2014-15 Colloquia talks

On the Convergence of Hermite-Padé Approximants for Rational Perturbations of a Nikishin System

Abstract: In the recent past multiple orthogonal polynomials have attracted great attention. They appear in simultaneous rational approximation, simultaneous quadrature rules, number theory, and more recently in the study of certain random matrix models. These are sequences of polynomials which share orthogonality conditions with respect to a system of measures. A central role in the development of this theory is played by the so called Nikishin systems of measures for which many results of the standard theory of orthogonal polynomials have been extended. In this regard, we present some results on the convergence of type I and type II Hermite-Padé approximation for a class of meromorphic functions obtained by adding vector rational functions with real coefficients to a Nikishin system of functions (the Cauchy transforms of a Nikishin system of measures).

Graphs are to Matroids as Embedded Graphs are to What?

Abstract: A matroid is a mathematical structure that generalises the notion of linear independence in vector spaces. There is a natural way to associate a matroid with a graph, and this results in a very close connection between graph theory and matroid theory. This is beneficial in two ways: graph theory can serve as an excellent guide for studying matroids; and matroid theory can lead to new, and more general, results about graphs. In fact, matroid theory may be best thought of as a generalisation of graph theory.

In many applications of graph theory, graphs come equipped with a drawing on a surface. However, the (graphic) matroid associated with such a graph records absolutely no information about how it is drawn. Thus matroids do not appear to provide a 'correct' generalisation of graphs in surfaces. This leads to the question if matroids don't, what do? In this talk, after giving a gentle introduction to matroids, I will propose an answer to this question.

This is joint work with Carolyn Chun, Steven Noble and Ralf Rueckriemen.

A Bayesian Predictive Approach to Design Studies for Comparing Biomarkers

Abstract: Finding efficient biomarker is critically important for disease detection. Furthermore, rigorous evaluation of biomarkers is essential to guarantee that the tests that are developed are sufficiently accurate and beneficial to the patient. Here we propose a Bayesian predictive approach for determining sample size to compare efficiencies of two binary biomarkers. The operational criteria include classification accuracy, sensitivity, and specificity. In the frequentist approach, one usually estimates the operational criterion from the training data for each marker and using the estimate as the 'true' value to select the sample size corresponding to a desirable power, say 80%. However, due to uncertainty to the training data estimates, the designed study could be under powered when the estimates are treated as true. This may result in a substantially underpowered study while the sample size for the training data is small. In the Bayesian predictive approach the sample size is determined taking into account the uncertainty of the training data estimates. Through simulation studies we show the effectiveness of the proposed approach. We can also extend this approach in case of continuous biomarkers.

Invariants of Legendrian Graphs

Abstract: A contact structure on a 3-dimensional manifold is an everywhere non-integrable plane field. In this talk, we look at Legendrian graphs in the standard contact structure of the 3-dimensional Euclidian space. A Legendrian graph is a graph embedded in such a way that its edges are everywhere tangent to the contact planes. We extend classical invariants of Legendrian knots like the Thurston-Bennequin number and the rotation number to Legendrian graphs. We look at various questions which arise in this context. In particular, we show that a graph G can be Legendrian realized with all cycles unknots of maximal Thurston-Bennequin number if and only if G doesn't contain K₄, the complete graph with four vertices, as a minor. This talk is based on joint work with Danielle O’Donnol.

Counting Derangements in Primitive Groups

Abstract: The study of the fixed points of permutations is a common topic in Probability Theory, Number Theory and Algebraic Geometry. In this talk we shall focus on the Group Theory machinery that lurks behind the curtains. We shall introduce the audience to basic combinatorial techniques and notions needed to tackle the following question:

Let A be a primitive permutation group and G a normal subgroup of A such that A/G is cyclic. Let xG be a generating coset for A/G. Motivated by questions arising in connection to coverings of smooth connected projective curves, we study the proportion of derangements in the coset xG. We use the Aschbacher-O'Nan-Scott theorem for primitive groups to partition the problem and provide answers in several cases.

Properties and Applications of Apéry-Like Numbers

Abstract: Apéry-like numbers are special integer sequences, going back to Beukers and Zagier, which are modeled after and share many of the properties of the numbers that underly Apéry's proof of the irrationality of ζ(3). Among their remarkable properties are connections with modular forms and so-called supercongruences, some of which remain conjectural. In the course of several examples, we demonstrate how these numbers and their connection with modular forms feature in various, apparently unrelated, problems. The examples are taken from personal research of the speaker and include the theories of short random walks, binomial congruences, series for 1/π, and positivity of rational functions. Finally, we return to the discussion of supercongruences and report on new perspectives and recent progress.

