## Person: Gunnells, Paul

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##### Research Projects

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##### Job Title

Professor, Department of Mathematics and Statistics

##### Last Name

Gunnells

##### First Name

Paul

##### Discipline

Algebraic Geometry

##### Expertise

Algebraic Geometry

Number Theory

Topology of Singular Spaces

Number Theory

Topology of Singular Spaces

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## Search Results

Now showing 1 - 10 of 26

Publication A characterization of Dynkin elements(2003-01-01) Gunnells, PE; Sommers, EWe give a characterization of the Dynkin elements of a simple Lie algebra. Namely, we prove that one-half of a Dynkin element is the unique point of minimal length in its $N$-region. In type $A_n$ this translates into a statement about the regions determined by the canonical left Kazhdan-Lusztig cells, which leads to some conjectures in representation theory.Publication Evaluation of Dedekind sums, Eisenstein cocycles, and special values of L-functions(2003-01-01) Gunnells, PE; Sczech, RWe define higher-dimensional Dedekind sums that generalize the classical Dedekind-Rademacher sums as well as Zagier's sums, and we show how to compute them effectively using a generalization of the continued-fraction algorithm. We present two applications. First, we show how to express special values of partial zeta functions associated to totally real number fields in terms of these sums via the Eisenstein cocycle introduced by R. Sczech. Hence we obtain a polynomial time algorithm for computing these special values. Second, we show how to use our techniques to compute certain special values of the Witten zeta function, and we compute some explicit examples.Publication Weyl group multiple Dirichlet series constructed from quadratic characters(2007-01-01) Chinta, G; Gunnells, PEWe construct multiple Dirichlet series in several complex variables whose coefficients involve quadratic residue symbols. The series are shown to have an analytic continuation and satisfy a certain group of functional equations. These are the first examples of an infinite collection of unstable Weyl group multiple Dirichlet series in greater than two variables having the properties predicted in [2].Publication Modular forms and elliptic curves over the field of fifth roots of unity(2010-01-01) Gunnells, PE; Hajir, F; Yasaki, DanLet F be the cyclotomic field of fifth roots of unity. We computationally investigate modularity of elliptic curves over FPublication Hecke operators and Q-groups associated to self-adjoint homogeneous cones(2003-01-01) Gunnells, PE; McConnell, MLet G be a reductive algebraic group associated to a self-adjoint homogeneous cone defined over , and let Î“G be an appropriate neat arithmetic subgroup. We present two algorithms to compute the action of the Hecke operators on for all i. This simultaneously generalizes the modular symbol algorithm of Ash-Rudolph (Invent. Math. 55 (1979) 241) to a larger class of groups, and proposes techniques to compute the Hecke-module structure of previously inaccessible cohomology groups.Publication CONSTRUCTING WEYL GROUP MULTIPLE DIRICHLET SERIES(2010-01-01) Chinta, G; Gunnells, PELet be a reduced root system of rank . A Weyl group multiple Dirichlet series for is a Dirichlet series in complex variables , initially converging for sufficiently large, that has meromorphic continuation to and satisfies functional equations under the transformations of corresponding to the Weyl group of . A heuristic definition of such a series was given by Brubaker, Bump, Chinta, Friedberg, and Hoffstein, and they have been investigated in certain special cases by others. In this paper we generalize results by Chinta and Gunnells to construct Weyl group multiple Dirichlet series by a uniform method and show in all cases that they have the expected properties.Publication TORSION IN THE COHOMOLOGY OF CONGRUENCE SUBGROUPS OF SL(4, Z) AND GALOIS REPRESENTATIONS(2010-01-01) Ash, A; Gunnells, PEWe report on the computation of torsion in certain homology the-ories of congruence subgroups of SL(4, Z). Among these are the usual group cohomology, the Tate-Farrell cohomology, and the homology of the sharbly complex. All of these theories yield Hecke modules. We conjecture that the Hecke eigenclasses in these theories have attached Galois representations. The interpretation of our computations at the torsion primes 2,3,5 is explained. We provide evidence for our conjecture in the 15 cases of odd torsion that we found in levels 31.Publication On the p-parts of quadratic Weyl group multiple Dirichlet series(2008-01-01) Chinta, G; Friedberg, S; Gunnells, PELet Î¦ be a reduced root system of rank r. A Weyl group multiple Dirichlet series for Î¦ is a Dirichlet series in r complex variables s1,â€¦, sr , initially converging for (si) sufficiently large, which has meromorphic continuation to r and satisfies functional equations under the transformations of r corresponding to the Weyl group of Î¦. Two constructions of such series are available, one [B. Brubaker, D. Bump, G. Chinta, S. Friedberg, and J. Hoffstein, Weyl group multiple Dirichlet series I, in: Multiple Dirichlet Series, Automorphic Forms, and Analytic Number Theory, S. Friedberg, D. Bump, D. Goldfeld, and J. Hoffstein, eds., Proc. Symp. Pure Math. 75 (2006), 91â€“114.] [B. Brubaker, D. Bump, and S. Friedberg, Twisted Weyl group multiple Dirichlet series: the stable case, in: Eisenstein Series and Applications, Gan, Kudla, and Tschinkel, eds., Progr. Math. 258 (2008), 1â€“26.] [B. Brubaker, D. Bump, S. Friedberg, and J. Hoffstein, Weyl group multiple Dirichlet series III: Eisenstein series and twisted unstable Ar, Ann. Math. 166 (2007), 293â€“316.] [B. Brubaker, D. Bump, and S. Friedberg, Weyl group multiple Dirichlet series II, The stable case, Invent. Math. 165 (2006), no. 2, 325â€“355.] based on summing products of n-th order Gauss sums, the second [G. Chinta and P. E. Gunnells, Weyl group multiple Dirichlet series constructed from quadratic characters, Invent. Math. 167 (2007), no. 2, 327â€“353.] based on averaging a certain group action over the Weyl group. In each case, the essential work occurs at a generic prime p; the local factors, satisfying local functional equations, are then pieced into a global object. In this paper we study these constructions and the relationship between them. First we extend the averaging construction to obtain twisted Weyl group multiple Dirichlet series, whose p-parts are given by evaluating certain rational functions in r variables. Then we develop properties of such a rational function, giving its precise denominator, showing that the nonzero coefficients of its numerator are indexed by points that are contained in a certain convex polytope, determining the coefficients corresponding to the vertices, and showing that in the untwisted case the rational function is uniquely determined from its polar behavior and the local functional equations. We also give evidence that in the case Î¦ = Ar, the p-part obtained here exactly matches the p-part of the twisted multiple Dirichlet series introduced in [B. Brubaker, D. Bump, S. Friedberg, and J. Hoffstein, Weyl group multiple Dirichlet series III: Eisenstein series and twisted unstable Ar, Ann. Math. 166 (2007), 293â€“316.] when n = 2.Publication Toric modular forms of higher weight(2003-01-01) Borisov, LA; Gunnells, PEPublication CELLS IN COXETER GROUPS I(2010-01-01) Belolipetsky, M; Gunnells, PE