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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 centers around problems
relating commutative algebra and algebraic combinatorics.
I received a B.S. in Mathematics and a B.S. in Physics
from Christian Brothers University in 2013.

Contact information:

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

Office Hours:

Thursday 1:30 PM - 2:25 PM or by appointment.

Teaching


I am the Gateway Coordinator for Spring 2016:

Classes taught previously:

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

Research


Research Interests:

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.

Publications:

[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 (2017)
[3]. On the Diameter of Dual Graphs of Stanley-Reisner Rings with Serre (S_2) Property and Hirsch Type Bounds on Abstractions of Polytopes. Preprint (2016)
[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).

Curriculum Vitae:

Find my Curriculum Vitae here.

Supplemental Material:

Proof of Proposition 3.6 from [2].