Douglas M Pickrell
 Associate Professor, Mathematics
 Associate Professor, Applied Mathematics  GIDP
 Member of the Graduate Faculty
Contact
 (520) 6214767
 Mathematics, Rm. 703
 Tucson, AZ 85721
 pickrell@arizona.edu
Bio
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Interests
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Courses
202425 Courses

Complex Analysis
MATH 520A (Fall 2024) 
Directed Research
MATH 492 (Fall 2024) 
Theory of Probability
MATH 464 (Fall 2024)
202324 Courses

Topological Spaces
MATH 432 (Spring 2024) 
Topological Spaces
MATH 532 (Spring 2024) 
Calculus I
MATH 125 (Fall 2023) 
Independent Study
MATH 599 (Fall 2023) 
Intro Math Physics
MATH 541 (Fall 2023) 
Intro Math Physics
PHYS 541 (Fall 2023)
202122 Courses

Dissertation
MATH 920 (Spring 2022) 
Independent Study
MATH 499 (Spring 2022) 
Real Analysis
MATH 523B (Spring 2022) 
Calculus II
MATH 129 (Fall 2021) 
Directed Research
MATH 492 (Fall 2021) 
Dissertation
MATH 920 (Fall 2021) 
Real Analysis
MATH 523A (Fall 2021)
202021 Courses

Independent Study
MATH 499 (Spring 2021) 
Independent Study
MATH 599 (Spring 2021) 
TopologyGeometry
MATH 534B (Spring 2021) 
Calculus II
MATH 129 (Fall 2020) 
Independent Study
MATH 599 (Fall 2020) 
Internship
MATH 593 (Fall 2020) 
TopologyGeometry
MATH 534A (Fall 2020)
201920 Courses

Independent Study
MATH 599 (Spring 2020) 
Theory of Probability
MATH 464 (Spring 2020) 
Adv Applied Mathematics
MATH 422 (Fall 2019) 
Independent Study
MATH 599 (Fall 2019) 
Internship
MATH 593 (Fall 2019)
201819 Courses

Honors Thesis
MATH 498H (Spring 2019) 
Real Analysis
MATH 523B (Spring 2019) 
Calculus II
MATH 129 (Fall 2018) 
Honors Thesis
MATH 498H (Fall 2018) 
Real Analysis
MATH 523A (Fall 2018)
201718 Courses

Directed Research
MATH 492 (Spring 2018) 
Independent Study
MATH 499 (Spring 2018) 
Independent Study
MATH 599 (Spring 2018) 
Theory of Probability
MATH 464 (Spring 2018) 
Adv Applied Mathematics
MATH 422 (Fall 2017) 
Adv Applied Mathematics
MATH 522 (Fall 2017) 
Directed Research
MATH 492 (Fall 2017) 
Intro Math Physics
MATH 541 (Fall 2017) 
Intro Math Physics
PHYS 541 (Fall 2017)
201617 Courses

Theory of Probability
MATH 464 (Spring 2017) 
Calculus II
MATH 129 (Fall 2016) 
Linear Algebra
MATH 513 (Fall 2016)
201516 Courses

