Cohomological rigidity of smooth toric Fano fourfolds
We prove that any two smooth toric Fano fourfolds whose integral cohomology rings are isomorphic as graded rings are diffeomorphic.
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arXiv subjects
Suyoung Choi
Publications and source records attributed to Suyoung Choi.
Complement minimally non-totally unimodular matrices
We prove that, up to row and column permutations and complement operations, the only complement minimally non-totally unimodular matrices are the cycle matrices $C_3$ and $C_5$. This settles a conjecture of Chervet, Grappe, and Vallée. As a consequence, every simplicial cone generated by the rows of a totally equimodular matrix admits a regular unimodular Hilbert triangulation.
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On small covers over Bier spheres
The Bier sphere of a simplicial complex $K$ is defined as the deleted join of $K$ and its combinatorial Alexander dual. We focus on the class of Bier spheres of the skeleta of a simplex. Since these Bier spheres are known to be polytopal, they give rise to small covers. We classify small covers over these Bier spheres up to Davis--Januszkiewicz equivalence. As applications, for all $m \geq 4$, we determine the homeomorphism types of small covers over the Bier spheres of the $0$-skeleton and the $(m-3)$-skeleton of an $(m-1)$-simplex. For the remaining cases $0<r<m-3$, we compute their rational Betti numbers.
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On the toric lifting properties for simplicial $3$-spheres
We study the lifting problem for mod $2$ characteristic maps over simplicial $3$-spheres. Using a bad-block partition of the universal complex $X(\mathbb{Z}_2^4)$, we prove an avoidance criterion for liftability. We show that every simplicial $3$-sphere with at most $20$ vertices has the toric lifting property. We also obtain image-size and join-type results, and prove sharpness of the image-size bound in the universal-complex sense.
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Enumerating Toric-Colorable Seeds of Picard Number Five via Binary Matroids
We introduce a binary matroid framework for the enumeration of mod $2$ toric-colorable seeds of fixed Picard number. Working with binary matroids up to isomorphism, we organize them through their contraction structure and recursively enumerate weak pseudomanifold subcomplexes by a dynamic programming algorithm. The resulting method combines these structural reductions with a Gray-code traversal of the mod $2$ kernel of the ridge--facet incidence matrix. Using this framework, we complete the classification of mod $2$ toric-colorable seeds of dimension four and Picard number five, proving that there are exactly $198{,}846$ isomorphism classes. This case was computationally infeasible for the GPU-based algorithm previously used by Choi, Jang, and Vallée to treat Picard number four. We further verify that each of these seeds admits an integral characteristic map. As a validation of the method, we also reproduce that Picard number four classification: the weak pseudomanifold enumeration stage drops from over ten days on a GPU to ten minutes on a single CPU.
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Symplectic and projective small covers over products of polygons
We study symplectic and projective structures on small covers over products of polygons. We introduce the factor-compatible class for small covers over products of polygons and prove that every factor-compatible small cover admits a smooth projective model as a finite quotient of a product of curves. Furthermore, we show that the graded mod~$2$ cohomology ring determines the Hodge diamond of the associated projective model. We also prove that every factor-compatible small cover admits an iterated equivariant bundle structure.
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Symplectic small covers in dimension four
We study symplectic structures on four-dimensional small covers. Our main result shows that every symplectic four-dimensional small cover is aspherical. We then classify symplectic small covers over products of two polygons, proving that symplecticity is equivalent to factor-compatibility. We also classify them up to diffeomorphism. Finally, we construct a symplectic four-dimensional small cover whose orbit polytope is not combinatorially equivalent to a product of two polygons.
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Small covers as pullbacks from the simplex
We introduce and study small covers that are pullbacks from the simplex, extending pullbacks from the linear model. Our main result gives several equivalent characterizations of this class, including torsion-freeness of odd-degree integral cohomology, vanishing of the first Steenrod square on even-degree mod $2$ cohomology, and relations among integral and mod $2$ Betti numbers.
