scalable practical algorithm and demonstrate results on dense meshes, which approximate those obtained by standard functional map algorithms at the fraction of the computation time. Moreover, our approach outperforms the standard acceleration procedures by a large margin, leading to accurate results even in challenging cases.
7 Bilateral contracts and grants with industry
7.1 Bilateral contracts with industry
7.1.1 Contract with Sanofi Inc.

Title:
Machine learning approaches for cryoEM

Partner Institution(s):
Sanofi Inc.

Date/Duration:
20232024

Additionnal info/keywords:
Cryogenic electron microscopy (cryoEM) allows the structure determination of antibody fragments bound to pharmaceutically relevant targets to accelerate drug discovery. The process of cryoEM data analysis is time consuming and requires user input. To accelerate the rate of structure solution by cryoEM, this project investigates machine learning approachesto fit and model the atomic coordinates of antibody fragments into the cryoEM density.
The project funds one postdoctoral researcher for 2 years, jointly between Sanofi Inc., and Ecole Polytechnique (the employer of Maks Ovsjanikov).
7.1.2 Contract with DASSAULT SYSTEMES

Title:
Generative Models for the Guided Synthesis of Complex and Functional 3D Scenes

Partner Institution(s):
DASSAULT SYSTEMES

Date/Duration:
20232026

Additionnal info/keywords:
This thesis focuses on machine learning applied to 3D computer vision, specifically addressing challenges related to the automatic synthesis of 3D environments.
The project funds one PhD student for 3 years.
7.1.3 MEDITWIN with DASSAULT SYSTEMES

Title:
MEDITWIN: Virtual human twins for medical applications

Partner Institution(s):
DASSAULT SYSTEMES

Date/Duration:
20232028

Additionnal info/keywords:
In the context of IPCEI on Health called MEDITWIN, Geomerix has started working on geometric measure theory and reduced models (Desbrun) and nonrigid registration (Ovsjanikov), with one student and two postdocs to be hired soon.
8 Partnerships and cooperations
8.1 International research visitors
8.1.1 Visits of international scientists
Other international visits to the team
Shreyas Samaga

Status
PhD

Institution of origin:
Purdue University

Country:
USA

Dates:
MayJuly 2023

Context of the visit:
collaboration with Steve Oudot

Mobility program/type of mobility:
research stay
8.1.2 Visits to international teams
Research stays abroad
Steve Oudot

Visited institution:
Center for Advanced Study

Country:
Norway

Dates:
JanuaryFebruary 2023

Context of the visit:
participation in the program Representation Theory: Combinatorial Aspects and Applications

Mobility program/type of mobility:
research stay as a fellow
Pooran Memari

Visited institution:
Technische Universität Berlin

Country:
Germany

Dates:
FebruaryJune 2023

Context of the visit:
Sabbatical leave, Visiting TU Berlin CG group,

Mobility program/type of mobility:
research stay for collaboration initiation
8.2 European initiatives
8.2.1 Horizon Europe

Title:
Exploring Relations in Structured Data with Functional Maps

Partner Institution(s):
 European Research Commission (ERC) Starting Grant

Date/Duration:
20182023

Additionnal info/keywords:
We propose to lay the theoretical foundations and design efficient computational methods for analyzing, quantifying and exploring relations and variability in structured data sets, such as collections of geometric shapes, point clouds, and large networks or graphs, among others. In particular, we propose to depart from the standard representations of objects as collections of primitives, such as points or triangles, and instead to treat them as functional spaces that can be easily manipulated and analyzed. Since realvalued functions can be defined on a wide variety of data representations and as they enjoy a rich algebraic structure, such an approach can provide a completely novel unified framework for representing and processing different types of data. Key to our study is the exploration of relations and variability between objects, which can be expressed as operators acting on functions and thus treated and analyzed as objects in their own right using the vast number of tools from functional analysis in theory and numerical linear algebra in practice.

