arXiv subjects
Publications and source records attributed to Jinhyung Park.
Using Boij-Söderberg theory, we give a quick new approach to the results of Han-Kwak and Ahn-Han-Kwak on upper bounds for graded betti numbers of projective schemes in the first nontrivial strand.
Let $C$ be a smooth projective complex curve of genus $g$ and gonality $k$, and $L$ be a very ample line bundle on $C$. When $L$ has sufficiently large degree, the vanishing and nonvanishing of the Koszul cohomology groups $K_{p,q}(C,L)$ have been determined previously, but the exact values of the graded Betti numbers $κ_{p,q}(C, L)$ remain largely unknown. In this paper, we give explicit closed formulas for all graded Betti numbers $κ_{p,q}(C, L)$ when the Brill--Noether locus $W_k^1(C)$ has the expected dimension and $H^1(C, L \otimes ω_C^{-1})=0$. Consequently, we determine the complete Betti table for a general curve when $°L \geq 4g-3$ or when $°L \geq 3g-3$ and $L$ is general. We also explicitly compute the Boij--Söderberg coefficient of the section ring $R(C, L)$ governing asymptotic purity, and show eventual monotonicity of the remaining coefficients: they decrease for hyperelliptic curves and increase under a natural generic reducedness assumption on the relevant Brill--Noether loci.
Despite the rise of billion-parameter foundation models trained across thousands of graphical processing units (GPUs), similar scaling gains have not been shown for humanoid control. Current neural controllers for humanoids remain modest in size, target a limited set of behaviors, and are trained on a handful of GPUs. We show that scaling model capacity, data, and compute yields a generalist humanoid controller capable of natural, robust whole-body movements. We position motion tracking as a scalable task for humanoid control, leveraging dense supervision from diverse motion-capture data to acquire human motion priors without manual reward engineering. We build a foundation model for motion tracking by scaling along three axes: network size (1.2M to 42M parameters), dataset volume (100M+ frames from 700 hours of motion capture), and compute (21k GPU hours). Beyond demonstrating the benefits of scale, we further show downstream utility through a real-time kinematic planner that bridges motion tracking to tasks such as navigation, enabling natural and interactive control, as well as a unified token space that supports virtual reality (VR) teleoperation and vision-language-action (VLA) models with a single policy. Through this interface, we demonstrate autonomous VLA-driven whole-body loco-manipulation requiring coordinated hand and foot placement. Scaling motion tracking exhibits favorable properties: performance improves steadily with compute and data diversity, and learned policies generalize to unseen motions, establishing motion tracking at scale as a practical foundation for humanoid control.
We show that higher secant varieties of smooth projective curves have symmetric admissible determinantal representations with symmetric Ulrich sheaves of rank one if the embedding is sufficiently ample. For secant varieties of real curves, we give conditions for the representing matrices to be positive definite. This allows us, under mild (conjecturally vacuous) conditions, to represent the convex hull of the curve as a spectrahedron whenever it is a hyperbolicity cone. For secant varieties of rational normal curves, we derive very explicit representations in terms of Littlewood--Richardson coefficients. One key tool that we use are the higher Szegő kernels and the higher Scorza correspondences associated to a non-effective theta characteristic on the curve.
In joint work with Lacini and Sheridan, we proved that the first and second secant varieties of a smooth projective complex variety embedded by the complete linear system of a sufficiently positive line bundle are projectively normal. The purpose of this paper is to establish effective results on how positive the embedding line bundle must be for this result to hold. We also provide effective conditions under which the defining ideal of the first secant variety is generated by cubics, and furthermore, generated by $3 \times 3$-minors of a matrix of linear forms. The latter result gives an effective version of a theorem of Agostini and the second author.
Scaling humanoid loco-manipulation requires robot-compatible demonstrations across diverse objects, whole-body motions, and scene geometries, but teleoperation and motion capture are difficult to scale because each collection depends on physical setups, instrumented actors, and robot operation. We present GRAIL, a digital generation pipeline that remains fully virtual until deployment: it composes 3D assets, simulator-ready scenes, and priors from video foundation models (VFMs) to synthesize interactions without rebuilding physical environments or teleoperating the robot. Rather than reconstructing unconstrained in-the-wild videos, GRAIL starts from fully specified 3D configurations in which object geometry, camera parameters, metric scale, environment depth, and a robot-proportioned character are known before video generation and reused during reconstruction. This privileged setup better conditions 4D recovery, allowing model-based object tracking, human motion estimation, and interaction-aware optimization to reconstruct metric 4D human-object interaction (HOI) trajectories with reduced depth ambiguity and morphology mismatch. We retarget the recovered motions to a humanoid robot and train complementary task-general trackers: an object-aware latent adaptor for manipulation and a scene-aware tracker for terrain traversal. GRAIL produces over 20,000 sequences spanning pick-up, object manipulation, sitting, and terrain traversal. Using only GRAIL-generated data, we train egocentric visual policies through a sim-to-real pipeline and deploy them on a Unitree G1 humanoid, achieving 84\% real-world success on diverse object pick-up and 90\% success on stair-climbing.
