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We prove that Schubert varieties in flag manifolds are uniquely determined by their equivariant cohomology classes, as well as a stronger result that replaces Schubert varieties with closures of Bialynicki-Birula cells under suitable conditions. This is used to prove that any two-pointed curve neighborhood representing a quantum cohomology product with a Seidel class is a Schubert variety.

A homology class $d \in H_2(X)$ of a complex flag variety $X = G/P$ is called a line degree if the moduli space $\overline{M}_{0,0}(X,d)$ of 0-pointed stable maps to $X$ of degree $d$ is also a flag variety $G/P'$. We prove a quantum equals classical formula stating that any $n$-pointed (equivariant, K-theoretic, genus zero) Gromov-Witten invariant of line degree on $X$ is equal to a classical intersection number computed on the flag variety $G/P'$. We also prove an $n$-pointed analogue of the Peterson comparison formula stating that these invariants coincide with Gromov-Witten invariants of the variety of complete flags $G/B$. Our formulas make it straightforward to compute the big quantum K-theory ring of $X$ modulo degrees larger than line degrees., Comment: To appear in the proceedings of the conference GLSM@30 (Stony Brook, May 2023)

We prove a collection of formulas for products of Schubert classes in the quantum $K$-theory ring $QK(X)$ of a cominuscule flag variety $X$. This includes a $K$-theory version of the Seidel representation, stating that the quantum product of a Seidel class with an arbitrary Schubert class is equal to a single Schubert class times a power of the deformation parameter $q$. We also prove new Pieri formulas for the quantum $K$-theory of maximal orthogonal Grassmannians and Lagrangian Grassmannians, and give a new proof of the known Pieri formula for the quantum $K$-theory of Grassmannians of type A. Our formulas have simple statements in terms of quantum shapes that represent the natural basis elements $q^d[{\mathcal O}_{X^u}]$ of $QK(X)$. Along the way we give a simple formula for $K$-theoretic Gromov-Witten invariants of Pieri type for Lagrangian Grassmannians, and prove a rationality result for the points in a Richardson variety in a symplectic Grassmannian that are perpendicular to a point in projective space.

The estimation of 6D poses of rigid objects is a fundamental problem in computer vision. Traditionally pose estimation is concerned with the determination of a single best estimate. However, a single estimate is unable to express visual ambiguity, which in many cases is unavoidable due to object symmetries or occlusion of identifying features. Inability to account for ambiguities in pose can lead to failure in subsequent methods, which is unacceptable when the cost of failure is high. Estimates of full pose distributions are, contrary to single estimates, well suited for expressing uncertainty on pose. Motivated by this, we propose a novel pose distribution estimation method. An implicit formulation of the probability distribution over object pose is derived from an intermediary representation of an object as a set of keypoints. This ensures that the pose distribution estimates have a high level of interpretability. Furthermore, our method is based on conservative approximations, which leads to reliable estimates. The method has been evaluated on the task of rotation distribution estimation on the YCB-V and T-LESS datasets and performs reliably on all objects., Comment: 11 pages, 2 figures

Many industrial assembly tasks involve peg-in-hole like insertions with sub-millimeter tolerances which are challenging, even in highly calibrated robot cells. Visual servoing can be employed to increase the robustness towards uncertainties in the system, however, state of the art methods either rely on accurate 3D models for synthetic renderings or manual involvement in acquisition of training data. We present a novel self-supervised visual servoing method for high precision peg-in-hole insertion, which is fully automated and does not rely on synthetic data. We demonstrate its applicability for insertion of electronic components into a printed circuit board with tight tolerances. We show that peg-in-hole insertion can be drastically sped up by preceding a robust but slow force-based insertion strategy with our proposed visual servoing method, the configuration of which is fully autonomous., Comment: Accepted at IEEE CASE 2022

Fast, robust, and flexible part feeding is essential for enabling automation of low volume, high variance assembly tasks. An actuated vision-based solution on a traditional vibratory feeder, referred to here as a vision trap, should in principle be able to meet these demands for a wide range of parts. However, in practice, the flexibility of such a trap is limited as an expert is needed to both identify manageable tasks and to configure the vision system. We propose a novel approach to vision trap design in which the identification of manageable tasks is automatic and the configuration of these tasks can be delegated to an automated feeder design system. We show that the trap's capabilities can be formalized in such a way that it integrates seamlessly into the ecosystem of automated feeder design. Our results on six canonical parts show great promise for autonomous configuration of feeder systems., Comment: IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS 2022)

