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Projects: Stefan Forcey, students and collaborators.
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New Hopf Structures on Binary Trees featuring multiplihedron modules and Hopf algebras. To appear in : DMTCS Proceedings, Formal Power Series and Algebraic Combinatorics 2009, with Aaron Lauve and Frank Sottile . New paper: Geometric combinatorial algebra featuring the cyclohedron and simplex. New proposal: Geometric combinatorial Hopf Algebras and modules Old Summary and Old proposal. Slides from the AMS sectional April 09. Cyclohedron algebras: slides Student: D. Springfield. |
Notes about the combinatorial Hopf Zoo: introduction. Part 1. Part 1b. Part 2. Part 3. Part 4. Part 5. Part 6. Part 7. Here are the products in the Hopf algebras by Aaron Lauve and Frank Sottile. Here's the specific data about the multiplihedra Hopf module . |
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Marked tubes and the graph multiplihedron with
S. Devadoss . Published at: Algebraic and Geometric Topology . Also available from the arXiv . Graph multiplihedra -- early notes. These contain alternate realizations of the polytopes. Colored tubes = marked tubes with the following translation: red = thick, blue = thin, red with blue interior =broken. Generalizing the multiplihedron: colored nested sets, colored cluster complexes and other definitions in progress. A possible type B_2 multiplihedron. Resources: cluster algebra portal ; associahedra and noncrossing partitions . |
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Quotients of the multiplihedron as categorified associahedra. Published in Homotopy, Homology and Applications . Also available from arXiv . OEIS entry here . CT06 talk on composihedra, enriched bicategories, and realization. |
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Notes about the Multigraph associahedra, with M. Carr and S. Devadoss . Part 1 Part 2. and here are 3d examples. Part 3. |
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Convex hull realizations for the multiplihedra. Published in Topology and its Applications . Also available from the arXiv . OEIS entry here . AMS chalkboard talk. Here is Sikimeti Mau's paper with Chris Woodward about using the multiplihedron to define morphisms of cohomological field theories. |
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Make your own composihedra: pdf cut out, fold on interior lines, use triangles to fill in final quadrilateral. Make your own composihedra: color jpg |
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Grant proposal: Student Research Natural number operads Student: A. Kheder. Crystals and Operads: SECA4 slides. Powerpoint version of above. |
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Derived Braids
The solution to some problems posed above: families of interchanging braids. Published in Algebraic and Geometric Topology: Classification of braids which give rise to interchange. |
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Higher morphisms for enriched categories: enriched 2-natural transformations, enriched modifications and all that!
Vertically Iterated Classical Enrichment Published in Theory and Applications of Categories Also available from the arXiv . |
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Ideas for n-dendroidal sets: start with n=2.
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Operads and cellular automata--work in progress |
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Enrichment as Categorical Delooping I: Enrichment Over Iterated Monoidal Categories Also available from the arXiv |
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Published (in AGT) version of above: Enrichment Over Iterated Monoidal Categories |
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Higher Dimensional Enrichment Also available from the arXiv |
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Introduction
Grope Constrained Braids Proposal version of above: some new ideas in section on string links. Bibliography for above. |