School of Mathematics
Persistent link for this communityhttps://hdl.handle.net/11299/42978
The University of Minnesota's School of Mathematics is a large and active department with over 70 faculty members, in addition to many graduate students and visitors. Related programs and centers are the Minnesota Center for Industrial Math (MCIM), the Program in Applied, Computational, Industrial Math (PACIM), the Institute for Mathematics and its Applications (IMA) (which has its own preprint series at http://conservancy.umn.edu/handle/683 ), and the IT Center for Educational Programs (ITCEP). Faculty research areas include algebraic geometry; algebraic topology, homotopy theory; applied mathematics; automorphic forms and l-functions; combinatorics; commutative algebra, homological algebra; complex analysis; differential geometry, Riemannian geometry; mathematical biology; mathematical fluid mechanics; mathematical logic; mathematical physics; mathematics education; number theory; numerical analysis; dynamical systems and differential equations; partial differential equations; probability and stochastic processes; real, harmonic, and functional analysis; topological dynamics.
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Item Abstract Algebra(online, and also CRC Press, 2007-07) Garrett, PaulItem Buildings and Classical Groups(online + Chapman-Hall, 1992) Garrett, PaulItem Differential Independence of Meromorphic Functions(University of Minnesota. School of Mathematics, 2003-01) Markus, LawrenceItem Differential invariants of maximally symmetric submanifolds(2008-09-16) Olver, Peter J.Let $G$ be a Lie group acting smoothly on a manifold $M$. A closed, nonsingular submanifold $S \subset M$ is called \is{maximally symmetric} if its symmetry subgroup $\sym S \subset G$ has the maximal possible dimension, namely $\dim \sym S = \dim S$, and hence $S = \sym S \cdot z_0$ is an orbit of $\sym S$. Maximally symmetric submanifolds are characterized by the property that all their differential invariants are constant. In this paper, we explain how to directly compute the numerical values of the differential invariants of a maximally symmetric submanifold from the infinitesimal generators of its symmetry group. The equivariant method of moving frames is applied to significantly simplify the resulting formulae. The method is illustrated by examples of curves and surfaces in various classical geometries.Item Dynamics for Ginzburg-Landau vortices under a mixed flow(2008-11-07) Kurzke, Matthias; Melcher, Christof; Moser, Roger; Spirn, DanielWe consider a complex Ginzburg-Landau equation that contains a Schrodinger term and a damping term that is proportional to the time derivative. Given well-prepared initial conditions that correspond to quantized vortices, we establish the vortex motion law until collision time.Item Existence and stability of viscoelastic shock profiles(2010-06-09) Barker, Blake; Lewicka, Marta; Zumbrun, KevinWe investigate existence and stability of viscoelastic shock profiles for a class of planar models including the incompressible shear case studied by Antman and Malek-Madani. We establish that the resulting equations fall into the class of symmetrizable hyperbolic--parabolic systems, hence spectral stability implies linearized and nonlinear stability with sharp rates of decay. The new contributions are treatment of the compressible case, formulation of a rigorous nonlinear stability theory, including verification of stability of small-amplitude Lax shocks, and the systematic incorporation in our investigations of numerical Evans function computations determining stability of large-amplitude and or nonclassical type shock profiles.Item The Foppl-von Karman equations for plates with incompatible strains(2010-02-24) Lewicka, Marta; Mahadevan, L.; Pakzad, RezaWe provide a derivation of the Foppl-von Karman equations for the shape of and stresses in an elastic plate with residual strains. These might arise from a range of causes: inhomogeneous growth, plastic deformation, swelling or shrinkage driven by solvent absorption. Our analysis gives rigorous bounds on the convergence of the three dimensional equations of elasticity to the low-dimensional description embodied in the plate-like description of laminae and thus justifies a recent formulation of the problem to the shape of growing leaves. It also formalizes a procedure that can be used to derive other low-dimensional descriptions of active materials.Item The Global Theory of Ordinary Differential Equations, Lecture Notes, 1964-1965(University of Minnesota. Institute of Technology. Department of Mathematics, 1965) Markus, LawrenceItem Hamiltonian Dynamics and Symplectic Manifolds(University of Minnesota, Institute of Technology, School of Mathematics, 1973) Markus, LawrenceItem The infinite hierarchy of elastic shell models: some recent results and a conjecture(2009-07-09) Lewicka, Marta; Pakzad, RezaWe summarize some recent results of the authors and their collaborators, regarding the derivation of thin elastic shell models (for shells with mid-surface of arbitrary geometry) from the variational theory of $3$d nonlinear elasticity. We also formulate a conjecture on the form and validity of infinitely many limiting $2$d models, each corresponding to its proper scaling range of the body forces in terms of the shell thicknessItem Kolmogorov nonlinear diffusion equation(1982) Bramson, MauryThe Kolmogorov nonlinear diffusion equation, u_t=\frac{1}{2}u_{xx}+f(u), was first investigated by Kolmogorov, Petrovsky, and Piscounov in their celebrated paper in 1937. After a long pause, there has been renewed interest in this equation over the past decade. In extending this equation to more general settings, various tools from both analysis and probability theory have been applied, including phase plane analysis, the maximum principle, Brownian motion estimates, and the Feynman-Kac formula. It is the intent of these notes to summarize much of the work done on the Kolmogorov equation to date, while emphasizing the interaction between analytic and probabilistic techniques. These notes are to a large extent based on a course given by Don Aronson and myself last spring at the University of Minnesota in Minneapolis. (Laboratoire de Probabilités, Université de Paris VI, Spring 1982)Item Lectures on moving frames(2009-01-21) Olver, Peter J.This article surveys