About

Paul Gunnells is a full professor in the Department of Mathematics and Statistics at the University of Massachusetts Amherst. He earned his Ph.D. from the Massachusetts Institute of Technology in 1994 and his B.A. from Stanford University in 1989. His research interests include number theory, with specific focus on elliptic curves, Weyl group multiple Dirichlet series, Coxeter groups, and the geometry of mathematical structures such as tetrahedron spaces. Gunnells has contributed to the understanding of Dedekind sums, Eisenstein cocycles, and special values of L-functions, among other topics. His work involves constructing and analyzing complex mathematical objects, often in collaboration with other researchers, and he is involved in teaching, research, and interdisciplinary programs within a diverse and inclusive academic community.

Research topics

• Computer Security
• Computer Science
• Mathematics
• Computer engineering
• Algorithm
• Pure mathematics
• Computer hardware
• Combinatorics
• Theoretical computer science
• Computer network

Selected publications

• Sector length distributions of recursively definable graph states through analytic combinatorics

  arXiv (Cornell University) · 2026-04-10

  preprintOpen access

  The sector length distribution or Shor-Laflamme distribution (SLD) of quantum states is governed by the $k$-body correlations amongst the different systems, and has been used to study entanglement and error correction. A succinct description of a quantum state's SLD can be obtained by representing it through the coefficients of an appropriate weight enumerator polynomial, yielding bounds on fidelity under depolarizing noise and on multipartite entanglement. However, such expressions quickly grow out of hand and are generally difficult to achieve analytically, reflecting the computational hardness of the SLD. We sidestep this problem and, instead of a single state's SLDs, encode a family of quantum state's SLD as a generating function. We then find closed-form expressions for a large class of graph states which we call `recursively definable' and which include many common graphs such as path graphs, cycle graphs, star graphs, grid graphs, and more. As direct corollary, we obtain analytical expressions for such graph states' concentratable entanglement, bounds on their depolarizing fidelity, and a multipartite entanglement criterion. Our work opens up the use of generating functions and more generally analytic combinatorics to solve problems in quantum information theory.

  PublisherDOI

• Black-white polynomials of graphs and generating functions

  arXiv (Cornell University) · 2026-04-12

  preprintOpen accessSenior author

  Let G be a graph. The black-white polynomial W_G(t) enumerates colorings of the vertices of G with two colors (black and white), where the power of t keeps track of how many white vertices have an even number of black neighbors. Such polynomials appear in quantum information theory, where they are used to capture properties of the entanglement in certain quantum states described by graphs. In this paper we describe how to use generating functions to compute these polynomials for various families X of graphs. Our main results are the following: (i) we describe some constructions under which X leads to a rational generating function; (ii) we use a matrix model to construct the exponential generating function of the black-white polynomials of all graphs; and (iii) we generalize a construction of Wright to build exponential generating functions of black-white polynomials for graphs of a given loop number.

  PublisherDOI

• Sector length distributions of recursively definable graph states through analytic combinatorics

  ArXiv.org · 2026-04-10

  articleOpen access

  The sector length distribution or Shor-Laflamme distribution (SLD) of quantum states is governed by the $k$-body correlations amongst the different systems, and has been used to study entanglement and error correction. A succinct description of a quantum state's SLD can be obtained by representing it through the coefficients of an appropriate weight enumerator polynomial, yielding bounds on fidelity under depolarizing noise and on multipartite entanglement. However, such expressions quickly grow out of hand and are generally difficult to achieve analytically, reflecting the computational hardness of the SLD. We sidestep this problem and, instead of a single state's SLDs, encode a family of quantum state's SLD as a generating function. We then find closed-form expressions for a large class of graph states which we call `recursively definable' and which include many common graphs such as path graphs, cycle graphs, star graphs, grid graphs, and more. As direct corollary, we obtain analytical expressions for such graph states' concentratable entanglement, bounds on their depolarizing fidelity, and a multipartite entanglement criterion. Our work opens up the use of generating functions and more generally analytic combinatorics to solve problems in quantum information theory.

