2012-13 Colloquia talks

Abstract: The number of periodic points of a given period in a shift space X is a conjugacy invariant. Let Per(X) denote the periodic points in X. Two shift spaces X and Y are essentially conjugate if there exists a conjugacy from X\Per(X) to Y\Per(Y). We will focus on finding essential conjugacies where we know conjugacies do not exist. In 1990, South Alabama's own Susan Williams presented an example of a sofic shift that is not topologically conjugate to a renewal system. I will show that this example is essentially conjugate to a renewal system. I will also present an example of a renewal system that is essentially conjugate to a shift of finite type but not conjugate to a shift of finite type. If time permits, we will discuss generalizations of the above examples and related problems on one-sided shift spaces.

Abstract: In this talk, we will first review some classical isoperimetric inequalities and their applications in the real world. Then we will take a closer look at some analytical inequalities and their interplay with geometric inequalities. Finally, we will discuss some generalizations of these geometric and analytic inequalities in different directions.

Abstract: Let X_i be a random variable with distribution function F_i, i = 1, 2, ..., K. Also let pi be the set of all permutations of the numbers (1,2,...,K). Then P(X_i1 < ... < X_iK) is a precedence probability if (i₁,...,i_K) belongs to pi.

Rank based inference using independent random samples to compare K > 1 continuous distributions, called the K-sample problem, based on precedence probabilities is developed and explored. There are many parametric and nonparametric approaches, most dealing with hypothesis testing, to this important, classical problem. Most existing tests are designed to detect differences among the location parameters of different distributions. Best known and most widely used of these is the F-test, which assumes normality. A comparable nonparametric test was developed by Kruskal and Wallis (1948).

When dealing with location-scale families of distributions, both of these tests can perform poorly if the differences among the distributions are among their scale parameters along with their location parameters. Overall, existing tests are not effective in detecting changes in both location and scale. I propose a new class of rank-based, asymptotically distribution free tests that are effective in detecting changes in both location and scale based on precedence probabilities. Properties of these of tests are developed using the theory of U-statistics (Hoeffding, 1948) and that of permutation tests.

Some of these new tests are related to volumes under ROC (Receiver Operating Characteristic) surfaces, which are of particular interest in clinical trials whose goal is to use a score to separate subjects into diagnostic groups. In a related problem, Properties of precedence probabilities are obtained and a bootstrap algorithm is used to estimate an interval for them.

Using the estimates of precedence probabilities, I also propose new index measures of separation or similarity among two or more distributions. These indices may be used as effect sizes. Some basic details of these measures will also be discussed.

Abstract: Characterizing the spread of avian influenza within migratory birds has been a meaningful and interesting topic from both biological and mathematical respects. In this talk, we start with a simple ecological model of bird migration in two patches and approximate its dynamic threshold via a novel finite dimensional reduction method and asymptotic techniques. Next, we incorporate the ecological model with an epidemiological process and approximate the dynamic thresholds for the resulted epidemiological system. Finally, we discuss some possible extensions of our methods and results to the study of avian influenza spreading between migratory birds and domestic poultry.

Abstract: Let C_r(N_n) be the set of all commuting r-tuples of nilpotent n x n matrices. This object is called a nilpotent commuting variety when we consider it as a zero locus of a collection of polynomials. It has applications to many branches of modern mathematics such as representation theory and algebraic geometry. There are many open questions related to the structure of this variety. For example, one might like to know whether it is reducible, i.e. whether it can be written as a union of smaller varieties. Indeed, it has been shown that when r = 2, n > 0, the variety is not reducible. For r > 2, it has been conjectured that the nilpotent commuting variety is reducible for most of the values of n, but no one has published any proof. In this talk, we present our verifications of the reducibility of C_r(N_n) for various values of r and n. We also discuss a connection between the nilpotent commuting variety and the representation theory for certain subgroups of the special linear group SL_n.

Abstract: Multiple orthogonal polynomials are powerful tools in the study of different areas in Analysis, which include simultaneous rational approximation, spectral theory of operators, analytic number theory, and recently several applications have been found in the study of random matrices and other probabilistic models. In this talk I will describe a family of multiple orthogonal polynomials associated with a system of analytic weights defined on a star-like set, and discuss its connection with a model of random normal matrices. It turns out that the asymptotic behavior of the zeros of the multiple orthogonal polynomials is closely related to the asymptotic behavior of the eigenvalues of the normal matrices, and this relation can be expressed in terms of logarithmic potentials and other interesting tools in complex analysis. This talk is based on a joint work with A.B.J. Kuijlaars.