A General Hybrid Clustering Technique

Abstract: Here, we propose a clustering technique for general clustering problems including those that have non-convex clusters. For a given desired number of clusters K, we use three stages to find a clustering. The first stage uses a hybrid clustering technique to produce a series of clusterings of various sizes (randomly selected). They key steps are to find a K-means clustering using K2-clusters where K2 ≫ K and then joins these small clusters by using single linkage clustering. The second stage stabilizes the result of stage one by reclustering via the 'membership matrix' under Hamming distance to generate a dendrogram. The third stage is to cut the dendrogram to get K3 clusters where K3 ≥ K and then prune back to K to give a final clustering. A variant on our technique also gives a reasonable estimate for KT, the true number of clusters. We provide a series of arguments to justify the steps in the stages of our methods and we provide numerous examples involving real and simulated data to compare our technique with other related techniques.

The Mathematics and Statistics of Wind

Abstract: We all experience wind in our daily lives, but we never think about it as a mathematical quantity. Establishing a climatology of any meteorological property, requires the calculation of means, standard deviations, extremes, and other statically properties. Wind is no different. But because wind has both speed and direction, it has to be quantified as a vector. Wind speed can easily be averaged the normal arithmetic way, but wind direction presents a problem due to its circular nature (a northerly wind is direction 0, easterly is 90, southerly 180, and so forth back to 360 degrees for a northerly wind). In this seminar, several ways of calculating wind speed and direction averages will be presented. Wind speed measurements from South Alabama Mesonet weather stations (http://chiliweb.southalabama.edu/) will be used to statistically investigate the differences between these methods.

Cluster Algebras - Why the Fuss?

Abstract: Cluster algebras were invented in 2000 by S. Fomin and A. Zelevinsky as constructively defined commutative algebras with a distinguished set of generators (cluster variables) grouped into overlapping subsets (clusters) of fixed cardinality. Both the generators and the relations among them are not given from the outset, but are produced by an iterative process of successive mutations. Although this procedure appears murky at first, it turns out to encode a surprisingly widespread range of phenomena.

We illustrate in this talk one particularly nice example of how certain cluster algebras are related to classical objects in mathematics, and we hope this helps explaining the explosive development of the subject in recent years.

Arrangements of Lines: When the Combinatorics Fails to Understand the Topology

Abstract: Zariski gave a pair of sextics with the same types of singularities but whose complements have different fundamental groups. This motivates the search for a similar "Zariski pair" of line arrangements: two with the same combinatorial intersection data but whose (complex projective) complements have different fundamental groups. Only one minimal case has been found so far: Rybnikov produced one with thirteen lines in 1998 by gluing two smaller arrangements together. No such pair exists on nine or fewer lines. Together with Amram, Sun, Teicher, Ye, and Zarkh, we investigate arrangements of ten lines.

This talk is accessible to those without backgrounds in combinatorics, topology, or algebraic geometry.

Orthogonal Polynomials over the Unit Disk: Some Examples and Asymptotic Results

Abstract: Generally speaking, orthogonal polynomials refers to a sequence of polynomials p₀(z), p₁(z), ..., p_n(z), ..., of a complex variable z, each p_n of degree n, that are orthogonal with respect to some inner product (orthogonal polynomials of several variables can also be considered). Typically, the inner product is given by an integral with respect to a (positive) measure whose support is a compact subset of the complex plane. Two particular cases stand out for the richness of the theory: orthogonal polynomials over the real line (the support of the measure is contained in R), and orthogonal polynomials over the unit circle (the support of the measure is contained in {z:|z|=1}). The richness is due to some special features of the real line and the circle. In contrast, when the orthogonality measure is no longer supported on the line or the circle, the wealth of results is much more limited, and in some aspects, nonexistent. We will discuss some recent results in this more general context, mainly concerning the behavior of p_n(z) as n approaching infinity for planar type orthogonality measures supported on the unit disk.

The Lyapunov-Schmidt Reduction in Nonlinear Analysis

Abstract: We introduce and give a survey of a functional analytic approach to solving problems in bifurcation theory and nonlinear analysis called the Lyapunov-Schmidt reduction. We illustrate how this method works in the context of a Hopf bifurcation in a system of ordinary differential equations, and indicate how it applies to other problems in nonlinear analysis. If time permits, we discuss some limitations of the method in the context of partial differential equations. The talk is aimed to be accessible to students and outsiders of the field.

Signed-Rank with Responses Missing at Random

Abstract: In this talk, we will start by introducing some asymptotic results on nonparametric kernel estimation. Next we will show how kernel estimation can be used in the study of the signed-rank estimator of the regression coefficients under the assumption that some responses are missing at random in the regression model. Strong consistency and asymptotic normality of the proposed estimator will be established under mild conditions. To demonstrate the performance of the signed-rank estimator, a simulation study under different settings of errors’ distributions will show that the proposed estimator is more efficient than the least squares estimator whenever the error distribution is heavy tailed or contaminated. When the errors follow a normal distribution, the simulation experiment also will show that the signed-rank estimator is more efficient than its least squares counterpart whenever a large proportion of the responses are missing.

