2,133 research outputs found
Moment-angle complexes, monomial ideals, and Massey products
Associated to every finite simplicial complex K there is a "moment-angle"
finite CW-complex, Z_K; if K is a triangulation of a sphere, Z_K is a smooth,
compact manifold. Building on work of Buchstaber, Panov, and Baskakov, we study
the cohomology ring, the homotopy groups, and the triple Massey products of a
moment-angle complex, relating these topological invariants to the algebraic
combinatorics of the underlying simplicial complex. Applications to the study
of non-formal manifolds and subspace arrangements are given.Comment: 30 pages. Published versio
Associahedra via spines
An associahedron is a polytope whose vertices correspond to triangulations of
a convex polygon and whose edges correspond to flips between them. Using
labeled polygons, C. Hohlweg and C. Lange constructed various realizations of
the associahedron with relevant properties related to the symmetric group and
the classical permutahedron. We introduce the spine of a triangulation as its
dual tree together with a labeling and an orientation. This notion extends the
classical understanding of the associahedron via binary trees, introduces a new
perspective on C. Hohlweg and C. Lange's construction closer to J.-L. Loday's
original approach, and sheds light upon the combinatorial and geometric
properties of the resulting realizations of the associahedron. It also leads to
noteworthy proofs which shorten and simplify previous approaches.Comment: 27 pages, 11 figures. Version 5: minor correction
Brown's moduli spaces of curves and the gravity operad
This paper is built on the following observation: the purity of the mixed
Hodge structure on the cohomology of Brown's moduli spaces is essentially
equivalent to the freeness of the dihedral operad underlying the gravity
operad. We prove these two facts by relying on both the geometric and the
algebraic aspects of the problem: the complete geometric description of the
cohomology of Brown's moduli spaces and the coradical filtration of cofree
cooperads. This gives a conceptual proof of an identity of Bergstr\"om-Brown
which expresses the Betti numbers of Brown's moduli spaces via the inversion of
a generating series. This also generalizes the Salvatore-Tauraso theorem on the
nonsymmetric Lie operad.Comment: 26 pages; corrected Figure
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Guanidinium-Rich ROMP Polymers Drive Phase, Charge, and Curvature-Specific Interactions with Phospholipid Membranes
Protein transduction domains (PTDs) and their and their synthetic mimics are short sequences capable of unusually high uptake in cells. Several varieties of these molecules, including the arginine-rich Tat peptide from HIV, have been extensively used as vectors for protein, DNA, and siRNA delivery into cells. Despite the wide-ranging utility of PTDs and their mimics, their uptake mechanism is still under considerable debate. How the molecules are able to cross phospholipid membranes, and what structural components are necessary for optimal activity are poorly understood. This thesis explores how PTDMs interact with phospholipid membrane phase, anionic lipid content and negative Gaussian curvature generation along the structural variables of polymer length, charge density, hydrophobicity, aromaticity, and architecture. From these it is demonstrated that PTDM-membrane interaction is primarily controlled by balancing solubility of polymer in solution and in the lipid. Barriers to activity, such as anionic lipid content and membrane rigidity, can be overcome by the addition of hydrophobicity, but this has to be balanced by considerations of water solubility and toxicity
The Deligne-Mostow 9-ball, and the monster
The "monstrous proposal" of the first author is that the quotient of a
certain 13-dimensional complex hyperbolic braid group, by the relations that
its natural generators have order 2, is the bimonster" (M x M)semidirect Z/2.
Here M is the monster simple group. We prove that this quotient is either the
bimonster or Z/2. In the process, we give new information about the isomorphism
found by Deligne-Mostow, between the moduli space of 12-tuples in CP1 and a
quotient of the complex 9-ball. Namely, we identify which loops in the 9-ball
quotient correspond to the standard braid generators
Geometric Graph Theory and Wireless Sensor Networks
In this work, we apply geometric and combinatorial methods to explore a variety of problems motivated by wireless sensor networks. Imagine sensors capable of communicating along straight lines except through obstacles like buildings or barriers, such that the communication network topology of the sensors is their visibility graph. Using a standard distributed algorithm, the sensors can build common knowledge of their network topology.
We first study the following inverse visibility problem: What positions of sensors and obstacles define the computed visibility graph, with fewest obstacles? This is the problem of finding a minimum obstacle representation of a graph. This minimum number is the obstacle number of the graph. Using tools from extremal graph theory and discrete geometry, we obtain for every constant h that the number of n-vertex graphs that admit representations with h obstacles is 2o(n2). We improve this bound to show that graphs requiring Ω(n / log2 n) obstacles exist.
We also study restrictions to convex obstacles, and to obstacles that are line segments. For example, we show that every outerplanar graph admits a representation with five convex obstacles, and that allowing obstacles to intersect sometimes decreases their required number.
Finally, we study the corresponding problem for sensors equipped with GPS. Positional information allows sensors to establish common knowledge of their communication network geometry, hence we wish to compute a minimum obstacle representation of a given straight-line graph drawing. We prove that this problem is NP-complete, and provide a O(logOPT)-factor approximation algorithm by showing that the corresponding hypergraph family has bounded Vapnik-Chervonenkis dimension
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