In this paper, we give a description of the cohomology groups of the symmetric powers of the tautological bundle associated with a sufficiently positive line bundle on the Hilbert scheme of 2 or 3 points on a smooth projective complex variety. The dimensions of these cohomology groups are expressed in terms of the Hilbert polynomials of the first and second secant varieties of the embedding given by the sufficiently positive line bundle. We also compute these Hilbert polynomials completely.
The main goal of this paper is to study some local and global properties of secant varieties of algebraic curves. These results complement our previous work [8] by addressing issues given therein and providing solutions to problems raised subsequently. Specifically, we show a description of tangent cones of secant varieties of curves, and compute the cohomology groups of secant sheaves on symmetric products of curves, which answers a question posed in [8] and leads to a recursive formula for Hilbert polynomials of secant varieties of curves. In the appendix, we present a cohomological approach to arithmetical Cohen--Macaulayness of secant varieties of curves, completing the proof in [8].
Vision Transformers (ViTs) partition input images into uniformly sized patches regardless of their content, resulting in long input sequence lengths for high-resolution images. We present Adaptive Patch Transformers (APT), which addresses this by using multiple different patch sizes within the same image. APT reduces the total number of input tokens by allocating larger patch sizes in more homogeneous areas and smaller patches in more complex ones. APT achieves a drastic speedup in ViT inference and training, increasing throughput by 40% on ViT-L and 50% on ViT-H while maintaining downstream performance, and can be applied to a previously fine-tuned ViT, converging in as little as 1 epoch. It also significantly reduces training and inference time without loss of performance in high-resolution dense visual tasks, achieving up to 30\% faster training and inference in visual QA, object detection, and semantic segmentation.
We present DuoMo, a generative method that recovers human motion in world-space coordinates from unconstrained videos with noisy or incomplete observations. Reconstructing such motion requires solving a fundamental trade-off: generalizing from diverse and noisy video inputs while maintaining global motion consistency. Our approach addresses this problem by factorizing motion learning into two diffusion models. The camera-space model first estimates motion from videos in camera coordinates. The world-space model then lifts this initial estimate into world coordinates and refines it to be globally consistent. Together, the two models can reconstruct motion across diverse scenes and trajectories, even from highly noisy or incomplete observations. Moreover, our formulation is general, generating the motion of mesh vertices directly and bypassing parametric models. DuoMo achieves state-of-the-art performance. On EMDB, our method obtains a 16% reduction in world-space reconstruction error while maintaining low foot skating. On RICH, it obtains a 30% reduction in world-space error. Project page: https://yufu-wang.github.io/duomo/
We introduce SAM 3D Body (3DB), a promptable model for single-image full-body 3D human mesh recovery (HMR) that demonstrates state-of-the-art performance, with strong generalization and consistent accuracy in diverse in-the-wild conditions. 3DB estimates the human pose of the body, feet, and hands. It is the first model to use a new parametric mesh representation, Momentum Human Rig (MHR), which decouples skeletal structure and surface shape. 3DB employs an encoder-decoder architecture and supports auxiliary prompts, including 2D keypoints and masks, enabling user-guided inference similar to the SAM family of models. We derive high-quality annotations from a multi-stage annotation pipeline that uses various combinations of manual keypoint annotation, differentiable optimization, multi-view geometry, and dense keypoint detection. Our data engine efficiently selects and processes data to ensure data diversity, collecting unusual poses and rare imaging conditions. We present a new evaluation dataset organized by pose and appearance categories, enabling nuanced analysis of model behavior. Our experiments demonstrate superior generalization and substantial improvements over prior methods in both qualitative user preference studies and traditional quantitative analysis. Both 3DB and MHR are open-source.
We present MHR, a parametric human body model that combines the decoupled skeleton/shape paradigm of ATLAS with a flexible, modern rig and pose corrective system inspired by the Momentum library. Our model enables expressive, anatomically plausible human animation, supporting non-linear pose correctives, and is designed for robust integration in AR/VR and graphics pipelines.