Title:
VEGA: Universal Geometric Transfer Learning

Partner Institution(s):
 European Research Commission (ERC) Consolidator Grant

Date/Duration:
20242028

Additionnal info/keywords:
In this project, we propose to develop a theoretical and practical framework for transfer learning with geometric 3D data. Most existing learningbased approaches, aimed at analyzing 3D data, are based on training neural networks from scratch for each data modality and application. Our main goal will be to develop universallyapplicable methods by combining powerful pretrainable modules with effective multiscale analysis and finetuning, given minimal taskspecific data. The overall key to our study will be analyzing rigorous ways, both theoretically and in practice, in which solutions can be transferred and adapted across problems, semantic categories and geometric data types.
8.2.2 H2020 projects

Title:
Creating Lively Interactive Populated Environments

Partner Institution(s):
 University of Cyprus, Universitat Politecnica de Catalunya, University College London, Trinity College Dublin, Max Planck Institute for Intelligent Systems, KTH Royal Institute of Technology.

Date/Duration:
20202024

Additionnal info/keywords:
This project designs new techniques to create and control interactive virtual worlds and characters, benefiting from opportunities open by the wide availability of emergent technologies in the domains of human digitization and artificial intelligence.
8.3 National initiatives
AEx PreMediT

Title:
Precision Medicine using Topology

Partner Institution(s):
 CRESS, HôtelDieu, France

Date/Duration:
20222025

Additionnal info/keywords:
While recent advances in machine learning are opening promising prospects for precision medicine, the sometimes small size, sparsity, or partly categorical nature of the data involved pose some crucial challenges. The goal of PreMediT is to address these challenges by integrating information about the geometric and topological structure of the data into the machine learning pipelines.
ANR AI Chair AIGRETTE