TopologyGeometry
MATH 534B (Spring 2016)
Scholarly Contributions
Journals/Publications
 Pickrell, D. M., & Dalthorp, M. (2019). Homeomorphisms of a circle and factorization. IMRN, 38.
 Pickrell, D. M., & LorraineHubach, A. (2019). sobolev mapping groups and curvature. Letters in Mathematical Physics, 13. doi:10.1007/s1100501801152w
 Pickrell, D. M. (2018). complex groups and factorization. Journal of Lie Theory, 28(4), 10951118.
 Pickrell, D. M., & Basor, E. (2018). Loops in SL(2,C) and factorization. Random Matrices: Theory and Application, 7(3), 26.
 Pickrell, D. M., & Caine, A. (2017). Noncompact Groups of Hermitian Symmetric Type and Factorization. Transformation Groups, 20. doi:doi:10.1007/s0003101794202
 Pickrell, D. M., & Basor, E. (2016). Loops in SU(2), Riemann surfaces, and factorization. SIGMA, 12(025), 29.
 Caine, A. (2015). Loops in noncompact groups and factorization. Journal of generalized Lie theory and applications, 9(2), 30.
 Pickrell, D. M., & Caine, A. (2015). Loops in groups of Hermitian symmetric type and factorization. Journal of generalized Lie theory and applications, 9, 30. doi:10.4172/17364337
 Chavez, A. (2014). Werner's measure on selfavoiding loops and welding. SIGMA, 10(081), 42.
 Pickrell, D. (2011). Loops in SU(2) and factorization. Journal of Functional Analysis, 260(8), 21912221.More infoAbstract: We discuss analytic issues associated with a refinement of triangular factorization for the loop group of SU(2). This factorization is of interest because (1) Toeplitz determinants factor in the associated coordinates, and (2) the factorization is intimately related to the critical degree of smoothness for loops, W1/2,L2. © 2011 Elsevier Inc.
 Pickrell, D., & PittmanPolletta, B. (2010). Unitary loop groups and factorization. Journal of Lie Theory, 20(1), 93112.More infoAbstract: We discuss a refinement of triangular factorization for unitary matrixvalued functions on S1 . © 2010 Heldermann Verlag.
 Caine, A., & Pickrell, D. (2009). Homogeneous Poisson structures on symmetric spaces. International Mathematics Research Notices, 2009(1), 98140.More infoAbstract: We calculate, in a relatively explicit way, the Hamiltonian systems that arise from the EvensLu construction of homogeneous Poisson structures on both compact and noncompacttype symmetric spaces. A corollary is that the Hamiltonian system arising in the noncompact case is isomorphic to the generic Hamiltonian system arising in the compact case. In the group case, these systems are also isomorphic to those arising from the BruhatPoisson structure on the flag space, and hence, by results of Lu, can be completely factored. © 2008 The Author. Published by Oxford University Press. All rights reserved.
 Pickrell, D. (2009). Consistency of regularization for free scalar fields. Letters in Mathematical Physics, 87(3), 283291.More infoAbstract: In two dimensional constructive quantum field theory for scalar fields, it is necessary to regularize both the action and the total (Gaussian) volume. In this paper we consider the compatibility of these regularizations. © 2009 Springer.
 Pickrell, D. (2008). Homogeneous Poisson structures on loop spaces of symmetric spaces. Symmetry, Integrability and Geometry: Methods and Applications (SIGMA), 4.More infoAbstract: This paper is a sequel to [Caine A., Pickrell D., Int. Math. Res. Not., to appear, arXiv:0710.4484], where we studied the Hamiltonian systems which arise from the EvensLu construction of homogeneous Poisson structures on both compact and noncompact type symmetric spaces. In this paper we consider loop space analogues. Many of the results extend in a relatively routine way to the loop space setting, but new issues emerge. The main point of this paper is to spell out the meaning of the results, especially in the SU(2) case. Applications include integral formulas and factorizations for Toeplitz determinants.
 Pickrell, D. (2008). P(φ)_{2} quantum field theories and Segal's axioms. Communications in Mathematical Physics, 280(2), 403425.More infoAbstract: The purpose of this paper is to show that P 2 Euclidean quantum field theories satisfy axioms of the type advocated by Graeme Segal. © 2008 SpringerVerlag.
 Pickrell, D. (2006). An invariant measure for the loop space of a simply connected compact symmetric space. Journal of Functional Analysis, 234(2), 321363.More infoAbstract: Let X denote a simply connected compact Riemannian symmetric space, U the universal covering of the identity component of the group of automorphisms of X, and LU the loop group of U. In this paper we prove the existence (and conjecture the uniqueness) of an LUinvariant probability measure on a distributional completion of the loop space of X. © 2005 Elsevier Inc. All rights reserved.
