Mathematics Colloquia Archives - Fall 2013
Mechanics of Soft Tissues: Reflecting On Roads Traveled and The Road Just Taken
- Professor Alan Freed (Saginaw Valley State University)
- September 12, 2013
- Abstract: About three-quarters of a century ago saw the emergence of a finite strain theory suitable for modeling natural rubber, which was critical to the war effort. This theory was derived from thermodynamics and has a solid foundation in statistical mechanics. With an application of invariant theory in the 1950's, the explicit theory for rubber elasticity became complete. In the decades that have ensued, rubber elasticity (hyperelasticity) has been applied to a broad selection of materials - some successfully, some not. Of particular interest for this talk are the soft solids of biologic origin. The presentation will reexamine the road well traveled in our attempts to secure a mathematical description for these materials, and it will show that a new fork in the road is worth exploring and developing. In 2003, Prof. K. Rajagopal from Texas A&M introduced an idea that the thermodynamic structure of an elastic material might be implicit in its dependence upon state. This simple notion is causing a profound revolution in our thinking of what an elastic solid is or can be - a revolution that is, unfortunately, slowed by the inertia of bias. Much of this presentation will present what such a theory is capable of predicting. Implicit elasticity today is where explicit elasticity was in the 1950's. There are great opportunities that lie ahead in the continued development of this theory, and in the construction of tools that can apply this technology for use in engineering applications.
Some Simple Ideas and Methods in Mathematics and their Recent Developments
- Professor Shihshu Walter Wei (University of Oklahoma)
- September 19, 2013
- Abstract: In this talk we'll discuss several elegant ideas and simple methods in mathematics that have made strong impacts on the study of algebra, analysis, geometry, topology, differential equations, mathematical physics, applied mathematics and their interconnectedness. We'll begin with a very simple idea in middle school algebra, or a philosophy of Lao-Zi, then we'll quickly move to some ideas in calculus and in classical geometry. Along these frame works, we'll explore their roles in real and complex geometry, several complex variables, algebraic topology, calculus of variations, optimization problems, Gauge theory, Lie group representation theory and transcendental algebraic geometry. Some related problems, further applications, and recent developments, especially in p-harmonic geometry will also be addressed. It is our effort to make the talk comprehensible to graduate students of any fields.
On a Measure of the Distance to ℓ1(2)
- Professor Richard J. Fleming (Central Michigan University)
- October 3, 2013
- Abstract: For a Banach space X, let λ0(X) denote the infimum of the Banach-Mazur distances between the two-dimensional subspaces of X and the two-dimensional L1-space ℓ1(2). The condition that λ0(X∗) > 1 has been used to establish certain isomorphic Banach-Stone theorems. We investigate some elementary properties of λ0(X) and consider conditions on X that imply that λ0(X) > 1 or λ0(X∗) > 1.
Boundary Crossing Based Threshold Regression Models for Lifetime Data
- Professor Mei-Ling Ting Lee (University of Maryland)
- October 15, 2013
- Abstract: Cox regression methods are well-known. It has, however, a strong proportional hazards assumption. In many medical contexts, a disease progresses until a failure event (such as death) is triggered when the health level first reaches a failure threshold. I'll present a model for the health process that requires few assumptions and, hence, is quite general in its potential application. Distribution-free methods for estimation and prediction are developed. Computational aspects of the approach are straightforward. Case examples are presented that demonstrate the methodology and its practical use. The methodology provides medical researchers and biostatisticians with new and robust statistical tools for estimating treatment effects and assessing a survivor's remaining life. This is a joint work with G.A. Whitmore of McGill University.
Fourier Bases on Fractals
- Dr. Keri Kornelson (University of Oklahoma)
- November 7, 2013
- Abstract: In this talk, we use an iterated function system (IFS) approach to describe the Bernoulli convolution measures. These measures are supported on Cantor sets which are subsets of the real line, and have a variety of interesting properties. In particular, one can find examples in which there exist Fourier bases for the associated L2 Hilbert space. We describe some of these bases, and the associated operators that arise from them. One interesting such operator exhibits its own fractal-like self-similarity.