Characteristic Classes in Group Cohomology

Abstract: I will introduce the theory of characteristic classes for permutation representations of groups, and illustrate the technique by playing around with the natural representation of S₄ and constructing generators for its "mod-2 cohomology ring." Then we will meet the Dickson invariants, and construct some nonzero cohomology classes on the general linear groups over F₂. Lastly, I'll describe how finding a nonzero Chern class on GL_nF_p inspired my recent work on cohomology of finite groups of Lie type.

Meridional and Non-Meridional Epimorphisms between Knot Groups

Abstract: We will consider epimorphisms between knot groups. Especially, we will focus on the image of a meridian under such an epimorphism. A homomorphism between knot groups is called meridional if it preserves their meridians. The existence of a meridional epimorphism introduces a partial order on the set of prime knots. We will determine the pairs of prime knots with up to 11 crossings which admit meridional epimorphisms between their knot groups. Moreover, we will describe some examples of non-meridional epimorphisms explicitly.

Constructing Partially Balanced Incomplete Block Designs from Strongly Regular Graphs

Abstract: An undirected graph without loops, is strongly regular when each vertex is of equal degree with any two vertices with an edge are joined with exactly m common vertices and any two vertices without an edge are joined to n common vertices. R.C. Bose, in his 1963 paper, has shown that a two class association scheme can be expressed as a strongly regular graph. And thus strongly regular graphs has close connections with two classes of partially balanced incomplete block designs (PBIBD). Basic concepts, definitions, interrelationships of graph theory and design theory will be discussed along with a few construction theorems for partially balanced incomplete block designs.

Optimal Properties of Variance Balanced Designs

Abstract: It is well known that for a proper block design the combinatorial property of pairwise balance is sufficient to ensure the statistical property of variance balance. The variance balance property of a block design implies the complete symmetry of the information matrix. Using these facts we discuss the optimality in a class of proper variance balanced designs with unequal replications. Further unequal replications force the variance balanced designs to be non-binary designs. Hence instead of using the usual optimality criteria given by J. Keifer, we compare the designs with respect to the functions based on efficiency factors of the design, namely eigenvalues of the matrix (RinverseC).

Nonparametric Methods for Classification and Feature Selection

Abstract: In this talk, I will discuss some nonparametric max-central classifiers as well as methods for selecting features that are relevant for discrimination. The feature selection method rewards features for information towards discrimination but penalizes them for their similarity to already selected features. Monte Carlo simulation studies demonstrate that there are several situations where the proposed procedures provide lower misclassification error rates than classical methods. Finally, I will discuss results of application of the proposed procedures on food safety and gene expression data.

What is a Scissors Congruence?

Abstract: We will define what it means to be Scissors Congruent for polygons and polyhedra. We'll see a scissors congruence proof of the Pythagorean Theorem and show that scissors congruence is an invariant of area. Next, we'll understand scissors congruence in terms of the distributive law and various other properties. We will define the Dehn Invariant. Noting that volume is an invariant of scissors congruence we'll recall Hilbert's Third Problem and see a simple proof that the converse is not true. We will explore a scissors congruence proof of Heron's Formula. Finally, if we have time, we will define scissors congruence in terms of the group of isometries and explore this notion in a group theoretic context.

Fractalicious Dogs and Potential Theory

Abstract: We give a brief introduction to potential theory. Then we explain how to use balayage to get beautiful fractalicious images such as dogs.

Sierpinski Figures in all Dimensions, the Chaos Game, and Multinomial Coefficients

Abstract: It is well-known among mathematicians that the result of the Chaos Game on 3 equidistant vertices yields a figure that approximates the Sierpinski triangle. It is also known that the binomial coefficients when read modulo 2 resemble this figure. In this talk, I want to show you some interesting n-dimensional generalizations of these phenomena.

Plutarch’s Box, O’Halloran Numbers, and the Riemann Hypothesis

Abstract: During our last summer camp some Mobile County six graders were given a rectangular prism of size 5 by 2 by 2. They were asked to calculate its surface area and then find another rectangular prism with integer dimensions and identical surface area. While daydreaming in the back of the class I started thinking about these rectangular “integer prisms”. Are there any such prisms whose surface area equals their volume? Can every (large) even number be realized as the surface area of some rectangular prism with integer dimension? The answer to these questions leads to some heavy-duty mathematics. Even the Riemann Hypothesis appears.

Dimer Coverings

Abstract: A dimer covering (also called a perfect matching) of a graph is a collection of edges that covers each vertex exactly once. The term “dimer” comes from chemistry: a dimer is a polymer with only two atoms. If we think of vertices as univalent atoms, then dimer coverings provide simple models for studying certain phase transitions. We explain how dimer coverings arise in both topology and algebraic dynamical systems.