 Pickrell, D. (2006). The diagonal distribution for the invariant measure of a unitary type symmetric space. Transformation Groups, 11(4), 705724.More infoAbstract: Let Θ denote an involution for a simply connected compact Lie group U let K denote the fixed point set and let μ denote the Uinvariant probability measure on U/K. Consider the geodesic embedding φ :U/K → U:u → uu Θ of Cartan. In this paper we compute the Fourier transform of the diagonal distribution for φ*μ relative to a compatible triangular decomposition of G the complexification of U. This boils down to a DuistermaatHeckman exact stationary phase calculation involving a Poisson structure on the dual symmetric space G0/K discovered by Evens and Lu. © Birkhauser Boston 2006.
 Dai, J., & Pickrell, D. (2004). Coadjoint orbits for the central extension of Diff^{+}(S ^{1}) and their representatives. Acta Mathematica Scientia, 24(2), 185205.More infoAbstract: According to Kirillov's idea, the irreducible unitary representations of a Lie group G roughly correspond to the coadjoint orbits script O sign. In the forward direction one applies the methods of geometric quantization to produce a representation, and in the reverse direction one computes a transform of the character of a representation, to obtain a coadjoint orbit. The method of orbits in the representations of Lie groups suggests the detailed study of coadjoint orbits of a Lie group G in the space script G sign* dual to the Lie algebra script G sign of G. In this paper, two primary goals are achieved: one is to completely classify the smooth coadjoint orbits of Virasoro group for nonzero central charge c; the other is to find representatives for coadjoint orbits. These questions have been considered previously by Segal, Kirillov, and Witten, but their results are not quite complete. To accomplish this, the authors start by describing the coadjoint action of Dthe Lie group of all orientation preserving diffeomorphisms on the circle S1, and its central extension D̄, then the authors will give a complete classification of smooth coadjoint orbits. In fact, they can be parameterized by a subspace of conjugacy classes of PSU(1, 1). Finally, the authors will show how to find representatives of coadjoint orbits by analyzing the vector fields stabilizing the orbits, and describe the amazing connection between the characteristic (trace) of conjugacy classes of PSU(1, 1) and that of vector fields stabilizing orbits.
 Pickrell, D. (2004). H^{4}(BK,Z) and Operator Algebras. Journal of Lie Theory, 14(1), 199213.More infoAbstract: There is a wellknown interpretation of group cohomology in terms of (generalized) group extensions. For a connected semisimple compact Lie group K, we prove that the extensions corresponding to classes in H 4(BK,ℤ) can be interpreted in terms of automorphisms of a pair consisting of a type II1 von Neumann algebra and a Cartan subalgebra.
 Pickrell, D. (2004). The radial part of the zeromode Hamiltonian for sigma models with group target space. Reviews in Mathematical Physics, 16(5), 603628.More infoAbstract: In this note, we use geometric arguments to derive a possible form for the radial part of the "zeromode Hamiltonian" for the twodimensional sigma model with target space S3, or more generally a compact simply connected Lie group.
 Dai, J., & Pickrell, D. (2003). The orbit method and the Virasoro extension of Diff^{+} (S^{1}). I. Orbital integrals. Journal of Geometry and Physics, 44(4), 623653.More infoAbstract: Using the finite dimensional example of PSU(1, 1), the universal covering of PSU(1, 1), as a guide, we revisit the orbit method as it applies to D̂, the universal central extension of D = Diff+ (S1). We clarify some aspects of the classification of coadjoint orbits, determine boundedness properties of the natural height function on these orbits, and calculate orbital integrals. © 2002 Elsevier Science B.V. All rights reserved.
 Pickrell, D., & Xia, E. Z. (2003). Ergodicity of mapping class group actions on representation varieties, II. Surfaces with boundary. Transformation Groups, 8(4), 397402.More infoAbstract: The mapping class group of a compact oriented surface of genus greater than one with boundary acts ergodically on connected components of the representation moduli corresponding to a connected compact Lie group, for every choice of conjugacy class boundary condition.
 Pickrell, D., & Xia, E. Z. (2002). Ergodicity of mapping class group actions on representation varieties, I. Closed surfaces. Commentarii Mathematici Helvetici, 77(2), 339362.More infoAbstract: We prove that the mapping class group of a closed surface acts ergodically on connected components of the representation variety corresponding to a connected compact Lie group.
 Neeb, K., & Pickrell, D. (2000). Supplements to the papers entitled "On a Theorem of S. Banach" and "the separable representations of U(H)". Journal of Lie Theory, 10(1), 107109.More infoAbstract: We explain how the theorem of Banach discussed in [Ne97] can be generalized to target spaces which are not necessarily path connected. Moreover, we correct errors in the literature concerning the classification of separable unitary representations for the Banach Lie group U(H) [Pi88, Pi90].