We prove that the Schubert structure constants of the quantum $K$-theory ring of any minuscule flag variety or quadric hypersurface have signs that alternate with codimension. We also prove that the powers of the deformation parameter $q$ that occur in the product of two Schubert classes in the quantum cohomology or quantum $K$-theory ring of a cominuscule flag variety form an integer interval. Our proofs are based on several new results, including an explicit description of the most general non-empty intersection of two Schubert varieties in an arbitrary flag manifold, and a computation of the cohomology groups of any negative line bundle restricted to a Richardson variety in a cominuscule flag variety. We also give a type-uniform proof of the quantum-to-classical theorem, which asserts that the (3-point, genus 0) Gromov-Witten invariants of any cominuscule flag variety are classical triple-intersection numbers on an associated flag variety. Finally, we prove several new results about the geometry and combinatorics related to this theorem.

Pose estimation is the task of determining the 6D position of an object in a scene. Pose estimation aid the abilities and flexibility of robotic set-ups. However, the system must be configured towards the use case to perform adequately. This configuration is time-consuming and limits the usability of pose estimation and, thereby, robotic systems. Deep learning is a method to overcome this configuration procedure by learning parameters directly from the dataset. However, obtaining this training data can also be very time-consuming. The use of synthetic training data avoids this data collection problem, but a configuration of the training procedure is necessary to overcome the domain gap problem. Additionally, the pose estimation parameters also need to be configured. This configuration is jokingly known as grad student descent as parameters are manually adjusted until satisfactory results are obtained. This paper presents a method for automatic configuration using only synthetic data. This is accomplished by learning the domain randomization during network training, and then using the domain randomization to optimize the pose estimation parameters. The developed approach shows state-of-the-art performance of 82.0 % recall on the challenging OCCLUSION dataset, outperforming all previous methods with a large margin. These results prove the validity of automatic set-up of pose estimation using purely synthetic data., Comment: 8 pages, 4 figures, 6 tables

For a nonsingular projective variety $X$, the virtual Tevelev degree in Gromov-Witten theory is defined as the virtual degree of the morphism from $M_{g,n}(X,d)$ to the product $M_{g,n} \times X^n$. After proving a simple formula for the virtual Tevelev degree in the (small) quantum cohomology ring of $X$ using the quantum Euler class, we provide several exact calculations for flag varieties and complete intersections. In the cominuscule case (including Grassmannians, Lagrangian Grassmannians, and maximal orthogonal Grassmannians), the virtual Tevelev degrees are calculated in terms of the eigenvalues of an associated self-adjoint linear endomorphism of the quantum cohomology ring. For complete intersections of low degree (compared to dimension), we prove a product formula. The calculation for complete intersections involves the primitive cohomology. Virtual Tevelev degrees are better behaved than arbitrary Gromov-Witten invariants, and, by recent results of Lian and Pandharipande, are much more likely to be enumerative.

We present an approach to learn dense, continuous 2D-3D correspondence distributions over the surface of objects from data with no prior knowledge of visual ambiguities like symmetry. We also present a new method for 6D pose estimation of rigid objects using the learnt distributions to sample, score and refine pose hypotheses. The correspondence distributions are learnt with a contrastive loss, represented in object-specific latent spaces by an encoder-decoder query model and a small fully connected key model. Our method is unsupervised with respect to visual ambiguities, yet we show that the query- and key models learn to represent accurate multi-modal surface distributions. Our pose estimation method improves the state-of-the-art significantly on the comprehensive BOP Challenge, trained purely on synthetic data, even compared with methods trained on real data. The project site is at https://surfemb.github.io/ .

