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Projects: Stefan Forcey, students and collaborators.





Introduction:
Kron reduction and Kalmanson networks.

Resistance : electric circuits :: mutation : genetic history.

Examples and questions:
Finding the Kron reduction,
    aka the Dirichlet-to-Neumann map,
     aka the Response matrix
...and the reduced Resistance matrix

Papers: Kalmanson metrics and networks
Resources:

Balanced Minimal Evolution Polytopes for Phylogenetic Networks

Slides:
Galois connections for phylogenetic networks and their polytopes  [ slides]

Papers:
Level-1 Phylogenetic Networks and their Balanced Minimum Evolution Polytopes.  (with C. Durell)
    2019.  [ preprint ] [submitted to journal] [arxiv]

Split Network Polytopes and Network Spaces.  (with S. Devadoss, C. Durell)
    DMTCS Proceedings, FPSAC 31, 2019.  [ preprint ] [ extended abstract] [arxiv]

Galois connections and duality between phylogenetic network spaces and polytopes  [Draft] paper with Drew Scalzo


FPSAC 2019, University of Ljubljana, Slovenia Split network polytopes and network spaces  [ poster]

Polytopes:
Encyclopedia entry: BME polytope

Encyclopedia entry: STSP polytope

Resources:

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

A space of Phylogenetic Networks, Satyan Devadoss and Samantha Petti.

Hamilton Mathematics Institute: Geometry and combinatorics of associativity, Dublin 2017
Clades and tubes: facets of graph associahedra and phylogenetic polytopes. Slides:  [[1]], [[2]], [[3]]

Short intro: notes and project ideas.

Theses:
Cassandra Durell's thesis

Balanced Minimal Evolution Polytopes Slides:
Facets of Balanced Minimal Evolution polytopes. [slides]

Using 5 dimensions to identify a first cousin once removed. [ slides]

Recursive Linear Programming on the Balanced Minimal Evolution Polytope. [slides]

Papers:
Book chapter (with Gabriela Hamerlinck, Logan Keefe, William Sands)

(with L. Keefe, W. Sands) Facets of the Balanced Minimum Evolution Polytopes.


(with L. Keefe, W. Sands) Split-facets for Balanced Minimum Evolution Polytopes and the Permutoassociahedron.

Polytopes:
Encyclopedia entry: BME polytope

Encyclopedia entry: Splitohedron with Matlab code.

Resources:
State of the art on BME linear programming from D. Catanzaro, including Enumerating Vertices of the Balanced Minimum Evolution Polytope and On the Balanced Minimum Evolution Polytope

Theses:

Logan Keefe's thesis

William Sand's thesis



Game theory for environmental protection.

Paper: When does compromise prevent more pollution?
    (with C. Clemons, J. Cossey, M. Ferrara, T. Norfolk, G. Obeng, D. Ricciardi, G. Young)
    Notices of the A.M.S, 59(9), pp 1223-1234, 2012. [ journal ]  [ preprint ]

Use excel to find games for the examples in the above paper.
SAEP's
D. Crawford's thesis: Minimizing Pollution Through Semi-Antagonistic Equilibrium Points.

A. Fernandes's thesis: A Carbon Credits Game, under advisor Jerzy Filar.

New work with Joe Johnson and Francesco Renna: A sequential game with monitoring.

Use excel to find games for the case of optional monitoring, after the fact.

Use wolfram alpha to solve for the four percentages in mixed equilibria.
Application to food safety: G. Obeng's thesis.


New poset polytopes, new algebras, and phylogenetic interpretations.

Publication (with S. Devadoss) Poset associahedra

SAGE files for computing poset associahedra.
Preprint: Facets of the Balanced minimal evolution Polytope .
Stephen Reisdorf's thesis
Patrick Shower's thesis
Lisa Berry's thesis [copy]

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

Draft: pterahedra




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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