 Pickrell, D. (2000). Invariant measures for unitary groups associated to KacMoody Lie algebras. Memoirs of the American Mathematical Society, 146(693).More infoAbstract: The main purpose of this paper is to prove the existence, and in some cases the uniqueness, of unitarily invariant measures on formal completions of groups associated to affine KacMoody algebras, and associated homogeneous spaces. The basic invariant measure is a natural generalization of Haar measure for a simply connected compact Lie group, and its projection to flag spaces is a generalization of the normalized invariant volume element. The other "invariant measures" are actually measures having values in line bundles over these spaces; these bundlevalued measures heuristically arise from coupling the basic invariant measure to Hermitian structures on associated line bundles, but in this infinite dimensional setting they are generally singular with respect to the basic invariant measure.
 Pickrell, D. (2000). On the action of the group of diffeomorphisms of a surface on sections of the determinant line bundle. Pacific Journal of Mathematics, 193(1), 177199.More infoAbstract: Let Σ denote a closed oriented surface. There is a natural action of the group Diff+(Σ) on sections of the chiral determinant line over the space of gauge equivalence classes of connections. The question we address is whether this action is unitarizable. We introduce a SDiffequivariant regularization, and we prove the existence of, and explicitly compute, the limit as the regularization is removed. The SDiff unitary representations that arise, both by regularization and after removing the regularization, appear to be new.
 Pickrell, D. (1996). On Y M_{2} measures and areapreserving diffeomorphisms. Journal of Geometry and Physics, 19(4), 315367.More infoAbstract: For a given gauge group and compact Riemannian twomanifold, it is known that the associated YangMills measure can be defined directly as a finitely additive measure on the space of connections, and this finitely additive measure is invariant with respect to SDiff, the group of all areapreserving diffeomorphisms of the surface. The first question we address is whether this symmetry essentially characterizes the projection of the YangMills measure to the space of gauge equivalence classes. The proper formulation of this question entails the construction of an SDiffequivariant completion of the space of continuous connections, such that the projection of the YangMills measure to the space of gauge equivalence classes has a countably additive extension. We also consider the coupling of the YangMills measure to determinants of Dirac operators. The basic problems are to prove that the coupled measure is absolutely continuous with respect to the background YangMills measure, to find a reasonable formula for the RadonNikodym derivative, and to analyze the action of SDiff.
 Pickrell, D. (1990). Separable representations for automorphism groups of infinite symmetric spaces. Journal of Functional Analysis, 90(1), 126.More infoAbstract: In this paper we consider the separable unitary representations for the automorphism groups of the classical infinite rank (Finsler) symmetric spaces defined by Schatten pclasses (often referred to as restricted groups). Following earlier work of Ol'shanskii and Voiculescu, it is shown that the spherical representations are always type I, the form of the irreducible spherical functions is determined, and their analyticity established. Using an intuitive geometric argument, it is shown that the real spherical functions extend to the HilbertSchmidt limit and never beyond. This yields a complete determination of the separable representations for groups corresponding to pclasses with p > 2. © 1990.
 Pickrell, D. (1989). On the MickelssonFaddeev extension and unitary representations. Communications in Mathematical Physics, 123(4), 617625.More infoAbstract: The MickelssonFaddeev extension is a 3space analogue of a KacMoody group, where the central charge is replaced by a space of functions of the gauge potential. This extension is a pullback of a universal extension, where the gauge potentials are replaced by operators in a Schatten ideal, as in noncommutative differential geometry. Our main result is that the universal extension cannot be faithfully represented by unitary operators on a separable Hilbert space. We also examine potential consequences of the existence of unitary representations for the MickelssonFaddeev extension. © 1989 SpringerVerlag.
 Pickrell, D. (1987). Measures on infinite dimensional Grassmann manifolds. Journal of Functional Analysis, 70(2), 323356.More infoAbstract: A 1parameter family of quasiinvariant measures is presented. These measures are cylinder measures in graph coordinates. Their characteristic functions are represented as integrals relative to an infinite product measure. This is applied to the problem of determining the support properties of the measures. One of the measures can be used to define the unitary structure for the basic representation of the affine extension of the restricted unitary group. © 1987.