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Shape-Aware Topological Representation for Pipeline Hyperbola Detection in GPR Data
Ground Penetrating Radar (GPR) is a widely used Non-Destructive Testing (NDT) technique for subsurface exploration, particularly in infrastructure inspection and maintenance. However, conventional interpretation methods are often limited by noise sensitivity and a lack of structural awareness. This study presents a novel framework that enhances the detection of underground utilities, especially pipelines, by integrating shape-aware topological features derived from B-scan GPR images using Topological Data Analysis (TDA), with the spatial detection capabilities of the YOLOv5 deep neural network (DNN). We propose a novel shape-aware topological representation that amplifies structural features in the input data, thereby improving the model's responsiveness to the geometrical features of buried objects. To address the scarcity of annotated real-world data, we employ a Sim2Real strategy that generates diverse and realistic synthetic datasets, effectively bridging the gap between simulated and real-world domains. Experimental results demonstrate significant improvements in mean Average Precision (mAP), validating the robustness and efficacy of our approach. This approach underscores the potential of TDA-enhanced learning in achieving reliable, real-time subsurface object detection, with broad applications in urban planning, safety inspection, and infrastructure management.
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Full subcomplexes of Bier spheres
Full subcomplexes of a simplicial complex encode essential structure for understanding the complex itself. For a simplicial complex $K$, possibly with a ghost vertex, the Bier sphere of $K$ is a simplicial sphere obtained as the deleted join of $K$ and its combinatorial Alexander dual. In this paper, we determine the homotopy types of all full subcomplexes of Bier spheres. As applications, we provide a formula for the bigraded Betti numbers of the Bier sphere of $K$ in terms of full subcomplexes of $K$, and we explicitly describe the cohomology of real toric manifolds associated with Bier spheres.
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Real toric manifolds associated with chordal nestohedra
This paper investigates the rational Betti numbers of real toric manifolds associated with chordal nestohedra. We consider the poset topology of a specific poset induced from a chordal building set, and show its EL-shellability. Based on this, we present an explicit description using alternating $\mathcal{B}$-permutations for a chordal building set $\mathcal{B}$, transforming the computing Betti numbers into a counting problem. This approach allows us to compute the $a$-number of a finite simple graph through permutation counting when the graph is chordal. In addition, we provide detailed computations for specific cases such as real Hochschild varieties corresponding to Hochschild polytopes.
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Reference Free Platform Adaptive Locomotion for Quadrupedal Robots using a Dynamics Conditioned Policy
This article presents Platform Adaptive Locomotion (PAL), a unified control method for quadrupedal robots with different morphologies and dynamics. We leverage deep reinforcement learning to train a single locomotion policy on procedurally generated robots. The policy maps proprioceptive robot state information and base velocity commands into desired joint actuation targets, which are conditioned using a latent embedding of the temporally local system dynamics. We explore two conditioning strategies - one using a GRU-based dynamics encoder and another using a morphology-based property estimator - and show that morphology-aware conditioning outperforms temporal dynamics encoding regarding velocity task tracking for our hardware test on ANYmal C. Our results demonstrate that both approaches achieve robust zero-shot transfer across multiple unseen simulated quadrupeds. Furthermore, we demonstrate the need for careful robot reference modelling during training: exposing the policy to a diverse set of robot morphologies and dynamics leads to improved generalization, reducing the velocity tracking error by up to 30% compared to the baseline method. Despite PAL not surpassing the best-performing reference-free controller in all cases, our analysis uncovers critical design choices and informs improvements to the state of the art.
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Complete non-singular toric varieties with Picard number 4
We classify all complete non-singular toric varieties with Picard number four via a combinatorial framework based on fanlike simplicial spheres and characteristic maps. This classification yields $59$ fanlike seeds with Picard number four, along with all toric manifolds supported by them. As a consequence, we resolve a conjecture of Gretenkort, Kleinschmidt, and Sturmfels by presenting the first known examples of toric manifolds supported by neighborly polytopes. We also answer a question of Batyrev concerning minimal non-faces of such spheres.
