My research is in discrete probability, combinatorics and graph theory; in particular, applications of combinatorial ideas to the study of phase transitions in statistical mechanics, algorithms in theoretical computer science and performance analysis of communications networks. You can find a 2012 summary of my research here. Below are links to my papers (not necessarily the final versions); most of these can also be found at arXiv.org.

For links to journal versions of these papers, please go here; note that a Notre Dame netID is needed to access this page.

- Independent sets in the discrete hypercube (an expository article)
- Independent set and matching permutations (with T. Ball, K. Hyry and K. Weingartner)
- The game of plates and olives (with T. Carroll),
*Electronic Journal of Combinatorics*, to appear - Phase Coexistence for the Hard-Core Model on Z^2 (with A. Blanca, Y. Chen, D. Randall and P. Tetali),
*Combinatorics, Probability and Computing*, to appear. The data sets used in this paper are available here - Restricted Stirling and Lah number matrices and their inverses (with J. Engbers and C. Smyth),
*Journal of Combinatorial Theory Series A***161**(2019), 271-298 - Total non-negativity of generalized Stirling matrices (with A. Pacurar), submitted
- Cutting lemma and Zarankiewicz's problem in distal structures (with A. Chernikov and S. Starchenko), submitted
- The independent set sequence of some
families of trees (with J. Hilyard),
*Australasian Journal of Combinatorics***70**(2018), 236-252 - Extremal H-colorings of trees and 2-connected graphs (with J. Engbers),
*Journal of Combinatorial Theory Series B***122**(2017), 800-814 - Guest editors' foreword (with. D. Chakrabarty),
*Theory of Computing***12**(2016), Article 8 (Special issue:**APPROX-RANDOM**2014) - On the independence ratio of distance graphs (with J. Carraher, S. Hartke, J. Radcliffe and D. Stolee),
*Discrete Mathematics***339**(2016), 3058-3072 - Asymptotic normality of some graph sequences,
*Graphs and Combinatorics***32**(2016), 639-647 - Phase coexistence and torpid mixing in the 3-coloring model on Z^d (with J. Kahn, D. Randall and G. Sorkin),
*SIAM Journal of Discrete Mathematics***29**(2015), 1223-1244 - Counting colorings of a regular graph,
*Graphs and Combinatorics***31**(2015), 629--638 - Combinatorially interpreting generalized Stirling numbers (with J. Engbers and J. Hilyard),
*European Journal of Combinatorics***43**(2015), 32--54 - Three tutorial lectures on entropy and counting (notes prepared to accompany a series of tutorial lectures given at the 1st Lake Michigan Workshop on Combinatorics and Graph Theory, Western Michigan University, March 15--16 2014), arXived
- Counting independent sets of a fixed size in graphs with given minimum degree (with J. Engbers),
*Journal of Graph Theory***76**(2014), 149--168 - Phase Coexistence and Slow Mixing for the Hard-Core Model on Z^2 (with A. Blanca, D. Randall and P. Tetali),
*Lecture Notes in Computer Science***8096**(*Proceedings of APPROX/RANDOM 2013*) (2013), 379--394. The data sets used in this paper are available here - Stirling numbers of forest and cycles (with Do Trong Thanh),
*Electronic Journal of Combinatorics***20**(2013), #P73 - Maximizing H-colorings of regular graphs,
*Journal of Graph Theory***73**(2013), 66--84 - H-coloring tori (with J. Engbers),
*Journal of Combinatorial Theory Series B***102**(2012), 1110-–1133 - The independent set sequence of regular bipartite graphs,
*Discrete Mathematics***312**(2012), 2881--2892 - H-colouring bipartite graphs (with John Engbers),
*Journal of Combinatorial Theory Series B***102**(2012), 726-–742 - Reverse Mathematics and infinite traceable graphs (with P. Cholak and R. Solomon),
*Mathematical Logic Quarterly***58**(2012), 18--28 - Two problems on independent sets in graphs,
*Discrete Mathematics***311**(2011), 2105--2112 - The multi-state hard core model on a regular tree (with Fabio Martinelli, Kavita Ramanan and Prasad Tetali),
*SIAM Journal of Discrete Mathematics***25**(2011), 894--916 - The number of independent sets in a graph with small maximum degree (with Yufei Zhao),
*Graphs and Combinatorics***27**(2011), 177--186. The java programs used in this paper (source files, compiled programs and documentation) are available here in a compressed (zipped) folder. - A threshold phenomenon for random independent sets in the discrete hypercube,
*Combinatorics, Probability and Computing***20**(2011), 27--51 - An upper bound for the number of independent sets in regular graphs,
*Discrete Mathematics***309**(2009), 6635--6640 - Matchings and Independent Sets of a Fixed Size in Regular Graphs (with Teena Carroll and Prasad Tetali),
*Journal of Combinatorial Theory Series A***116**(2009), 1219--1227) - Sampling independent sets in the discrete torus,
*Random Structures and Algorithms***33**(2008), 356--376 - Sampling 3-colourings of regular bipartite graphs,
*Electronic Journal of Probability***12**(2007), 481--497 - Torpid Mixing of Local Markov Chains on 3-Colorings of the Discrete Torus (with Dana Randall),
*Proceedings of the Eighteenth Annual ACM--SIAM Symposium on Discrete Algorithms*(2007), 376--384 - Bounding the partition function of spin systems,
*Electronic Journal of Combinatorics***13**(2006), #R72 - Global connectivity from local geometric constraints for sensor network with various wireless footprints (with Raissa D'Souza, Cris Moore and Dana Randall),
*Proceedings of the Fifth International Conference on Information Processing in Sensor Networks*(2006), 19--26 - Slow mixing of Glauber Dynamics for the hard-core model on regular bipartite graphs (with Prasad Tetali),
*Random Structures and Algorithms***28**(2006) 427--443 - Slow mixing of the Glauber Dynamics for the hard-core model on the Hamming cube (with Prasad Tetali),
*Proceedings of the Fifteenth Annual ACM--SIAM Symposium on Discrete Algorithms*(2004), 459--460 - On weighted graph homomorphisms (with Prasad Tetali),
*DIMACS series in discrete mathematics and theoretical computer science***63**(*Graphs, Morphisms and Statistical Physics*) (2004), 97--104 - Entropy and graph homomorphisms,
*Oberwolfach reports***1**(2004), 30--32 - On phase transition in the hard-core model on Z^d (with Jeff Kahn),
*Combinatorics, Probability and Computing***13**(2004), 137--164 - On homomorphisms from the Hamming cube to Z,
*Israel Journal of Mathematics***138**(2003), 189--213 - Two problems involving the notion of phase transition (PhD thesis written under the direction of Jeff Kahn, Rutgers University, October 2002), ProQuest Dissertations and Theses (2002)

2019

2018

2017

2016

2015

2014

2013

2012

2011

2009

2008

2007

2006

2004

2003

2002

- Ramanujan Graphs (Essay for Part III of Mathematical Tripos, University of Cambridge, June 1996)
- [4]^2 Tic-Tac-Toe is a draw
- Erdos's proof of Bertrand's postulate (an exposition of Erdos's elementary proof that there is always a prime between
*n*and*2n*) - Ultrafilters, with applications to analysis, social choice and combinatorics (an introduction to the basics of ultrafilters, with some applications)
- A topological approach to evasiveness (an introduction to the use of topological ideas in the study of evasive properties, based on a paper by Kahn, Saks and Sturtevant)
- Bregman's theorem and extensions (an exposition of Radhakrishnan's entropy proof of Bregman's theorem, and some results and conjectures around the theorem)

Back to my home page