Interests
Operator calculus on graphs, algebraic probability, algebraic combinatorics, symbolic computation, combinatorial properties and applications of Clifford algebras
Book
Papers Published or Accepted for Publication
- (with R. Schott) Connected components and evolution of random graphs: an algebraic approach, Journal of Algebraic Combinatorics, 35 (2012), 141-156.
DOI: 10.1007/s10801-011-0297-1
- (with R. Schott) Complexity of counting cycles using zeons
Computers & Mathematics with Applications, 62 (2011), 1828-1837.
DOI: 10.1016/j.camwa.2011.06.026
- (with G. Harris) Spinorial formulations of graph enumeration problems
Advances in Applied Clifford Algebras, (2011).
DOI: 10.1007/s00006-011-0298-0
- (with R. Schott) Nilpotent adjacency matrices and random graphs
Ars Combinatoria, 98 (2011), 225-239.
- (with R. Schott) Zeons, lattices of partitions, and free probability
Communications on Stochastic Analysis, 4 (2010), 311-334.
- (with R. Schott) Operator homology and cohomology in Clifford algebras
Cubo, A Mathematical Journal, 12 (2010), 299-326.
- (with R. Schott) Dynamic geometric graph processes: adjacency operator approach
Advances in Applied Clifford Algebras, 20 (2010), 893-921.
- (with R. Schott) Dynamic random walks in Clifford algebras
Advances in Pure and Applied Mathematics, 1 (2010), 81-115.
- (with R. Schott) Reductions in computational complexity using Clifford algebras
Advances in Applied Clifford Algebras, 20 (2010), 121-140.
- A new adjacency matrix for finite graphs
Advances in Applied Clifford Algebras, 18 (2008), 979-991.
- (with R. Schott) Nilpotent adjacency matrices, random graphs, and quantum random variables, Journal of Physics A: Mathematical and Theoretical, 41 (2008), 155205.
- (with R. Schott) Random walks in Clifford algebras of arbitrary
signature as walks on directed hypercubes,
Markov Processes and Related Fields, 14 (2008), 515-542.
- Norms and generating functions in Clifford algebras
Advances in Applied Clifford Algebras, 18 (2008), 75-92.
- (with R. Schott) Partitions and Clifford algebras
European Journal of Combinatorics, 29 (2008), 1133-1138.
- Graph-theoretic approach to stochastic integrals with Clifford algebras
Journal of Theoretical Probability, 20 (2007), 257-274.
- (with R. Schott) Operator calculus and Appell systems on Clifford algebras
International Journal of Pure and Applied Mathematics, 31 (2006), 427-446.
- Clifford-algebraic random walks on the hypercube
Advances in Applied Clifford Algebras, 15 (2005), 213-232.
Book Chapters
- (with R. Schott) On the Complexity of Cycle Enumeration for Simple Graphs
Guide to Geometric Algebra in Practice, L. Dorst and J. Lasenby (editors), Springer, 2011. ISBN 978-0-85729-810-2
- (with R. Schott) Computational Complexity Reductions Using Clifford Algebras
Geometric Algebra Computing for Engineering and Computer Science, E. Bayro-Corrochano and G. Scheuermann (editors), Springer Verlag, 2010.
Conference Proceedings
-
(with R. Schott) Clifford operator calculus: homology, cohomology, and Appell systems
Proceedings of 9th International Conference on Clifford Algebras and Their Applications in Mathematical Physics (ICCA9) Weimar, Germany, July 2011.
- (with R. Schott) Clifford algebras, random graphs, and quantum random
variables
QUANTUM STOCHASTICS AND INFORMATION (Statistics, Filtering and
Control), University of Nottingham, UK 15 - 22 July 2006
edited by V P Belavkin & M Guţă (University of
Nottingham, UK)
World Scientific Publishing, 2008.
Research Presentations at Conferences
- Clifford Operator Calculus: Homology, Cohomology, and Appell Systems
Invited plenary talk, 9th International Conference on Clifford Algebras and Their Applications
Weimar, Germany, July 2011.
- On the Complexity of Counting Cycles in Sparse Graphs using Nilpotent Adjacency Matrices
24th Midwest Conference on Combinatorics, Cryptography, and Computing
Illinois State University, Normal, Illinois, September 2010.
- On the Complexity of Cycle Enumeration using Zeons
Applications of Geometric Algebras in
Computer Science and Engineering (AGACSE 2010)
Amsterdam, The Netherlands, June 2010.
- Graph-Theoretic Free Cumulants using Zeons
30th Conference on Quantum Probability and Related Topics
Santiago, Chile, November 2009.
- Reductions in Computational Complexity Using Clifford Algebras
Applications of Geometric Algebras in Computer Science and Engineering (AGACSE 2008)
Leipzig, Germany, August 2008.
- Clifford Algebras, Graph Problems, and Computational Complexity
Invited plenary talk, 8th International Conference on Clifford Algebras and Their Applications
Campinas, Brazil, May 2008.
- Three Algebras Related to Random Graphs and Quantum Probability
4th International Conference of Applied Mathematics and Computing
Plovdiv, Bulgaria, August 2007.
- How Can NP Graph Problems Be Moved Into P?
20th Midwest Conference on Combinatorics, Cryptography, and Computing
Wichita State University, Wichita, Kansas, October 2006.
- Clifford Algebras, Random Graphs, and Quantum Probability
27th International Conference on Quantum Probability and Infinite Dimensional Analysis
Nottingham, England, July 2006.
- Clifford Algebras and Random Graphs
12th International Conference on Random Structures and Algorithms
Poznan, Poland, August 2005.
- Clifford-Algebraic Generating Functions and a Frobenius Norm for Clifford Matrices
7th International Conference on Clifford Algebras and Their Applications
Toulouse, France, May 2005.
Seminars & Colloquia
- Wireless Networks and Random Graphs: An Operator Calculus Approach
INRIA, Paris, November 2010.
- Wireless Networks and Graded Semigroup Algebras
Southern Illinois University Carbondale, November 2010.
- Wireless Networks, Random Graphs, and Clifford Algebras
Institut Élie Cartan, Université Henri Poincaré-Nancy I, Nancy, France, October 2010.
- Clifford Algebras, Finite Graphs, and Random Walks
Southern Illinois University Carbondale, November 2008.
- Clifford-Algebraic Generating Functions and a Frobenius Norm for Clifford Matrices
Tennessee Technological University, Cookeville, Tennessee, November 2004.
Prepublications
- (with R. Schott) On the role of blade factorization in constructing Clifford Appell systems
- (with R. Schott) Operator calculus in generalized zeon algebras: theory and application to multi-constrained path problems
- (with R. Schott) Operator calculus and invertible Clifford Appell systems: theory and application to the n-particle fermion algebra.
- (with H. Cruz-Sanchez, R. Schott, Y-Q. Song) Operator calculus approach to minimal paths: precomputed routing in the Orbcomm satellite constellation.
- (with H. Cruz-Sanchez, R. Schott, Y-Q. Song) Operator calculus approach to minimal paths: precomputed routing in a store-and-forward satellite constellation.
Professional Affiliations
- American Mathematical Society (AMS)
- Association for Quantum Probability and Infinite Dimensional Analysis (AQPIDA)
- Clifford Algebras International Research Open Studies Center (CAIROS)
- International Advisory Board: 10th International Conference on Clifford Algebras and Their Applications
Software
Below are links to some Mathematica notebooks I’ve been working
on. It’s all open source, and I welcome comments and suggestions
for improvements.
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