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The cohomology rings of real permutohedral varieties
A permutohedral variety is a remarkable object in various areas of mathematics, and its topological invariants are widely recognized. However, only little is known about a real permutohedral variety, that is, the real locus of a permutohedral variety. The rational Betti numbers of real permutohedral varieties were computed in terms of alternating permutations in 2012. In this paper, we provide explicit descriptions of the cohomology ring of real permutohedral varieties. In particular, we describe the multiplicative structure in terms of alternating permutations.
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Stellar subdivisions, wedges and Buchstaber numbers
A seed is a PL sphere that is not obtainable by a wedge operation from any other PL sphere. In this paper, we study two operations on PL spheres, known as the stellar subdivision and the wedge, that preserve the maximality of Buchstaber numbers and polytopality. We construct a new polytopal toric colorable seed from these two operations. As a corollary, we prove that the toric colorable seed inequality established by Choi and Park is tight.
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Toric wedge induction and toric lifting property for piecewise linear spheres with a few vertices
Let $K$ be an $(n-1)$-dimensional piecewise linear sphere on $[m]$, where $m\leq n+4$. There are a canonical action of $m$-dimensional torus $T^m$ on the moment-angle complex $\mathcal{Z}_K$, and a canonical action of $\mathbb{Z}_2^m$ on the real moment-angle complex $\mathbb{R}\mathcal{Z}_K$, where $\mathbb{Z}_2$ is the additive group with two elements. We prove that any subgroup of $\mathbb{Z}_2^m$ acting freely on $\mathbb{R}\mathcal{Z}_K$ is induced by a subtorus of $T^m$ acting freely on $\mathcal{Z}_K$. The proof primarily utilizes a suitably modified method of toric wedge induction and the combinatorial structure of a specific binary matroid of rank $4$.
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The characterization of $(n-1)$-spheres with $n+4$ vertices having maximal Buchstaber number
We present a computationally efficient algorithm that is suitable for graphic processing unit implementation. This algorithm enables the identification of all weak pseudo-manifolds that meet specific facet conditions, drawn from a given input set. We employ this approach to enumerate toric colorable seeds. Consequently, we achieve a comprehensive characterization of $(n-1)$-dimensional PL spheres with $n+4$ vertices that possess a maximal Buchstaber number. A primary focus of this research is the fundamental categorization of non-singular complete toric varieties of Picard number $4$. This classification serves as a valuable tool for addressing questions related to toric manifolds of Picard number $4$. Notably, we have determined which of these manifolds satisfy equality within an inequality regarding the number of minimal components in their rational curve space. This addresses a question posed by Chen, Fu, and Hwang in 2014 for this specific case.
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Legged Robot State Estimation With Invariant Extended Kalman Filter Using Neural Measurement Network
This paper introduces a novel proprioceptive state estimator for legged robots that combines model-based filters and deep neural networks. Recent studies have shown that neural networks such as multi-layer perceptron or recurrent neural networks can estimate the robot states, including contact probability and linear velocity. Inspired by this, we develop a state estimation framework that integrates a neural measurement network (NMN) with an invariant extended Kalman filter. We show that our framework improves estimation performance in various terrains. Existing studies that combine model-based filters and learning-based approaches typically use real-world data. However, our approach relies solely on simulation data, as it allows us to easily obtain extensive data. This difference leads to a gap between the learning and the inference domain, commonly referred to as a sim-to-real gap. We address this challenge by adapting existing learning techniques and regularization. To validate our proposed method, we conduct experiments using a quadruped robot on four types of terrain: \textit{flat}, \textit{debris}, \textit{soft}, and \textit{slippery}. We observe that our approach significantly reduces position drift compared to the existing model-based state estimator.
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