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On the Minimum Rank Among Positive Semidefinite Matrices with a Given Graph

- Matt Booth, Philip Hackney,
+8 authors Wendy Wang - MathematicsSIAM J. Matrix Anal. Appl.
- 1 May 2008

TLDR

Infinity Properads and Infinity Wheeled Properads

- Philip Hackney, M. Robertson, Donald Yau
- Mathematics, Engineering
- 24 October 2014

Introduction.- Graphs.- Properads.- Symmetric Monoidal Closed Structure on Properads.- Graphical Properads.- Properadic Graphical Category.- Properadic Graphical Sets and Infinity Properads.-… Expand

Linearly independent vertices and minimum semidefinite rank

- Philip Hackney, Benjamin Harris, Margaret Lay, Lon H. Mitchell, S. Narayan, A. Pascoe
- Mathematics
- 1 September 2009

On the minimum semidefinite rank of a simple graph

- Matt Booth, Philip Hackney,
+7 authors B. D. Sutton - Mathematics, Computer Science
- 3 March 2011

The minimum semidefinite rank (msr) of a graph is defined to be the minimum rank among all positive semidefinite matrices whose zero/nonzero pattern corresponds to that graph. We recall some known… Expand

Higher cyclic operads

- Philip Hackney, M. Robertson, Donald Yau
- MathematicsAlgebraic & Geometric Topology
- 8 November 2016

We introduce a convenient definition for weak cyclic operads, which is based on unrooted trees and Segal conditions. More specifically, we introduce a category $\Xi$ of trees, which carries a tight… Expand

On the Category of Props

- Philip Hackney, M. Robertson
- MathematicsAppl. Categorical Struct.
- 11 July 2012

The category of (colored) props is an enhancement of the category of colored operads, and thus of the category of small categories. The titular category has nice formal properties: it is bicomplete… Expand

A graphical category for higher modular operads

- Philip Hackney, M. Robertson, Donald Yau
- Mathematics
- 4 June 2019

Group actions on Segal operads

- J. Bergner, Philip Hackney
- Mathematics
- 14 July 2012

We give a Quillen equivalence between model structures for simplicial operads, described via the theory of operads, and Segal operads, thought of as certain reduced dendroidal spaces. We then extend… Expand

Relative left properness of colored operads

- Philip Hackney, M. Robertson, Donald Yau
- Mathematics
- 17 November 2014

The category of $\mathfrak{C}$-colored symmetric operads admits a cofibrantly generated model category structure. In this paper, we show that this model structure satisfies a relative left properness… Expand

On factorizations of graphical maps

- Philip Hackney, M. Robertson, Donald Yau
- Mathematics
- 23 May 2017

We study the categories governing infinity (wheeled) properads. The graphical category, which was already known to be generalized Reedy, is in fact an Eilenberg-Zilber category. A minor alteration to… Expand

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