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Projects: Stefan Forcey, students and collaborators.





New poset polytopes, new algebras, and phylogenetic interpretations.

Submitted paper (with S. Devadoss) Poset associahedra

SAGE files for computing poset associahedra.

Stephen Reisdorf's thesis
Patrick Shower's thesis
Lisa Berry's thesis

Encyclopedia entry: Pterahedron
Pterahedra vertices:
1, 2, 6, 22, 94, ... Catalan transform of the factorials [ OEIS ?]

Draft: pterahedra


Resource: The Neighbor-Net Algorithm, Dan Levy, Lior Pachter, Section 2: The Mathematics.

Ideas for n-dendroidal sets: start with n=2.

Ittay Weiss has written a great introduction.


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.

Old proposal: Geometric combinatorial Hopf Algebras and modules

Old Summary and Older 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 .

Ideas about polyhexes, benzene, and the composihedron. Proposal.

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.

Notes about the Multigraph associahedra, with M. Carr and S. Devadoss .
Part 1
Part 2.
and here are 3d examples.
Part 3.

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


Collaborator: Jesse Siehler
Talk on operads from Streetfest
Operads in iterated monoidal categories. Also available from JHRS.
Even more fun--higher products for Young diagrams.
Work in progress on twisted interchangers.

Grant proposal: Student Research

Natural number operads Student: A. Kheder.

Crystals and Operads: SECA4 slides.
Powerpoint version of above.

Enrichment using an operad bimodule.
CT06 talk on composihedra, enriched bicategories, and realization.
New project description, including operad bimodule ideas.
Bibliography for above.
Talks on Weak Higher Dimensional Enrichment

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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.

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 .

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

Published (in AGT) version of above: Enrichment Over Iterated Monoidal Categories

Higher Dimensional Enrichment Also available from the arXiv

Introduction
Grope Constrained Braids
Proposal version of above: some new ideas in section on string links.
Bibliography for above.
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