Since the introduction of modern deep learning methods for object pose estimation, test accuracy and efficiency has increased significantly. For training, however, large amounts of annotated training data are required for good performance. While the use of synthetic training data prevents the need for manual annotation, there is currently a large performance gap between methods trained on real and synthetic data. This paper introduces a new method, which bridges this gap. Most methods trained on synthetic data use 2D images, as domain randomization in 2D is more developed. To obtain precise poses, many of these methods perform a final refinement using 3D data. Our method integrates the 3D data into the network to increase the accuracy of the pose estimation. To allow for domain randomization in 3D, a sensor-based data augmentation has been developed. Additionally, we introduce the SparseEdge feature, which uses a wider search space during point cloud propagation to avoid relying on specific features without increasing run-time. Experiments on three large pose estimation benchmarks show that the presented method outperforms previous methods trained on synthetic data and achieves comparable results to existing methods trained on real data., Comment: 10 pages, 5 figures, 7 tables

This paper demonstrates a visual servoing method which is robust towards uncertainties related to system calibration and grasping, while significantly reducing the peg-in-hole time compared to classical methods and recent attempts based on deep learning. The proposed visual servoing method is based on peg and hole point estimates from a deep neural network in a multi-cam setup, where the model is trained on purely synthetic data. Empirical results show that the learnt model generalizes to the real world, allowing for higher success rates and lower cycle times than existing approaches.

We discover a family of surfaces of general type with $K^2=3$ and $p=q=0$ as free $C_{13}$ quotients of special linear cuts of the octonionic projective plane $\mathbb O \mathbb P^2$. A special member of the family has $3$ singularities of type $A_2$, and is a quotient of a fake projective plane. We use the techniques of \cite{BF20} to define this fake projective plane by explicit equations in its bicanonical embedding., Comment: 6 pages + ancillary files

We present a learning-based method for 6 DoF pose estimation of rigid objects in point cloud data. Many recent learning-based approaches use primarily RGB information for detecting objects, in some cases with an added refinement step using depth data. Our method consumes unordered point sets with/without RGB information, from initial detection to the final transformation estimation stage. This allows us to achieve accurate pose estimates, in some cases surpassing state of the art methods trained on the same data., Comment: 5 pages

We prove that if X is a Grassmannian of type A, then the Schubert basis of the (small) quantum cohomology ring QH(X) is the only homogeneous deformation of the Schubert basis of the ordinary cohomology ring of X that multiplies with non-negative structure constants. This implies that the (three point, genus zero) Gromov-Witten invariants of X are uniquely determined by Witten's presentation of QH(X) and the fact that they are non-negative. We conjecture that the same is true for any flag variety X = G/P of simply laced Lie type. For the variety GL(n)/B of complete flags, this conjecture is equivalent to Fomin, Gelfand, and Postnikov's conjecture that the quantum Schubert polynomials of type A are uniquely determined by positivity properties. Our proof for Grassmannians answers a question of Fulton., Comment: 16 pages

We prove that the sheaf Euler characteristic of the product of a Schubert class and an opposite Schubert class in the quantum $K$-theory ring of a (generalized) flag variety $G/P$ is equal to $q^d$, where $d$ is the smallest degree of a rational curve joining the two Schubert varieties. This implies that the sum of the structure constants of any product of Schubert classes is equal to 1. Along the way, we provide a description of the smallest degree $d$ in terms of its projections to flag varieties defined by maximal parabolic subgroups., Comment: 10 pages

We propose a benchmark for 6D pose estimation of a rigid object from a single RGB-D input image. The training data consists of a texture-mapped 3D object model or images of the object in known 6D poses. The benchmark comprises of: i) eight datasets in a unified format that cover different practical scenarios, including two new datasets focusing on varying lighting conditions, ii) an evaluation methodology with a pose-error function that deals with pose ambiguities, iii) a comprehensive evaluation of 15 diverse recent methods that captures the status quo of the field, and iv) an online evaluation system that is open for continuous submission of new results. The evaluation shows that methods based on point-pair features currently perform best, outperforming template matching methods, learning-based methods and methods based on 3D local features. The project website is available at bop.felk.cvut.cz., Comment: ECCV 2018