A System of Hyperbolic Balance Laws Arising from Chemotaxis
- Dr. Kun Zhao (Tulane University)
- November 14, 2013
- Abstract: In contrast to diffusion (random diffusion without orientation), chemotaxis is the biased movement of cells/particles toward the region that contains higher concentration of beneficial or lower concentration of unfavorable chemicals. The former often refers to the attractive chemotaxis and latter to the repulsive chemotaxis. Chemotaxis has been advocated as a leading mechanism to account for the morphogenesis and self-organization of a variety of biological coherent structures such as aggregates, fruiting bodies, clusters, spirals, spots, rings, labyrinthine patterns and stripes, which have been observed in experiments. In this talk, I will present recent results on the rigorous analysis of a partial differential equation model arising from repulsive chemotaxis, which is a system of hyperbolic balance laws consisting of nonlinear and coupled parabolic and hyperbolic type PDEs. In particular, global wellposedness, large-time asymptotic behavior of classical solutions to such model are obtained which indicate that chemorepulsion problem with non-diffusible chemical signal and logarithmic chemotactic sensitivity exhibits strong tendency against pattern formation. The results are consistent with general results for classical repulsive chemotaxis models.
From Modular Forms to Crystal Bases
- Dr. Philip Lombardo (Saint Joseph's College of New York)
- November 22, 2013
- Abstract: Riemann's 1859 paper introducing his zeta function as a means of proving the prime number theorem marked the beginning of a beautiful interaction between complex analysis and number theory. As the study of L-functions and modular forms progressed into a more general theory of automorphic forms, representation theory soon became an important part of the story. The Casselman-Shalika formula is one example of the strong connection between automorphic forms and representation theory. In this talk, I will highlight some important ideas in the development of the theory of automorphic forms while providing context for a more recent result connecting the Casselman-Shalika formula to the combinatorics of crystal bases.
S-estimators for Functional Principal Component Analysis
- Dr. Matías Salibián-Barrera (University of British Columbia)
- December 3, 2013
- Abstract: Principal components analysis is a widely used technique that provides an optimal lower-dimensional approximation to multivariate observations. In the functional case, a new and simple characterization of elliptical distributions on separable Hilbert spaces allows us to obtain an equivalent stochastic optimality property for the principal component subspaces associated with elliptically distributed random elements. This property holds even when second moments do not exist. These lower-dimensional approximations can be very useful in identifying potential outliers among high-dimensional or functional observations. In this talk we propose a new class of robust estimators for principal components. For a fixed dimension q, we robustly estimate the q-dimensional linear space that best fits the data, in the sense of minimizing the sum of coordinate-wise robust residual scale estimators. The extension to the infinite-dimensional case is also studied. In analogy to the linear regression case, we call this proposal S-estimators. Our method is consistent for elliptical random vectors, and is Fisher-consistent for elliptically distributed random elements on arbitrary Hilbert spaces. Numerical experiments show that our proposal is highly competitive when compared with other existing methods when the data are generated both by finite- or infinite-rank stochastic processes. We also illustrate our approach using two real functional data sets, where the robust estimator is able to discover atypical observations in the data that would have been missed otherwise. This talk is the result of recent collaborations with Graciela Boente and David Tyler.
Elementary Statistical Methods and Measurement Error
- Professor Marcus Jobe (Miami University)
- December 5, 2013
- Abstract: How the sources of physical variation interact with a data collection plan determines what can be learned from the resulting dataset, and in particular, how measurement error is reflected in the dataset. The implications of this fact are rarely given much attention in most statistics courses. Even the most elementary statistical methods have their practical effectiveness limited by measurement variation; and understanding how measurement variation interacts with data collection and the methods is helpful in quantifying the nature of measurement error. We illustrate how simple one- and two-sample statistical methods can be effectively used in introducing important concepts of metrology and the implications of those concepts when drawing conclusions from data.