  PublisherOA PDF

• Black-white polynomials of graphs and generating functions

  ArXiv.org · 2026-04-12

  articleOpen accessSenior author

  Let G be a graph. The black-white polynomial W_G(t) enumerates colorings of the vertices of G with two colors (black and white), where the power of t keeps track of how many white vertices have an even number of black neighbors. Such polynomials appear in quantum information theory, where they are used to capture properties of the entanglement in certain quantum states described by graphs. In this paper we describe how to use generating functions to compute these polynomials for various families X of graphs. Our main results are the following: (i) we describe some constructions under which X leads to a rational generating function; (ii) we use a matrix model to construct the exponential generating function of the black-white polynomials of all graphs; and (iii) we generalize a construction of Wright to build exponential generating functions of black-white polynomials for graphs of a given loop number.

  PublisherOA PDF

• Explicit sharbly cycles at the virtual cohomological dimension for $$\textrm{SL}_n(\mathbb {Z})$$

  Journal of Homotopy and Related Structures · 2025-07-09

  articleCorresponding
  PublisherDOI

• Cohomology with Sym coefficients for congruence subgroups of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msub><mml:mrow><mml:mi mathvariant="normal">SL</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi mathvariant="double-struck">Z</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> and Galois representations

  Journal of Algebra · 2025-04-17

  articleCorresponding
  PublisherDOI

• Cohomology with Sym^g coefficients for congruence subgroups of SL_4(Z) and Galois representations

  arXiv (Cornell University) · 2024-05-13

  preprintOpen access

  We extend the computations in our prior work to find the cohomology in degree five of a congruence subgroup Gamma of SL_4(Z) with coefficients in Sym^g(K^4), twisted by a nebentype character eta, along with the action of the Hecke algebra. This is the top cuspidal degree. In this paper we take K to be a finite field of large characteristic, as a proxy for the complex numbers. For each Hecke eigenclass found, we produce the unique Galois representation that appears to be attached to it. The computations require modifications to our previous algorithms to accommodate the fact that the coefficients are not one-dimensional.

  PublisherOA PDFDOI

• Explicit sharbly cycles at the virtual cohomological dimension for SL_n(Z)

  arXiv (Cornell University) · 2024-02-21

  preprintOpen access

  Denote the virtual cohomological dimension of SL_n(Z) by t=n(n-1)/2. Let St denote the Steinberg module of SL_n(Q) tensored with Q. Let Sh_* denote the sharbly resolution of the Steinberg module St. By Borel-Serre duality, the one-dimensional Q-vector space H^0(SL_n(Z), Q) is isomorphic to H_t(SL_n(Z),St). We find an explicit generator of H_t(SL_n(Z),St) in terms of sharbly cycles and cosharbly cocycles. These methods may extend to other degrees of cohomology of SL_n(Z).

  PublisherOA PDFDOI

• Hypergraph matrix models and generating functions

  Combinatorics and Number Theory · 2024-07-01

  article1st authorCorresponding
  PublisherDOI

• On the Cohomology of GL ₂ and SL ₂ over Imaginary Quadratic Fields

  Experimental Mathematics · 2024-09-01

  articleOpen accessCorresponding

  We report on computations of the cohomology of (Formula presented.) and (Formula presented.), where D < 0 is a fundamental discriminant. These computations go well beyond earlier results of Vogtmann and Scheutzow. We use the technique of homology of Voronoi complexes, and our computations recover the integral cohomology away from the primes 2, 3. We observed exponential growth in the torsion subgroup of H 2 as (Formula presented.) increases, and compared our data to bounds of Rohlfs.

  PublisherOA PDFDOI

Recent grants

• Number Theory, Algebraic Geometry & Representation Theory

  NSF · $106k · 2004–2008

• Problems in number theory and representation theory

  NSF · $150k · 2008–2012

• EAGER: Braid Statistics and Hard Problems in Braid Groups with Applications to Cryptography

  NSF · $150k · 2015–2019

• Multiple Dirichlet series, Whittaker functions, and the cohomology of arithmetic groups

  NSF · $150k · 2015–2021

• Problems in arithmetic groups and multiple Dirichlet series.

  NSF · $156k · 2011–2016

Frequent coauthors

• Dan Yasaki

  University of North Carolina at Greensboro

  23 shared

• Avner Ash

  Boston College

  21 shared

• Mark McConnell

  20 shared

• Dorian Goldfeld

  Columbia University

  18 shared

• Gautam Chinta

  17 shared

• Lev Borisov

  Rutgers, The State University of New Jersey

  15 shared

• Iris Anshel

  15 shared

• Richard A. Scott

  10 shared

Education

• PhD, Mathematics

  Massachusetts Institute of Technology

  1994

• BS, Mathematics

  Stanford University

  1989