(This page is currently under construction-- papers are being added. Let me know if there are particular papers you'd like me to add.)

• On the Relativized Alon Second Eigenvalue Conjecture VI: Sharp Bounds for Ramanujan Base Graphs, with David Kohler, November 13, 2019. arxiv posting.
• On the Relativized Alon Second Eigenvalue Conjecture V: Sharp Bounds for Ramanujan Base Graphs, with David Kohler, November 13, 2019. arxiv posting.
• A Relativized Alon Second Eigenvalue Conjecture for Regular Base Graphs IV: An Improved Sidestepping Theorem, with David Kohler, November 13, 2019. arxiv posting.
• On the Relativized Alon Second Eigenvalue Conjecture III: Asymptotic Expansions for Tangle-Free Hashimoto Traces, with David Kohler, November 13, 2019. arxiv posting.
• On the Relativized Alon Eigenvalue Conjecture II: Asymptotic Expansion Theorems for Walks, with David Kohler, November 13, 2019. arxiv posting.
• On the Relativized Alon Second Eigenvalue Conjecture I: Main Theorems, Examples, and Outline of Proof, with David Kohler, November 13, 2019. arxiv posting.
• Sheaves and Duality in the Two-Vertex Graph Riemann-Roch Theorem, with Nicolas Folinsbee, December 23, 2017, last modified June 27, 2019, Older arxiv posting.
• Formal Zeta Function Expansions and the Frequency of Ramanujan Graphs, June 17, 2014, last modified May 4, 2019. Older arxiv posting.

Older papers:

If a paper has figures, these figures will not be visible with the "dvi" version of the paper.

Here is a remarkable and much shorter proof of the Hanna Neumann Conjecture, which simplifies, generalizes, and clarifies aspects of my sheaf theoretic proof, by Warren Dicks, written entirely in the language of skew group rings.

• Linear Algebra and the Hanna Neumann Conjecture, July 28, 2011.
• Sheaves on Graphs, Their Homological Invariants, and a Proof of the Hanna Neumann Conjecture, May 30, 2011. Note: this paper is a merged version (with redundancies eliminated) of the two papers listed just below.
• Sheaves on Graphs and a Proof of the Hanna Neumann Conjecture, April 30, 2011.
• Sheaves on Graphs and Their Homological Invariants, April 13, 2011.
• The non-backtracking spectrum of the universal cover of a graph, with Omer Angel and Shlomo Hoory, last modified November 20, 2007.
• Cohomology in Grothendieck Topologies and Lower Bounds in Boolean Complexity II: A Simple Example (8 pages), last modified March 30, 2006.
• Cohomology in Grothendieck Topologies and Lower Bounds in Boolean Complexity (70 pages), last modified December 2, 2005.
• A Proof of Alon's Second Eigenvalue Conjecture (118 pages), last modified (third revision) May 3, 2004. Accepted to the Memoirs of the A.M.S.
• Generalized Alon-Boppana Theorems and Error-Correcting Codes, with Jean-Pierre Tillich, last modified May 27, 2002.
• Wave Equations for Graphs and The Edge-based Laplacian, with Jean-Pierre Tillich, last modified May 24, 2002, accepted to the Pacific Journal of Mathematics.
• Relative Expanders or Weakly Relatively Ramanujan Graphs, last modified April 8, 2002, Duke Mathematical Journal
• Recognizing more unsatisfiable random 3-SAT instances efficiently, with Andreas Goerdt, last modified January 26, 2001.
• Laplacian Eigenvalues and Distances Between Subsets of a Manifold, with Jean-Pierre Tillich, last modified November 22, 2000.
• On Cayley Graphs on the Symmetric Group Generated by Tranpositions, last modified March 27, 2000.
• The Action of a Few Permutations on r-tuples is Quickly Transitive, with Antoine Joux, Yuval Roichman, Jacques Stern, and Jean-Pierre Tillich, last modified January 23, 1998.
• Computing Betti Numbers via Combinatorial Laplacians, last modified June 4, 1997.
• On the Betti Numbers of Chessboard Complexes, with Phil Hanlon, last modified May 14, 1997.
• Relative Expansion and an Extremal Degree Two Cover of the Boolean Cube, last modified 1993--5.
• Some Geometric Aspects of Graphs and their Eigenfunctions, last modified October 26, 1992 (or earlier?).
• On the Bit Extraction Problem, last modified August 12, 1992.
• A Note on Matrix Rigidity, last modified October 25, 1991.
• The Spectra of Infinite Hypertrees, last modified September 13, 1990.
• On the Second Eigenvalue of Hypergraphs, last modified November 11, 1989.
• Some Graphs with Small Second Eigenvalue, last modified October 12, 1989.
• A Deterministic View of Random Sampling and its Use in Geometry, with Bernard Chazelle, last modified ???
• On the Second Eigenvalue and Random Walks in Random d-Regular Graphs, last modified May 22, 1989.
• On the Roadmap Coloring Problem, last modified January 1989.
• A Density Theorem for Purely Iterative Zero Finding Methods, last modified January 1989.
• On the Convergence of Newton's Method, last modified September 1988.
• Random Polynomials and Approximate Zeros of Newton's Method, last modified November 1987.