the equivariant method of moving frames, along with a variety of applications to geometry, differential equations, computer vision, numerical analysis, the calculus of variations, and invariant curve flows.Item The matching property of infinitesimal isometries on elliptic surfaces and elasticity of thin shells(2008-10-14) Lewicka, Marta; Mora, Maria Giovanna; Pakzad, Mohammad RezaUsing the notion of \Gamma-convergence, we discuss the limiting behavior of the 3d nonlinear elastic energy for thin elliptic shells, as their thickness h converges to zero, under the assumption that the elastic energy of deformations scales like h^\beta with 2 < \beta < 4. We establish that, for the given scaling regime, the limiting theory reduces to the linear pure bending. Two major ingredients of the proofs are: the density of smooth infinitesimal isometries in the space of W^{2,2} first order infinitesimal isometries, and a result on matching smooth infinitesimal isometries with exact isometric immersions on smooth elliptic surfaces.Item Models for the 3D singular isotropic oscillator quadratic algebra(2008-11-06) Kalnins, E.G.; Miller Jr., W.; Post, SarahWe give the first explicit construction of the quadratic algebra for a 3D quantum superintegrable system with nondegenerate (4-parameter) potential together with realizations of irreducible representations of the quadratic algebra in terms of differential-differential or differential-difference and difference-difference operators in two variables. The example is the singular isotropic oscillator. We point out that the quantum models arise naturally from models of the Poisson algebras for the corresponding classical superintegrable system. These techniques extend to quadratic algebras for superintegrable systems in $n$ dimensions and are closely related to Hecke algebras and multivariable orthogonal polynomials.Item Models of quadratic quantum algebras and their relation to classical superintegrable systems(2008-09-12) Kalnins, E.G.; Miller Jr., W.; Post, SarahWe show how to construct realizations (models) of quadratic algebras for 2 dimensional second order superintegrable systems in terms of differential or difference operators in one variable. We demonstrate how various models of the quantum algebras arise naturally from models of the Poisson algebras for the corresponding classical superintegrable system. These techniques extend to quadratic algebras related to superintegrable systems in n dimensions and are intimately related to multivariable orthogonal polynomials.Item A note on convergence of low energy critical points of nonlinear elasticity functionals, for thin shells of arbitrary geometry(2008-11-26) Lewicka, MartaWe prove that the critical points of the $3$d nonlinear elasticity functional over a thin shell of arbitrary geometry and of thickness $h$, as well as the weak solutions to the static equilibrium equations (formally the Euler Lagrange equations associated to the elasticity functional) converge, in the limit of vanishing thickness $h$, to the critical points of the generalized von Karman functional on the mid-surface, recently derived in [14]. This holds provided the elastic energy of the $3$d deformations scale like $h^4$ and the magnitude of the body forces scale like $h^3$.Item On the existence of traveling waves in the 3d Boussinesq system(2008-10-14) Lewicka, Marta; Mucha, Piotr B.We extend an earlier work on traveling waves in premixed ames in a gravitationally stratified medium, subject to the Boussinesq approximation. For three-dimensional channels not aligned with the gravity direction, and under the Dirichlet boundary conditions in the fluid velocity, it is shown that a non-planar traveling wave, corresponding to a non-zero reaction, exists, under an explicit condition relating the geometry of the crossection of the channel to the magnitude of the Prandtl and Rayleigh numbers, or when the advection term in the flow equations is neglected.Item Reduced theories in nonlinear elasticity(2010-06-09) Lewicka, MartaThe purpose of this note is to report on the recent development concerning the analysis and the rigorous derivation of thin film models for structures with nontrivial geometry. This includes: (i) shells with mid-surface of arbitrary curvature, and (ii) plates exhibiting residual stress at free equilibria. In the former setting, we derive a full range of models, some of them previously absent from the physics and engineering literature. The latter phenomenon has been observed in different contexts: growing leaves, torn plastic sheets and specifically engineered polymer gels. After reviewing available results, we list open problems with a promising angle of approach.Item Resolutions, relation modules and Schur multipliers for categories(2008-09-15) Webb, PeterWe show that the construction in group cohomology of the Gruenberg resolution associated to a free presentation and the resulting relation module can be copied in the context of representations of categories. We establish five-term exact sequences in the cohomology of categories and go on to show that the Schur multiplier of the category has properties which generalize those of the Schur multiplier of a group.Item Scaling laws for non-Euclidean plates and the W^{2,2} isometric immersions of Riemannian metrics(2009-07-09) Lewicka, Marta; Pakzad, RezaThis paper concerns the elastic structures which exhibit non-zero strain at free equilibria. Many growing tissues (leaves, flowers or marine invertebrates) attain complicated configurations during their free growth. Our study departs from the 3d incompatible elasticity theory, conjectured to explain the mechanism for the spontaneous formation of non-Euclidean metrics. Recall that a smooth Riemannian metric on a simply connected domain can be realized as the pull-back metric of an orientation preserving deformation if and only if the associated Riemann curvature tensor vanishes identically. When this condition fails, one seeks a deformation yielding the closest metric realization. We set up a variational formulation of this problem by introducing the non-Euclidean version of the nonlinear elasticity functional, and establish its $\Gamma$-convergence under the proper scaling. As a corollary, we obtain new necessary and sufficient conditions for existence of a $W^{2,2}$ isometric immersion of a given $2$d metric into $\mathbb R^3$.