Title:
Analyzing Large Scale Geometric Data Collections

Partner Institution(s):
 ANR

Date/Duration:
20202024

Additionnal info/keywords:
Motivated by the deluge of 3D data using geometric representations (point clouds, triangle, quad meshes, graphs...) that are illsuited for modern applications, we are developing efficient algorithms and mathematical tools for analyzing diverse geometric data collections.
9 Dissemination
9.1 Promoting scientific activities
9.1.1 Scientific events: organisation
General chair, scientific chair
 Mathieu Desbrun organized the Workshop Machine Learning for Geometry at the Institut Poincaré (see https://ml4geo.sciencesconf.org/).
Member of the organizing committees
 Steve Oudot coorganized the MiniSymposium Recent Developments in MultiParameter Persistence at the SIAM Conference on Applied Algebraic Geometry (https://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=77710).
9.1.2 Scientific events: selection
Chair of conference program committees
 Pooran Memari served as Program CoChair of Eurographics Symposium on Geometry Processing (SGP) 2023.
Member of the conference program committees
 Pooran Memari was a member of the Program Committee of Eurographics 2023 Conference.
 Mathieu Desbrun was a member of the Program Committee of Technical Papers for ACM SIGGRAPH Asia 2023.
9.1.3 Journal
Member of the editorial boards
 Steve Oudot is a member of the Editorial Board of the Journal of Computational Geometry.
 Maks Ovsjanikov is a member of the Editorial Board of the IEEE Transactions on Visualization and Computer Graphics journal.
 Pooran Memari is a member of the Editorial Board of the Journal of Computer Graphics Forum (CGF).
 Pooran Memari is also a member of the Editorial Board of Graphical Models Journal, Elsevier.
 Mathieu Desbrun ss a member of the Editorial Board of the Journal of Geometric Mechanics.
9.1.4 Invited talks
 Maks Ovsjanikov gave a keynote talk at the Symposium on 3D Object Retrieval 2023 (3DOR'23) held in Lille, France.
 Maks Ovsjanikov gave a keynote talk at Pacific Graphics 2023, held in Daejeon, South Korea.
 Maks Ovsjanikov gave a keynote talk at the Machine Learning for Geometry Workshop, held in Paris, France.
 Steve Oudot gave an invited talk in the special session The ubiquity of quivers and their representations at the 29th Nordic Congress of Mathematicians, Aalborg, Denmark.
 Pooran Memari gave an invited talk at the LIB colloquium (Laboratoire d’Informatique de Bourgogne) in Dijon, France, Nov. 2023.
 Mathieu Desbrun gave an invited talk at Ecole Normale Supérieure, France, Nov. 2023.
 Mathieu Desbrun gave an invited talk for the “Journée scientifique du groupe SMAISIGMA” at Jussieu, France, Dec. 2023.
9.1.5 Leadership within the scientific community
 Pooran Memari is the local coordinator for the GTMG (Modélisation Géométrique).
9.1.6 Research administration
 Steve Oudot is vicepresident of the Commission Scientifique at Inria Saclay.
 Steve Oudot is a member of the Comité de Département in the CS Department of École Polytechnique (DIX).
 Pooran Memari is a member of the comité de Web du LIX, École Polytechnique.
 Pooran Memari is a deputy member of the conseil de laboratoire du LIX, École Polytechnique.
 Mathieu Desbrun is a member of the Comité du Labo in the CS lab of École Polytechnique (LIX)
9.2 Teaching  Supervision  Juries
9.2.1 Teaching
 Master: Steve Oudot, Computational Geometry and Topology, 18h eqTD, M2, MPRI;
 Master: Steve Oudot, Topological data analysis, 45h eqTD, M1, École polytechnique, France;
 Master: Mathieu Desbrun, Digital Representation and Analysis of Shapes, M2, École polytechnique, France;
 Master: Pooran Memari, Artificial Intelligence and Advanced Visual Computing, and Digital Representation and Analysis of Shapes, M2, École polytechnique, France;
 Master: Maks Ovsjanikov, Artificial Intelligence and Advanced Visual Computing, École polytechnique, France;
 UndergradMaster: Steve Oudot, Algorithms for data analysis in C++, 22.5h eqTD, L3/M1, École Polytechnique, France.
 MasterPhD: Pooran Memari is a member of the Jury d’admission Masters & PhD Track IGD (Interaction, Graphics & Design), IPParis (20202023).
9.2.2 Supervision
 PhD: Vadim Lebovici, Deux Approches Complémentaires de la Persistance Multiparamétrique: Décompositions en Intervalles et Fonctions Constructibles, Université ParisSaclay. Defended in September 2023. Steve Oudot and François Petit (CRESS).
 PhD: Jiayi Wei, Robust Statistics for Geometry Processing – Detecting and Handling Discontinuities and Dissimilarities in 3D Pointset Denoising and Mesh Parameterization. Defended in December 2023. Pooran Memari and Damien Rohmer.
