Poset topology: Tools and
                    applications

Wachs, Michelle L.

doi:10.1090/pcms/013/09

Cited by 192 publications

(231 citation statements)

References 198 publications

(374 reference statements)

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“…We refer to [18,Subsection 3.2] for the notions and facts from poset topology used in the proof of the following lemma. This lemma is undoubtedly previously known, but we were unable to find it in the literature in the form needed for the subsequent theorem.…”

Section: Proposition 41 the Only Non-vanishing Homology Of A Shellamentioning

confidence: 99%

“…Our approach utilizesÀlvarez, García, and Zarzuela's computation of local cohomology with support in a subspace arrangement in [1]. Their formula is a local cohomology analogue of the celebrated Goresky-MacPherson Formula for the singular cohomology of the complement of a real subspace arrangement (see, e.g., [18,Theorem 1.3.8]); in this way, one can consider our results a link between the Goresky-MacPherson Formula and the Shephard-Todd Theorem.…”

Section: Introductionmentioning

confidence: 99%

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Separating invariants and local cohomology

Dufresne

Jeffries

2015

Advances in Mathematics

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Abstract. The study of separating invariants is a recent trend in invariant theory. For a finite group acting linearly on a vector space, a separating set is a set of invariants whose elements separate the orbits of G. In some ways, separating sets often exhibit better behavior than generating sets for the ring of invariants. We investigate the least possible cardinality of a separating set for a given G-action. Our main result is a lower bound that generalizes the classical result of Serre that if the ring of invariants is polynomial then the group action must be generated by pseudoreflections. We find these bounds to be sharp in a wide range of examples.

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Section: Proposition 41 the Only Non-vanishing Homology Of A Shellamentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Separating invariants and local cohomology

Dufresne

Jeffries

2015

Advances in Mathematics

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“…The order complex of a poset is the natural simplicial complex with simplices the finite totally-ordered chains of the poset. (Useful references include [5,6,25]. )…”

Section: Introductionmentioning

confidence: 99%

Arboreal singularities

Nadler

2017

Geom. Topol.

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Abstract. We introduce a class of combinatorial singularities of Lagrangian skeleta of symplectic manifolds. The link of each singularity is a finite regular cell complex homotopy equivalent to a bouquet of spheres. It is determined by its face poset which is naturally constructed starting from a tree (nonempty finite acyclic graph). The choice of a root vertex of the tree leads to a natural front projection of the singularity along with an orientation of the edges of the tree. Microlocal sheaves along the singularity, calculated via the front projection, are equivalent to modules over the quiver given by the directed tree.

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“…For instance, one may speak of the topology of a poset, implicitly meaning the topology of its order complex. We will not elaborate on this connection further, but refer the reader to [8,6,75].…”

Section: The Quillen Fiber Lemmamentioning

confidence: 99%

On the Topology of Discrete Strategies

Erdmann

2009

The International Journal of Robotics Research

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This paper explores a topological perspective of planning in the presence of uncertainty, focusing on tasks specified by goal states in discrete spaces. The paper introduces strategy complexes. A strategy complex is the collection of all plans for attaining all goals in a given space. Plans are like jigsaw pieces. Understanding how the pieces fit together in a strategy complex reveals structure. That structure characterizes the inherent capabilities of an uncertain system. By adjusting the jigsaw pieces in a design loop, one can build systems with desired competencies.The paper draws on representations from combinatorial topology, Markov chains, and polyhedral cones. Triangulating between these three perspectives produces a topological language for describing concisely the capabilities of uncertain systems, analogous to concepts of reachability and controllability in other disciplines. The major nouns in this language are topological spaces.Three key theorems (numbered 1, 11, 20 in the paper) illustrate the sentences in this language: (a) Goal Attainability: There exists a strategy for attaining a particular goal from anywhere in a system if and only if the strategy complex of a slightly modified system is homotopic to a sphere. (b) Full Controllability: A system can move between any two states despite control uncertainty precisely when its strategy complex is homotopic to a sphere of dimension two less than the number of states. (c) General Structure: Any system's strategy complex is homotopic to the product of a spherical part, modeling full controllability on subspaces, and a general part, modeling adversarial capabilities.The paper contains a number of additional results required as stepping stones, along with many examples. The paper provides algorithms for computing the key structures described. Finally, the paper shows that some interesting questions are hard. For instance, it is NP -complete to determine the most precisely attainable goal of a system with perfect sensing but uncertain control.

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Poset topology: Tools and applications

Cited by 192 publications

References 198 publications

Separating invariants and local cohomology

Separating invariants and local cohomology

Arboreal singularities

On the Topology of Discrete Strategies

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