Using Hilbert schemes of points, we establish a number of results for a smooth projective variety $X$ in a sufficiently ample embedding. If $X$ is a curve or a surface, we show that the ideals of higher secant varieties are determinantally presented, and we prove the same for the first secant variety if $X$ has arbitrary dimension. This completely settles a conjecture of Eisenbud-Koh-Stillman for curves and partially resolves a conjecture of Sidman-Smith in higher dimensions. If $X$ is a curve or a surface we also prove that the corresponding embedding of the Hilbert scheme of points $X^{[d]}$ into the Grassmannian is projectively normal. Finally, if $X$ is an arbitrary projective scheme in a sufficiently ample embedding, then we demonstrate that its homogeneous ideal is generated by quadrics of rank three, confirming a conjecture of Han-Lee-Moon-Park. Along the way, we check that the Hilbert scheme of three points on a smooth variety is the blow-up of the symmetric product along the big diagonal.
Parametric body models offer expressive 3D representation of humans across a wide range of poses, shapes, and facial expressions, typically derived by learning a basis over registered 3D meshes. However, existing human mesh modeling approaches struggle to capture detailed variations across diverse body poses and shapes, largely due to limited training data diversity and restrictive modeling assumptions. Moreover, the common paradigm first optimizes the external body surface using a linear basis, then regresses internal skeletal joints from surface vertices. This approach introduces problematic dependencies between internal skeleton and outer soft tissue, limiting direct control over body height and bone lengths. To address these issues, we present ATLAS, a high-fidelity body model learned from 600k high-resolution scans captured using 240 synchronized cameras. Unlike previous methods, we explicitly decouple the shape and skeleton bases by grounding our mesh representation in the human skeleton. This decoupling enables enhanced shape expressivity, fine-grained customization of body attributes, and keypoint fitting independent of external soft-tissue characteristics. ATLAS outperforms existing methods by fitting unseen subjects in diverse poses more accurately, and quantitative evaluations show that our non-linear pose correctives more effectively capture complex poses compared to linear models.
Despite the demonstrated efficiency and performance of sparse query-based representations for perception, state-of-the-art 3D occupancy prediction methods still rely on voxel-based or dense Gaussian-based 3D representations. However, dense representations are slow, and they lack flexibility in capturing the temporal dynamics of driving scenes. Distinct from prior work, we instead summarize the scene into a compact set of 3D queries which are propagated through time in an online, streaming fashion. These queries are then decoded into semantic Gaussians at each timestep. We couple our framework with a denoising rendering objective to guide the queries and their constituent Gaussians in effectively capturing scene geometry. Owing to its efficient, query-based representation, S2GO achieves state-of-the-art performance on the nuScenes and KITTI occupancy benchmarks, outperforming prior art (e.g., GaussianWorld) by 1.5 IoU with 5.9x faster inference.
We study the higher secant varieties of a smooth projective variety embedded in projective space. We prove that when the variety is a surface and the embedding line bundle is sufficiently positive, these varieties are normal with Du Bois singularities and the syzygies of their defining ideals are linear to the expected order. We show that the cohomology of the structure sheaf of the surface completely determines whether the singularities of its secant varieties are Cohen--Macaulay or rational. We also prove analogous results when the dimension of the original variety is higher and the secant order is low, and by contrast we prove a result that strongly implies these statements do not generalize to higher dimensional varieties when the secant order is high. Finally, we deduce a complementary result characterizing the ideal of secant varieties of a surface in terms of the symbolic powers of the ideal of the surface itself, and we include a theorem concerning the weight one syzygies of an embedded surface -- analogous to the gonality conjecture for curves -- which we discovered as a natural application of our techniques.
The non-isomorphic locus of a general projection from an embedded smooth projective variety to a hypersurface moves in a linear system of an effective divisor which we call the double point divisor. David Mumford proved that the double point divisor from outer projection is always base point free, and Bo Ilic proved that it is ample except for a Roth variety. The first aim of this paper is to show that the double point divisor from outer projection is very ample except in the Roth case. This answers a question of Bo Ilic. Unlike the case of outer projection, the double point divisor from inner projection may not be base point free nor ample. However, Atsushi Noma proved that it is semiample except when a variety is neither a Roth variety, a scroll over a curve, nor the second Veronese surface. In this paper, we investigate when the double point divisor from inner projection is base point free or big.
This is a survey paper on recent work on syzygies of algebraic varieties. We discuss the gonality conjecture on weight-one syzygies of algebraic curves, syzygies of secant varieties of algebraic curves, syzygies of tangent developable surfaces and Green's conjecture on syzygies of canonical curves, and asymptotic syzygies of algebraic varieties. All results considered in this paper were proven using the geometry of symmetric products of algebraic curves.