 PhD in progress: Julie Mordacq, Analyse Topologique des Données et Apprentissage Machine pour analyser et prédire des transitions de phase en ndimensions, Institut Polytechnique de Paris. Started Sept. 2022. Steve Oudot and Vicky Kalogeiton (Vista, LIX).
 PhD in progress: Jingyi Li, Invariants algébriques effectifs pour la persistance multiparamètre, Institut Polytechnique de Paris. Started Nov. 2023. Steve Oudot.
 PhD: Mariem Mezghanni, Structural and Functional Learning for Industrial Design Automatization. Defended in February 2023. Institut Polytechnique de Paris. Maks Ovsjanikov with Malika Boulkenafed (Dassault Systèmes).
 PhD: Nicolas Donati, Robust representations for supervised and unsupervised 3D shape matching. Defended in January 2023. Institut Polytechnique de Paris. Maks Ovsjanikov with Etienne Corman (CNRS).
 PhD in progress: Souhaib Attaiki, 3D shape analysis with methods based on Deep Learning, Institut Polytechnique de Paris. Started Nov. 2020. Maks Ovsjanikov
 PhD in progress: Robin Magnet, Robust Spectral Methods for Shape Analysis and Deformation Assessment, Institut Polytechnique de Paris. Started February 2021. Maks Ovsjanikov
 PhD in progress: Souhail Hadgi, Transfer learning for 3D data, Institut Polytechnique de Paris. Started January 2023. Maks Ovsjanikov
 PhD in progress: Ramana S Sundararaman, Analysis of large scale 3D shape collection with learning based approaches, Institut Polytechnique de Paris. Started October 2021. Maks Ovsjanikov
 PhD in progress: Tim Scheller, Capturing 4D Plant Growth, Institut Polytechnique de Paris. Started October 2021. Maks Ovsjanikov with MariePaule Cani (Ecole Polytechnique).
 PhD in progress: Nissim Maruani, Machine Learning for Geometric Modeling; Started October 2022. Mathieu Desbrun amd Pierre Alliez (INRIA SophiaAntipolis).
 PhD in progress: Lucas Brifault, Geometric Measure Theory for Geometric Modeling; Started April 2022. Mathieu Desbrun and David CohenSteiner (INRIA SophiaAntipolis).
 PhD in progress: Theo Braune, Discrete Bundledvalued Exterior Calculus; Started October 2022. Mathieu Desbrun.
9.2.3 Juries
 Pooran Memari was an examiner for the Ph.D. defense of Guillaume Coiffier, LORIA, Université de Lorraine, Dec. 2023.
 Mathieu Desbrun was an examiner for the PhD defense of Loï Paulin, LIRIS, Lyon, Apr. 2023.
10 Scientific production
Major publications
 1 articlePredicting highresolution turbulence details in space and time.ACM Transactions on Graphics406December 2021, 200HALDOI
 2 articleMultiscale Cholesky Preconditioning for Illconditioned Problems.ACM Transactions on Graphics404July 2021, Art. 91HALDOI
 3 inproceedingsDeep Geometric Functional Maps: Robust Feature Learning for Shape Correspondence.CVPRSeattle (virtual), United StatesJune 2020HAL
 4 articleAccurate Synthesis of MultiClass Disk Distributions.Computer Graphics Forum3822019HAL
 5 articlePCPNET Learning Local Shape Properties from Raw Point Clouds.Computer Graphics Forum372018HAL
 6 inproceedingsLarge Scale computation of Means and Clusters for Persistence Diagrams using Optimal Transport.NIPSMontreal, Canada2018HAL
 7 articleA Framework for Differential Calculus on Persistence Barcodes.Foundations of Computational Mathematics2021HALDOI
 8 articleEfficient kinetic simulation of twophase flows.ACM Transactions on Graphics4142022, 114HALDOI
 9 articleHodgeoptimized triangulations.ACM Transactions on Graphics304August 2011, 103,112HAL
 10 bookPersistence Theory: From Quiver Representations to Data Analysis.Mathematical Surveys and Monographs209American Mathematical Society2015, 218HAL
 11 inproceedingsUnsupervised Deep Learning for Structured Shape Matching.ICCVSEOUL, South KoreaOctober 2019HAL
10.1 Publications of the year
International journals
 12 articleLocal characterizations for decomposability of 2parameter persistence modules.Algebras and Representation Theory2023HALDOIback to text
 13 articlePatternshop: Editing Point Patterns by Image Manipulation.ACM Transactions on Graphics424July 2023, 114/53HALDOIback to text
 14 articleReVISOR: ResUNets with visibility and intensity for structured outlier removal.ISPRS Journal of Photogrammetry and Remote Sensing2022023, 184204HALDOIback to text
 15 articleA Gradient Sampling Algorithm for Stratified Maps with Applications to Topological Data Analysis.Mathematical Programming2022023, 199–239HALDOIback to text
 16 articleFluidSolid Coupling in Kinetic TwoPhase Flow Simulation.ACM Transactions on Graphics424July 2023, 1  14HALDOIback to text
 17 articleHighOrder MomentEncoded Kinetic Simulation of Turbulent Flows.ACM Transactions on Graphics426December 2023HALDOIback to text
 18 articleBuilding a Virtual WeaklyCompressible Wind Tunnel Testing Facility.ACM Transactions on Graphics424August 2023, 120HALDOIback to text