Abstract: The goal of this simulation study is to investigate the choice of working covariance structures in the analysis of spatially correlated data using a fixed-effect model. This study is motivated by the question of what impact does myocardial infarction (MI) have on the rotation of the left ventricle (LV). Rotation is a measurement of the wringing motion of the LV used to pump blood to the aorta. Rotation values may be obtained from 16 segments of the LV via cardiovascular MRI (cMRI), resulting in spatially correlated data. To achieve the goal of this study, we performed simulation studies based on the assumption that rotation follows a multivariate normal distribution with spatial covariance. We simulated under four spatial covariance structures: exponential, Gaussian, Matern, and spherical. For comparison, we also simulated under the unstructured and variance component (independent) covariance structures. We examine simulated type I error and power of tests that compares the mean rotation for two different populations (for instance, MI patients versus control patients) and the effect on the parameter estimate and its standard error. We also examine and compare the performance of goodness-of-fit indices (AIC and BIC) in the choice of the working covariance. Finally, we compare the fixed-effects model results to those of the popular two-sample t-test to determine what happens when we completely ignore the correlation within the data.

Abstract: A strategy for studying questions in topology, that was advocated by V.I. Arnold, is to use the symplectic geometry of the cotangent bundle to study the smooth topology of the base manifold. Knot contact homology (KCH) is a new invariant of knots that represents an application of this technique to knot theory. Some fundamental results show that KCH is both robust and computable. Recently a link to the (colored) HOMFLY-PT polynomial has been found, through a connection to string theory and mirror symmetry. While this connection remains mostly mysterious there is a parallel, and better understood, story for a specialization of KCH involving a relationship to the knot group and the character variety of the knot. In this talk we will discuss our work on this relationship, which benefits investigations both of KCH and the knot complement. Specifically, our work provides the most successful method to date for understanding augmentations of KCH; in addition, we obtain an application to the meridional rank of knot groups.

Abstract: Cyclic AMP (cAMP) and calcium are small molecules that act as important ësecond messengersí within cells. In concentrations of 105 molecules per picoliters, these messengers regulate activity in pulmonary endothelial cells that are micrometers in diameter. Millions of these cells then arrange and self-organize to form a semi-permeable lining of blood vessels in the lung. Endothelial cells must be elastic enough to stretch with sinusoidal changes in lung volume (breathing).

Changes in cAMP and calcium are known to regulate endothelial barrier permeability (the flux of water and proteins into the air space), vascular tone, gene transcription, and proliferation. We are interested in discovering how changes in these signaling pathways alter pulmonary function. Math modeling can be used to integrate over large spatial scales and population sizes to identify important interactions of components or make predictions regarding how the system will respond to infection, drugs, or injury. We will describe biological questions that my colleagues and I are currently trying to address with the interdisciplinary efforts of biologists, engineers, and mathematicians.

Abstract: Boson-fermion correspondence describes the equivalence of two representations of the Heisenberg algebra and uses the fact that the algebra of symmetric functions can be identified with a polynomial algebra. We will show that the main components of this famous correspondence can be interpreted as equally famous identities in symmetric functions - such as Cauchy identity, Jacoby-Trudi identity, relations between generating functions of homogeneous, elementary and power sum functions. This combinatorial interpretation allows to construct analogues of the boson-fermion correspondence in "variations on the theme of symmetric functions".

Abstract: The fundamental problem in topology is that of classifying spaces up to homeomorphism or some weaker equivalence. One way to study a space is to examine the different ways a subset can be embedded. For example, a basic property of 3-dimensional spaces is that strings tend to get entangled. Similarly, surfaces in 4-dimensional space can be knotted. In this talk, I will give a concise history of knotted surfaces and illustrate some of the local crossing information that can occur.

Abstract: We take a brief tour of the mathematical theory of information. Along the way, we discuss a strategy for playing Twenty Questions, discover why crossword puzzles are hard but possible, and learn how to say things twice without just repeating ourselves.