It is possible to associate a highly constrained subset of relative 6 DoF poses between two 3D shapes, as long as the local surface orientation, the normal vector, is available at every surface point. Local shape features can be used to find putative point correspondences between the models due to their ability to handle noisy and incomplete data. However, this correspondence set is usually contaminated by outliers in practical scenarios, which has led to many past contributions based on robust detectors such as the Hough transform or RANSAC. The key insight of our work is that a single correspondence between oriented points on the two models is constrained to cast votes in a 1 DoF rotational subgroup of the full group of poses, SE(3). Kernel density estimation allows combining the set of votes efficiently to determine a full 6 DoF candidate pose between the models. This modal pose with the highest density is stable under challenging conditions, such as noise, clutter, and occlusions, and provides the output estimate of our method. We first analyze the robustness of our method in relation to noise and show that it handles high outlier rates much better than RANSAC for the task of 6 DoF pose estimation. We then apply our method to four state of the art data sets for 3D object recognition that contain occluded and cluttered scenes. Our method achieves perfect recall on two LIDAR data sets and outperforms competing methods on two RGB-D data sets, thus setting a new standard for general 3D object recognition using point cloud data., Comment: Accepted for International Conference on Computer Vision (ICCV), 2017

We present a method for finding correspondence between 3D models. From an initial set of feature correspondences, our method uses a fast voting scheme to separate the inliers from the outliers. The novelty of our method lies in the use of a combination of local and global constraints to determine if a vote should be cast. On a local scale, we use simple, low-level geometric invariants. On a global scale, we apply covariant constraints for finding compatible correspondences. We guide the sampling for collecting voters by downward dependencies on previous voting stages. All of this together results in an accurate matching procedure. We evaluate our algorithm by controlled and comparative testing on different datasets, giving superior performance compared to state of the art methods. In a final experiment, we apply our method for 3D object detection, showing potential use of our method within higher-level vision.

We address the problem of estimating the alignment pose between two models using structure-specific local descriptors. Our descriptors are generated using a combination of 2D image data and 3D contextual shape data, resulting in a set of semi-local descriptors containing rich appearance and shape information for both edge and texture structures. This is achieved by defining feature space relations which describe the neighborhood of a descriptor. By quantitative evaluations, we show that our descriptors provide high discriminative power compared to state of the art approaches. In addition, we show how to utilize this for the estimation of the alignment pose between two point sets. We present experiments both in controlled and real-life scenarios to validate our approach.

We prove an identity relating the product of two opposite Schubert varieties in the (equivariant) quantum K-theory ring of a cominuscule flag variety to the minimal degree of a rational curve connecting the Schubert varieties. We deduce that the sum of the structure constants associated to any product of Schubert classes is equal to one. Equivalently, the sheaf Euler characteristic map extends to a ring homomorphism defined on the quantum K-theory ring., Comment: 5 pages

We prove a type-uniform Chevalley formula for multiplication with divisor classes in the equivariant quantum $K$-theory ring of any cominuscule flag variety $G/P$. We also prove that multiplication with divisor classes determines the equivariant quantum $K$-theory of arbitrary flag varieties. These results prove a conjecture of Gorbounov and Korff concerning the equivariant quantum $K$-theory of Grassmannians of Lie type A., Comment: version 2: 28 pages; few examples, and a simplified Chevalley formula in the minuscule case, are added

We introduce a mutation algorithm for puzzles that is a three-direction analogue of the classical jeu de taquin algorithm for semistandard tableaux. We apply this algorithm to prove our conjectured puzzle formula for the equivariant Schubert structure constants of two-step flag varieties. This formula gives an expression for the structure constants that is positive in the sense of Graham. Thanks to the equivariant version of the `quantum equals classical' result, our formula specializes to a Littlewood-Richardson rule for the equivariant quantum cohomology of Grassmannians., Comment: In this version illegal puzzle pieces have been renamed to temporary puzzle pieces. Conjecture 4.7 has been replaced with a counterexample. This is the final version to appear in Annals of Mathematics

We prove a conjecture of Knutson asserting that the Schubert structure constants of the cohomology ring of a two-step flag variety are equal to the number of puzzles with specified border labels that can be created using a list of eight puzzle pieces. As a consequence, we obtain a puzzle formula for the Gromov-Witten invariants defining the small quantum cohomology ring of a Grassmann variety of type A. The proof of the conjecture proceeds by showing that the puzzle formula defines an associative product on the cohomology ring of the two-step flag variety. It is based on an explicit bijection of gashed puzzles that is analogous to the jeu de taquin algorithm but more complicated., Comment: Final Version. 32 pages; 381 figures (best viewed in color)

