Research
////////////////////////////////////////////

My research interests include:

  • stochastic topology (random topological spaces, especially in combinatorial settings)
  • topological statistical mechanics (configuration spaces of particles, homological percolation)
  • geometric group theory (random groups, especially hyperbolicity and Kazhdan's Property (T))
  • random graphs (topological and geometric properties, spectral expansion)
  • graph coloring and graph homomorphisms (obstructions to graph maps from statistical physics and topology)
  • discrete geometry (unit-distance graphs, disk packing)

    My research is supported in part by the National Science Foundation (NSF). I am currently the PI on an NSF-DMS grant on "Stochastic topology and topological statistical mechanics" (2020–2023), a TRIPODS+X:RES grant on "Thermodynamic Phases and Configuration Space Topology" (2019–2022), and an RTG grant on "Algebraic Topology and its Applications" (2016–2023).

    Most of my papers are available in preprint form on the arXiv.


    Preprints

    Hannah Alpert, Matthew Kahle, Robert MacPherson. "Asymptotic Betti numbers for hard squares in the homological liquid regime." (submitted). arXiv:2207.13139.

    Benjamin Braun, Kaitlin Bruegge, Matthew Kahle. "Facets of Random Symmetric Edge Polytopes, Degree Sequences, and Clustering." (submitted). arXiv:2204.07239.

    Paul Duncan, Matthew Kahle, Benjamin Schweinhart. "Homological percolation on a torus: plaquettes and permutohedra." (submitted). arXiv:2011.11903.

    Sukhada Fadnavis, Matthew Kahle, Francisco Martinez-Figueroa. "Warmth and mobility of random graphs." (revised in 2021 and submitted). arXiv:1009.0792.

    Papers

    Hannah Alpert, Ulrich Bauer, Matthew Kahle, Robert MacPherson, Kelly Spendlove. "Homology of configuration spaces of hard squares in a rectangle." to appear in Algebraic & Geometric Topology. arXiv:2010.14480.

    Matthew Kahle, Elliot Paquette, Érika Roldán. "Topology of random 2-dimensional cubical complexes." to appear in Forum of Mathematics, Sigma. arxiv:2001.07812.

    Matthew Kahle, Andrew Newman. "Topology and geometry of random 2-dimensional hypertrees." to appear in Discrete & Computational Geometry. arXiv:2004.13572.

    Matthew Kahle, Francisco Martinez-Figueroa. "The chromatic number of random Borsuk graphs." to appear in Random Structures & Algorithms. arXiv:1901.08488.

    Hannah Alpert, Matthew Kahle, Robert MacPherson. "Configuration spaces of disks in an infinite strip." Journal of Applied and Computational Topology. May 2021. arXiv:1908.04241.

    Matthew Kahle, Érika Roldán. "Polyominoes with maximally many holes." Geombinatorics. Aug 2019. arXiv:1807.10231.

    Christopher Hoffman, Matthew Kahle, Elliot Paquette. "Spectral gaps of random graphs and applications." International Mathematics Research Notices (IMRN). May 2019. arXiv:1201.0425.

    Matthew Kahle. "Branko Grünbaum in many dimensions." Geombinatorics. Jan 2019. arXiv:1901.08622.

    Matthew Kahle, Frank Lutz, Andrew Newman, Kyle Parsons. "Cohen–Lenstra heuristics for torsion in homology of random complexes." Experimental Mathematics. Jun 2018. arXiv:1710.05683.

    Omer Bobrowski, Matthew Kahle. "Topology of random geometric complexes: a survey." Journal of Applied and Computational Topology. Jun 2018. arXiv:1409.4734.

    Dominic Dotterrer, Larry Guth, Matthew Kahle. "2-complexes with large 2-girth." Discrete & Computational Geometry. Apr 2018. arxiv:1509.03871.

    Matthew Kahle. "Random simplicial complexes." Book chapter for Handbook of Discrete & Computational Geometry (3rd edition). Nov 2017. arXiv:1607.07069.

    Omer Bobrowski, Matthew Kahle, Primoz Skraba. "Maximally Persistent Cycles in Random Geometric Complexes." Annals of Applied Probability. Oct 2017. arXiv:1509.04347.

    Christopher Hoffman, Matthew Kahle, Elliot Paquette. "The threshold for integer homology in random d-complexes." Discrete & Computational Geometry. May 2017. arXiv:1308.6232.

    Matthew Kahle, Elizabeth Meckes. "Erratum: Limit theorems for Betti numbers of random simplicial complexes." Homology, Homotopy and Applications. Jan 2016. arXiv:1501.03759.

    Matthew Kahle, Boris Pittel. "Inside the critical window for cohomology of random k-complexes." Random Structures & Algorithms . Jan 2016. arXiv:1301.1324.

    Matthew Kahle, Birra Taha. "New lower bounds on $\chi(R^d)$ for $d=8 \dots 12$." Geombinatorics . Jul 2015. arXiv 1409:1278.

    Matthew Kahle. "Topology of random simplicial complexes: a survey." AMS Contemporary Volumes in Mathematics, Algebraic Topology: Applications and New Directions. Nov 2014. arXiv:1301.7165.

    Matthew Kahle. "Sharp vanishing thresholds for cohomology of random flag complexes." Annals of Mathematics. Aug 2014. arXiv:1207.0149.

    Michael Davis, Matthew Kahle. "Random graph products of finite groups are rational duality groups." Journal of Topology. Feb 2014. arXiv:1210.4577.

    Matthew Kahle, Elizabeth Meckes. "Limit theorems for Betti numbers of random simplicial complexes." Homology, Homotopy and Applications. May 2013. arXiv:1009.4130.

    Yuliy Baryshnikov, Peter Bubenik, Matthew Kahle. "Min-type Morse theory for configuration spaces of hard spheres." International Mathematics Research Notices (IMRN). Feb 2013. arxiv:1108.3061.

    Dominic Dotterrer, Matthew Kahle. "Coboundary expanders." Journal of Topology and Analysis. Nov 2012. arXiv:1012.5316.

    Matthew Kahle. "Sparse locally-jammed disk packings." Annals of Combinatorics. Oct 2012.

    Gunnar Carlsson, Jackson Gorham, Matthew Kahle, Jeremy Mason. "Computational topology for configuration spaces of hard disks." Physical Review E. Jan 2012. arXiv:1108.5719.

    Matthew Kahle. "Random geometric complexes." Discrete & Computational Geometry. Jan 2011. arXiv:0910.1649.

    Eric Babson, Christopher Hoffman, Matthew Kahle. "The fundamental group of random 2-complexes." Journal of the American Mathematical Society. Jan 2011. arXiv:0711.2704.

    Matthew Kahle. "Points in a triangle forcing small triangles." Geombinatorics. Oct 2009. arXiv:0811.2449.

    Matthew Kahle. "Topology of random clique complexes." Discrete Math. Aug 2009. arXiv:0605536.

    Matthew Kahle. "The neighborhood complex of a random graph." Journal of Combinatorial Theory, Series A. Sep 2007. arXiv:math/0512077.

    Matthew Kahle. "Scatters, unavoidable shapes, and crystallization." Geombinatorics. Oct 2005.