# DE Seminars: Spring 2006

#### Monday March 13

Title |
The structure of hyperbolic sets |

Speaker |
Todd Fisher UMCP |

**Abstract:**

We will begin with a review of the fundamental concepts and examples of hyperbolic dynamical systems. We will then mention some recent results concerning: locally maximal hyperbolic sets, hyperbolic sets with interior, and hyperbolic attractors on surfaces.

#### Monday March 27

Title |
Geometric Control of Nonlinear Dynamical Systems |

Speaker |
James Dibble Department of Mathematics and Statistics UMBC |

**Abstract:**

The preliminaries of geometric control theory will be discussed, including the means by which the differential equations governing dynamical systems can be viewed as vector fields on differentiable manifolds. In control systems, the ability to manipulate those differential equations, and hence alter those vector fields, opens up questions about which states the system can reach given certain initial conditions. The use of Lie brackets to determine controllability and (strong) accessibility of control-affine systems will be explained, as will results concerning connected analytic systems.

#### Monday April 17

Title |
Recovering a space dependent parameter in a cable theory model of a dendritic fiber with spines |

Speaker |
Dan Wang Department of Mathematics and Statistics UMBC |

**Abstract:**

The aim of this presentation is to determine parameters in a distributed parameter model of a nerve fiber. I will begin with a review of the fundamental concepts and some mathematical models of a nerve fiber. I will then discuss a inverse problem of recovering a spatially distributed coefficient from a model involving a coupled equation system.

#### Monday April 24

Title |
Stability Results for Elastic Rods with Electrostatic Self-Repulsion |

Speaker |
Kathleen Hoffman UMBC |

**Abstract:**

Conjugate points, attributed to Jacobi, have been a part of the classical calculus of variations literature for over a century, however, the classical theory pertains only to the standard calculus of variations problems. In this talk, I will present a general theory of conjugate points for variational problems satisfying generic assumptions. The motivation for this work is to determine the stability of an elastic rod with an electrostatic self-repulsion. The singular, non-local repulsive potential makes the problem remarkably different from the standard calculus of variations problem, yet a theory of conjugate points can still be used to identify minima, or stable equilibria. Results for the two-dimensional elastic strut will be presented. This is joint work with R. Manning, Haverford College.

#### Monday May 1

Title |
Biological organisms and their mathematical descriptions: examples of the models we can build |

Speaker |
Mariajose Castellanos Chemical Engineering Department UMBC |

**Abstract:**

The rate of accumulation of biological information is increasing exponentially. This information is driven by new technologies and powerful data acquisition methods. These changes have revolutionized the nature of biology, allowing a greater intersection with engineering concepts and computation. Most of the new data is widely available, but the content is disconnected and primarily based on static and equilibrium measurements. In contrast, actual biological behavior is the product of integrative and dynamic interactions between system components. This behavior can be better understood with the development of structurally, biochemically, and physiologically detailed computational models based on experimental data. Mathematical models allow simulation of intracellular processes and the connection to the extracellular environment to predict system responses to variety of intrinsic perturbations (e.g. changes in cellular regulatory machinery) as well as environmental fluctuations (e.g. in nutrient levels). At the same time, models increase our understanding of the basic principles of systems functions and provide insight into the regulatory mechanisms controlling a biological system. Additionally, models can help in the design of future experiments and produce testable questions about the object of study.

#### Friday May 5

Title |
Oscillations and concentrations in sequences of gradients |

Speaker |
Martin Kruží Academy of Sciences of the Czech Republic |

**Abstract:**

Young measures proved to be an efficient tool to record statistical distributions of fast spatial oscillations in sequences of gradients living in a Lebesgue space. They are a key ingredient in mathematical models of shape-memory materials and modern calculus of variations. Concentration effects related to the lack of equiintegrability of the power of the gradient modulus however, cannot be treated by Young measures. We will introduce a suitable generalization of Young measures, called DiPerna-Majda measures, and give an explicit characterization of such measures generated by gradients. This result will be then used to obtain a new sequential weak lower semicontinuity theorem for integral functionals with a critical decay. The obtained results clearly explain examples of the failure of weak sequential continuity of determinants due to L. Tartar, J.M. Ball, and F. Murat.

#### Monday May 8

Title |
The analysis of curvatures of discrete surfaces with boundary |

Speaker |
Brian Krummel Department of Mathematics and Statistics UMBC |

**Abstract:**

Anatomical surfaces can be studied by extracting discrete surfaces from medical imaging data. Thus understanding the curvature of discrete surfaces is important for analyzing the structure of anatomical surfaces. One approach to studying the curvature of discrete surfaces is by deriving curvature measures which quantify the average curvature of a region. Recent work of Cohen-Steiner and Morvan has considered curvature measures for discrete surfaces without boundary using the concept of a normal cycle from geometric measure theory. I have extended this approach to discrete surfaces with boundary. In the talk, I will describe the normal cycles of surfaces with boundary and show how they can used to define curvature measures. Explicit formulae for the curvature measures of smooth and discrete surfaces will be described. I will conclude with a convergence theorem which gives an error bound on the difference the curvature of a discrete surface and the curvature of a smooth surface that it approximates.

#### Monday May 15

Title |
Existence of Minima in for Elastic Rod Problems |

Speaker |
Michael Childers Department of Mathematics and Statistics UMBC |

**Abstract:**

Hyperelastic rods can be formulated as a constrained caluclus of variations problem. With this formulation, existence of minima are proven under general conditions using techniques from functional analysis. Minima of the functional are related to stable configurations of the elastic rod. This is joint work with Prof. Hoffman and Prof. Seidman.