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Home Page of Brent Holmes

About Me:

I am currently pursuing my PhD in mathematics at the University of Kansas.
I am studying commutative algebra with Hailong Dao.  My research explores
the relationship between commutative algebra and combinatorics. In my research,
I have used Macaulay2 and Sage, a python based language for Combinatorics.
You can see my projects in python on my github.

Contact information:

Office: 560 Snow Hall  
Fax: (785) 864-5255
E-mail: brentholmes@ku.edu

Office Hours:

Thursday 11:00 AM - 12:00 PM or by appointment.



Publications:

[6]. Serre's Condition, Balanced Simplicial Complexes, and Higher Nerves (with J. Lyle) (2018)
[5]. Higher Nerves of Simplicial Complexes (with H.Dao, J.Doolittle, K.Duna, B.Goeckner, and J.Lyle) Preprint (2017)
[4]. A Generalized Serre's Condition Preprint, Submitted (2018)
[3]. On the Diameter of Dual Graphs of Stanley-Reisner Rings with Serre (S_2) Property and Hirsch Type Bounds on Abstractions of Polytopes. Accepted to Electronic Jounral of Combinatorics (2018)
[2]. Rainbow Colorings of some Geometrically Defined Uniform Hypergraphs in the Plane Geombinatorics 23 (2014), 158-169.
[1]. Two Kinds of Frobenius Problems in Z[ m  ] (with L. Beneish, P. Johnson, and T. Lai) International Journal of Mathematics and Computer Science, (7), no. 2, 93100 (2012).

Supplemental Material:

Proof of Proposition A.12 from [3].



Research Summary:


My research interest is in the expanding field of combinatorial commutative algebra, more specifically studying the connectivity of collections of finite sets using tools from commutative algebra and combinatorics. In particular, I have examined the relationship between homologies of simplicial complexes and the depth conditions satisfied by their associated Stanley-Reisner rings.

I have recently put a paper on the arXiv exploring a generalization of the famous nerve complex. We call these new generalized nerves, higher nerves. In particular, we show that for a simplicial complex Delta, the homologies of these higher nerves of Delta give full information about the depth and f-vector of Delta. My current research project is to examine the interplay between these nerve homologies and other algebraic properties such as Serre's condition and Buchsbaum condition. I am also working to create Sage code to help users interact with the nerves. The code will create nerves and nerve homology tables from input and provide information about depth properties and the f-vector and h-vector of the complex. Below, I provide a picture of a simplicial complex and the higher nerves of that complex. Note that blue colored figures are 3-dimensional simplices.


Teaching


I am the Gateway Coordinator for Spring 2018:

Information about the Gateway Exam.

To read about my teaching experience and philosophies:
see my Teaching Statement.

Classes taught previously:

  • Math 101
  • Math 115
  • Math 116
  • Math 125
  • Math 126

  • Coordinating Experience:

  • Math 116
  • Gateway Lab


  • Professional Links:

    Curriculum Vitae.

    Resume.


    Fun Facts About Me:

    I enjoy playing and watching basketball, writing and playing music on my guitar and bass, and playing strategy-based board games.