Abstract: We know from personal experience that it is possible to parallel park a car, even though one cannot directly move a car side-ways. The key is to concatenate simple forward and backward motions with suitable changes of the steering angle. Motivated by such examples, we present an introduction into the geometric and algebraic foundations of controllability of nonlinear controlled dynamical systems. A key feature is the lack of commutativity of the flows of the system. Combinatorial algebra comes into play when searching for more efficient formulas and calculations needed for example for automated path generation for nonlinear control systems.

Abstract: In contrast to the general theory of relativity, which requires the full apparatus of differential geometry, the special theory of relativity can be understood with a solid background in high-school mathematics and does not even require calculus. For many people, understanding this strange theory has been a life-long wish, but few find the necessary time to learn it. In this talk, we give an introduction to special relativity and explain from a different perspective why you stay younger and become thinner when you move, even though you gain mass. The talk will be elementary, but not trivial: All the necessary mathematical derivations will be presented in full technical detail.

Abstract: Most undergraduate math majors are often posed the question in an Abstract Algebra course, "Is normality a transitive relation in groups?" That is, if H is a normal subgroup of K and K is a normal subgroup of a group G, then is H a normal subgroup of G? A good student will quickly answer "no" and provide the dihedral group with eight elements, D₈, as a counterexample. This is because the subgroup generated by a reflection (thinking of the group as the symmetry group of the square) is a normal subgroup of the normal subgroup of D₈ generated by the same reflection and a rotation of 180 degrees. Yet it is easy to see that the subgroup generated by the reflection is not normal in D₈. Usually at this point in an algebra course one leaves behind the idea of normality being transitive and gives it no more thought. This begs the question, "For the groups in which normality is a transitive relation, what can we say about their structure?" The purpose of this talk is to answer this question at a level understandable by students with knowledge of basic group theory including the basics on solvable groups and the Sylow theorems. If time permits we will explore some related topics on the permutability of subgroups.

Abstract: Colorings provide the simplest effective invariants for knots. We present three equivalent ways to do this using the ideas of James Alexander, Max Dehn and Ralph Fox.

Abstract: In this paper, we consider a signed-rank estimator of nonlinear regression coefficients under stochastic errors. These errors include a wide array of applications in economic literature such as serial correlation, heteroscedasticity, autoregression, etc. General conditions for strong consistency and T-asymptotic normality of the resulting estimator are provided.

Abstract: We will take a visual stroll through the book "Indra's Pearls," by Mumford, Series, and Wright. The goal is to understand the action of certain groups of Mˆbius transformations on the extended complex plane. Such actions were first studied by several mathematicians, with significant advancement due to the work of Fricke and Klein in the late 1800s. Aspects of the subject are today quite accessible, especially visually, thanks to computer computation and animation. We will touch on connections to several key areas of mathematics, including analytic number theory and low-dimensional topology.

Abstract: How many pieces (not necessarily of the same size) can be made when you make ten cuts across a pizza? You tie a terrible knot in each of a friendís shoes: are these knots the same? Zoom into Pixarís Buzz Lightyearís digital representation, and all you see is a plane of triangles: how can we reduce this myriad of triangles to recover a more basic topological complex?

In this talk, I define these objects and explore topological results that can be obtained from combinatorics.

Abstract: The spider and the fly are sitting in the coordinate plane. he spider sits at (-2561,2353) and the fly sits at (3169,6903). The spider is no ordinary spider. It can move four different ways. Starting from a point (a,b), the spider can move to (a+b,b),(a-b,b), (a,a+b), or (a,a-b). The fly is terrified and sits perfectly still. Will the spider ever catch the fly?

After we catch all the spiders and flies, we will send them into orbit. Groups will jump into action. The Faithful will be presented and represented.

Abstract: Simple branched coverings of manifolds play an important role in various topological studies. For example, the 2-fold branched cover of the 3-sphere branched along a knot or link provides a space whose topological properties provide invariants of the knot or link. By generalizing an idea of Kamada, we construct immersions and embeddings of simple branched coverings in codimension 2.

The talk will be self-contained and indicate very intricate and detailed examples of our construction. Of course, there will be a lot of pictures.