 19 articleAssessing craniofacial growth and form without landmarks: A new automatic approach based on spectral methods.Journal of Morphology28482023, e21609HALDOIback to text
 20 articleScalable and Efficient Functional Map Computations on Dense Meshes.Computer Graphics Forum422May 2023, 89101HALDOIback to text
 21 articleRobust Pointset Denoising of PiecewiseSmooth Surfaces through Line Processes.Computer Graphics Forum4222023, 175189HALDOIback to text
International peerreviewed conferences
 22 inproceedingsGeneralizable Local Feature Pretraining for Deformable Shape Analysis.CVPR 2023  The IEEE/CVF Conference on Computer Vision and Pattern RecognitionVancouver, CanadaJune 2023HALback to text
 23 inproceedingsUnderstanding and Improving Features Learned in Deep Functional Maps.CVPR 2023  The IEEE/CVF Conference on Computer Vision and Pattern RecognitionVancouver, CanadaIEEEJune 2023HALDOIback to text
 24 inproceedingsSomigliana Coordinates: an elasticityderived approach for cage deformation.SIGGRAPH 2023  Special Interest Group on Computer Graphics and Interactive Techniques ConferenceLos Angeles, CA, United StatesACMJuly 2023, 18HALDOIback to text
 25 inproceedingsStable Vectorization of Multiparameter Persistent Homology using Signed Barcodes as Measures.NeurIPS 2023  36th Conference on Neural Information Processing SystemsAdvances in Neural Information Processing Systems 36New Orleans (LA), United StatesJune 2023HALback to text
 26 inproceedingsVoroMesh: Learning Watertight Surface Meshes with Voronoi Diagrams.ICCV 2023  International Conference on Computer VisionParis, FranceAugust 2023HALback to text
 27 inproceedingsFeatureSized Sampling for Vector Line Art.Pacific Graphics 2023  The 31th Pacific Conference on Computer Graphics and ApplicationsDaejeon, South Korea2023HALback to text
 28 inproceedingsBioSketch: A new medium for interactive storytelling illustrated by the phenomenon of infection.Eurographics Workshop on Visual Computing for Biology and MedicineVCBM 2023  13th Eurographics Workshop on Visual Computing for Biology and MedicineNorrköping, SwedenSeptember 2023, 11HALback to text
Conferences without proceedings
 29 inproceedingsZeroShot 3D Shape Correspondence.SIGGRAPH ASIA 2023  The 16th ACM SIGGRAPH Conference and Exhibition on Computer Graphics and Interactive TechniquesSydney, AustraliaDecember 2023HALDOIback to text
 30 inproceedingsSATR: ZeroShot Semantic Segmentation of 3D Shapes.ICCV 2023  International Conference on Computer VisionParis, FranceOctober 2023HALback to text
 31 inproceedingsAffection: Learning Affective Explanations for RealWorld Visual Data.CVPR 2023  IEEE/CVF Conference on Computer Vision and Pattern RecognitionVancouver, CanadaIEEEJune 2023HALDOIback to text
 32 inproceedingsShape Nonrigid Kinematics (SNK): A ZeroShot Method for NonRigid Shape Matching via Unsupervised Functional Map Regularized Reconstruction.NeurIPS 2023  37th Conference on Neural Information Processing SystemsNew Orleans (Louisiana), United StatesDecember 2023HALback to text
 33 inproceedingsTIDE: Time Derivative Diffusion for Deep Learning on Graphs.ICML 2023  The 40th annual International Conference on Machine LearningHonolulu, United StatesJuly 2023HALback to text
 34 inproceedingsSpatially and Spectrally Consistent Deep Functional Maps.ICCV 2023  International Conference on Computer VisionParis, FranceOctober 2023HALback to text
Doctoral dissertations and habilitation theses
 35 thesisTwo complementary approaches in multiparameter persistence : intervaldecompositions and constructible functions.Université ParisSaclaySeptember 2023HAL
Reports & preprints
10.2 Other
Cited publications
 38 articleMultiple correspondence analysis.Encyclopedia of measurement and statistics242007, 651657back to text
 39 articleReynoldsaveraged NavierStokes equations for turbulence modeling.Applied Mechanics Reviews6242009back to text
 40 articleDetecting coherent structures using braids.Physica D: Nonlinear Phenomena2412012, 95105back to text
 41 articleFinite element exterior calculus, homological techniques, and applications.Acta Numerica152006, 1–155back to text
 42 inproceedingsThe wave kernel signature: A quantum mechanical approach to shape analysis.2011 IEEE international conference on computer vision workshops (ICCV workshops)IEEE2011, 16261633back to text
 43 inproceedingsAn operator approach to tangent vector field processing.Computer Graphics Forum325Wiley Online Library2013, 7382back to text
 44 articleDiscrete derivatives of vector fields on surfacesan operator approach.ACM Trans. Graph.3432015, 113back to text