A projected Gromov-Witten variety is the union of all rational curves of fixed degree that meet two opposite Schubert varieties in a homogeneous space X = G/P. When X is cominuscule we prove that the map from a related Gromov-Witten variety is cohomologically trivial. This implies that all (3 point, genus zero) K-theoretic Gromov-Witten invariants of X are determined by the projected Gromov-Witten varieties, which extends an earlier result of Knutson, Lam, and Speyer. Our proof uses that any projected Gromov-Witten variety in a cominuscule space is also a projected Richardson variety., Comment: 13 pages

Based on Thomas and Yong's K-theoretic jeu de taquin algorithm, we prove a uniform Littlewood-Richardson rule for the K-theoretic Schubert structure constants of all minuscule homogeneous spaces. Our formula is new in all types. For the main examples of Grassmannians of type A and maximal orthogonal Grassmannians it has the advantage that the tableaux to be counted can be recognized without reference to the jeu de taquin algorithm., Comment: 37 pages

Let X be any generalized flag variety with Picard group of rank one. Given a degree d, consider the Gromov-Witten variety of rational curves of degree d in X that meet three general points. We prove that, if this Gromov-Witten variety is rationally connected for all large degrees d, then the structure constants of the small quantum K-theory ring of X vanish for large degrees., Comment: 6 pages

A previous result of the authors with Chaput and Perrin states that the union of all rational curves of fixed degree passing through a Schubert variety in a homogeneous space G/P is again a Schubert variety. In this paper we identify this Schubert variety explicitly in terms of the Hecke product of Weyl group elements. We apply our result to give an explicit formula for any two-point Gromov-Witten invariant as well as a new proof of the quantum Chevalley formula and its equivariant generalization. We also recover a formula for the minimal degree of a rational curve between two given points in a cominuscule variety., Comment: 24 pages, 1 figure, 1 table

Let X be an orthogonal Grassmannian parametrizing isotropic subspaces in an even dimensional vector space equipped with a nondegenerate symmetric form. We prove a Giambelli formula which expresses an arbitrary Schubert class in the singular and quantum cohomology ring of X as a polynomial in certain special Schubert classes. Our analysis reveals a surprising relation between the Schubert calculus on even and odd orthogonal Grassmannians. We also study eta polynomials, a family of polynomials defined using raising operators whose algebra agrees with the Schubert calculus on X., Comment: 28 pages; added appendix gives geometric description of Schubert varieties, correcting errors in arXiv:0809.4966

The product of two Schubert classes in the quantum K-theory ring of a homogeneous space X = G/P is a formal power series with coefficients in the Grothendieck ring of algebraic vector bundles on X. We show that if X is cominuscule, then this power series has only finitely many non-zero terms. The proof is based on a geometric study of boundary Gromov-Witten varieties in the Kontsevich moduli space, consisting of stable maps to X that take the marked points to general Schubert varieties and whose domains are reducible curves of genus zero. We show that all such varieties have rational singularities, and that boundary Gromov-Witten varieties defined by two Schubert varieties are either empty or unirational. We also prove a relative Kleiman-Bertini theorem for rational singularities, which is of independent interest. A key result is that when X is cominuscule, all boundary Gromov-Witten varieties defined by three single points in X are rationally connected., Comment: 16 pages; proofs slightly improved; explicit multiplications in QK(Cayley plane) from v1 no longer necessary

We prove Pieri formulas for the multiplication with special Schubert classes in the K-theory of all cominuscule Grassmannians. For Grassmannians of type A this gives a new proof of a formula of Lenart. Our formula is new for Lagrangian Grassmannians, and for orthogonal Grassmannians it proves a special case of a conjectural Littlewood-Richardson rule of Thomas and Yong. Recent work of Thomas and Yong and of E. Clifford has shown that the full Littlewood-Richardson rule for orthogonal Grassmannians follows from the Pieri case proved here. We describe the K-theoretic Pieri coefficients both as integers determined by positive recursive identities and as the number of certain tableaux. The proof is based on a computation of the sheaf Euler characteristic of triple intersections of Schubert varieties, where at least one Schubert variety is special.

Let X be a symplectic or odd orthogonal Grassmannian which parametrizes isotropic subspaces in a vector space equipped with a nondegenerate (skew) symmetric form. We prove quantum Giambelli formulas which express an arbitrary Schubert class in the small quantum cohomology ring of X as a polynomial in certain special Schubert classes, extending the cohomological Giambelli formulas of arXiv:0811.2781., Comment: 11 pages