The discrete mathematics group studies theoretical, algorithmic, and computational problems in the fields of graph theory, discrete optimization, combinatorics, and algebraic geometry, with applications in biology, computer science, physics, management sciences, and engineering. Network science, with fundamental concepts from graph theory and computational techniques from discrete optimization, is widely applied to problems arising in transportation/communication, distribution, and security of resources and information. Combinatorial search incorporates graph theory, set systems, and algorithms to tackle information-theoretic questions in topics such as message transmission, data compression, and identification of defective samples in a population. Computational algebra joins tools from algebraic geometry with randomized algorithms from discrete geometry to develop methods to solve systems of polynomial equations arising in statistical inference and mathematical modeling.
Faculty with primary interests in discrete applied mathematics
- Adam Rumpf
- William Schwartz
- Dane Wilburne
- J. A. De Loera, S. Petrovic, L. Silverstein, D. Stasi, and D. Wilburne. Random Monomial Ideals, Submitted, arXiv:1701.07130.
- S. Petrović, A. Thoma and M. Vladoiu. Bouquet algebra of toric ideals, Submitted, arXiv:1507.02740.
- J. N. Cooper and R. B. Ellis. Linearly bounded liars, adaptive covering codes, and deterministic random walks, J Combin, to appear.
- A. Corso, U. Nagel, S. Petrovic, and C. Yuen. Blow-up algebras, determinantal ideals, and Dedekind-Mertens-like formulas. Forum Mathematicum, to appear.
- V. Karwa, M. J. Pelsmajer, S. Petrovic, D. Stasi, and D. Wilburne. Statistical models for cores decomposition of an undirected random graph, Electr. J. Stat, to appear.
- R. Fulek, M.J. Pelsmajer, and M. Schaefer. Hanani-Tutte for Radial Planarity, J. Graph Algorithms and Appl. 21 (2017) no.1 pp.135-154.
- E. Gross, S. Petrovic, and D. Stasi. Goodness of fit for log-linear network models: Dynamic Markov bases using hypergraphs, Ann Inst Stat Math (2017) 69: 673.
- S. Petrovic. A survey of discrete methods in (algebraic) statistics for networks, Contemporary Mathematics, vol. 685, 260-281, American Mathematical Society, 2017.
- R. H. Sloan, D. Stasi and G. Túran. Hydras: Directed Hypergraphs and Horn Formulas. Theoretical Computer Science, Volume 658, Part B, 2017.
- J. A. De Loera, S. Petrovic, and D. Stasi. Random Sampling in Computational Algebra: Helly Numbers and Violator Spaces, J Symb Comp 77 (2016) pp.1-15.
- R. Fulek, M.J. Pelsmajer, and M. Schaefer. Hanani-Tutte for Radial Planarity II. In: Hu Y., Nöllenburg M. (eds) Graph Drawing and Network Visualization. GD 2016. LNCS 9801 (2016) pp. 468-481.
- S. Antaris, D. Stasi, M. Högqvist, G. Pallis, M. D. Dikaiakos. A Socio-Aware Decentralized Topology Construction Protocol, in the Proceedings of the Third IEEE Workshop on Hot Topics in Web Systems and Technologies (2015): 91-96.
- J. A. De Loera, S. Marguiles, M. Pernpeintner, E. Reidl, D. Rolnic, G. Spencer, D. Stasi, and J. Swenson. Graph-Coloring Ideals: Nullstellensatz Certificates, Gröbner Bases for Chordal Graphs, and Hardness of Gröbner Bases. In International Symposium on Symbolic and Algebraic Computation, 2015.
- R. Fulek, M.J. Pelsmajer, and M. Schaefer. Hanani-Tutte for Radial Planarity, In: Di Giacomo E., Lubiw A. (eds) Graph Drawing and Network Visualization LNCS 9411 (2015) pp.99-110.
- A. Kündgen, M.J. Pelsmajer, and R. Ramamurthi. Finding Minors in Graphs with a Given Path Structure, J. Graph Theory 79 (2015) no.1 pp. 30-47.
- S. Petrovic and D. Stasi. Toric algebra of hypergraphs, J Alg Comb 39 (2014) no. 1, 187-208.
- D. Stasi, K. Sadeghi, A. Rinaldo, S. Petrovic, and S. E. Fienberg. Beta models for random hypergraphs with a given degree sequence. In Proceedings of 21st International Conference on Computational Statistics, 2014.
Recent Research Grants
- NSF DMS-1522662 (PIs Petrovic and Stasi) Randomized algorithms in computational algebra, 2015-2018.
- NSA H98230-08-1-0043 Young Investigators Grant (PI Pelsmajer) Topological Approaches to Avoiding Crossings, 2008-2011.