 45 articleLinking combinatorial and classical dynamics: Conley index and Morse decompositions.Foundations of Computational Mathematics2052020, 9671012back to text
 46 articleCotorsion torsion triples and the representation theory of filtered hierarchical clustering.Advances in Mathematics3692020, 107171back to text
 47 articleVariational Principles for Circle Patterns and Koebe's Theorem.Trans. Amer. Math. Soc.3562003, 659689back to text
 48 articleDelaunay triangulation of manifolds.Foundations of Computational Mathematics1822018, 399431back to text
 49 articleHermite regularization of the lattice Boltzmann method for open source computational aeroacoustics.Journal of the Acoustical Society of America14242017, 23322345back to text
 50 articleGeometric Deep Learning: Going beyond Euclidean data.IEEE Signal Processing Magazine3442017, 1842DOIback to text
 51 inproceedingsExposition and Interpretation of the Topology of Neural Networks.2019 18th IEEE International Conference On Machine Learning And Applications (ICMLA)2019, 10691076back to text
 52 articleGeometry of the ergodic quotient reveals coherent structures in flows.Physica D: Nonlinear Phenomena241152012, 12551269back to text
 53 articleCoalescent embedding in the hyperbolic space unsupervisedly discloses the hidden geometry of the brain.arXiv preprint arXiv:1705.041922017back to text
 54 articleThe Theory of Multidimensional Persistence.Discrete and Computational Geometry4212009, 7193back to text
 55 inproceedingsOptimizing persistent homology based functions.Proc. International Conference on Machine Learning2021back to text
 56 inproceedingsPersLay: a neural network layer for persistence diagrams and new graph topological signatures.International Conference on Artificial Intelligence and Statistics (PMLR)2020, 27862796back to text
 57 articleShapenet: An informationrich 3d model repository.arXiv preprint arXiv:1512.030122015back to text
 58 articlePersistenceBased Clustering in Riemannian Manifolds.\bf Journal of the ACM6062013, 138URL: http://doi.acm.org/10.1145/2535927DOIback to textback to text
 59 articleStatistical inference using the MorseSmale complex.Electronic Journal of Statistics1112017, 13901433back to text
 60 inproceedingsA topological regularizer for classifiers via persistent homology.The 22nd International Conference on Artificial Intelligence and Statistics (PMLR)2019, 25732582back to text
 61 articleA Reflectionless Discrete Perfectly Matched Layer.Journal of Computational Physics3812019, 91109back to text
 62 bookIsolated invariant sets and the Morse index.38American Mathematical Soc.1978back to text
 63 articleRecursive regularization step for highorder lattice Boltzmann methods.Phys. Rev. E9632017, 033306back to text
 64 articleTrivial Connections on Discrete Surfaces.Computer Graphics Forum (\emph{Symposium on Geometry Processing})2952010, 15251533back to text
 65 articleHow Many Directions Determine a Shape and other Sufficiency Results for Two Topological Transforms.arXiv preprint arXiv:1805.097822018back to text
 66 articleDesign and analysis of largescale biological rhythm studies: a comparison of algorithms for detecting periodic signals in biological data.Bioinformatics29242013, 31743180back to text
 67 incollectionDiscrete Differential Forms for Computational Modeling.Discrete Differential GeometrySpringer2007back to text
 68 article back to text
 69 inproceedingsPersistence of the Conley Index in Combinatorial Dynamical Systems.36th International Symposium on Computational Geometry (SoCG 2020)164Schloss DagstuhlLeibnizZentrum für Informatik2020, 37:137:17back to text
 70 inproceedingsTopological data analysis for arrhythmia detection through modular neural networks.Canadian Conference on Artificial IntelligenceSpringer2020, 177188back to text
 71 inproceedingsDeep geometric functional maps: Robust feature learning for shape correspondence.Proceedings of the IEEE/CVF conference on Computer Vision and Pattern Recognition2020, 85928601back to text
 72 inproceedingsAccurate synthesis of multiclass disk distributions.Computer Graphics Forum38(2)Wiley Online Library2019, 157168back to text
 73 articleCombinatorial vector fields and dynamical systems.Mathematische Zeitschrift22841998, 629681back to text
 74 bookThe Geometry of Physics: An Introduction.Cambridge University Press2011back to text
 75 inproceedingsRecognizing Objects in Range Data Using Regional Point Descriptors.European Conference on Computer VisionSpringer Berlin Heidelberg2004, 224237back to text
 76 inproceedingsAlmostInvariant and FiniteTime Coherent Sets: Directionality, Duration, and Diffusion.Ergodic Theory, Open Dynamics, and Coherent StructuresSpringer New York2014, 171216back to text
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