2010-11 Colloquia talks

Abstract: Randomized response models, introduced by Warner (1965), are important data acquisition tools in survey sampling when the researcher is faced with sensitive questions. These models allow a respondent to provide a scrambled response and the researcher is able to unscramble the responses at an aggregate level but not at an individual level. Such models have applications in many fields, most notably in health and behavioral sciences. An optional randomized response model, introduced by Gupta, Gupta and Singh (2002), is a variation of the usual randomized response model and is based on the basic premise that a question may be sensitive for one respondent but may not be sensitive for another, and hence the choice to provide a truthful response or a scrambled response should be left to the respondent. Gupta, Shabbir and Sehra (2010) have recently introduced a two-stage optional RRT model that provides further improvement over the one-stage optional randomized response model. In this talk we will discuss, background of randomized response models, theoretical framework of the optional randomized response model, asymptotic properties of the parameter estimators, some optimality issues, and simulation results.

Abstract: D. Bar-Natan showed that the set of all chord diagrams has a Hopf algebra structure. In this talk, we will define a new multiplication and comultiplication on the set of chord diagrams and show that they give a new Hopf algebra structure.

Abstract: The hyper-reals are an ordered field containing the real numbers as well as infinitesimals and infinitely large numbers. They have languished for 50 years, spurned by most professionals. The situation recalls the slow acceptance of other extensions of the concept of number. For example, as late as the 1880's, Kronecker disputed the existence of irrational numbers. Recently hyper-reals have been appearing in many areas of math, in part because they offer conceptual simplification and shorter proofs. I will construct the hyper-reals, and it will become evident that, just from this simple definition, one can deduce most things one wants to know. I will explain how one can do calculus and geometry in this setting, and perhaps a bit of the mathematical logic underpinning it all.

Abstract: Consider the general linear group G=GL_n(k) where k is an algebraically closed field of characteristic p. Raising all matrix entries to the p-th power gives a group homomorphism G -> G, and thus induces a map on G-modules called the Frobenius twist M -> M⁽¹⁾. The result of this twisting on individual weight spaces of M is just multiplication by p. G-modules are closely related to modules for the symmetric group S_d, but there is nothing resembling a Frobenius twist on the symmetric group side. Nevertheless we have recently obtained results in several different situations that suggest something like a hidden twist map lurking in the background. These situations involved cohomology of irreducible, Specht and Young modules, homomorphisms between various modules, support varieties for Specht modules and the Mullineux map. This talk will begin with a general introduction to representation theory focusing on the symmetric group and on the differences between working in characteristic zero and characteristic p. Then we describe some of the results above and give some speculation as to what might be going on. It should be accessible to graduate students.

Abstract: Springer varieties are studied because their cohomology carries an action of the symmetric group, and their top-dimensional cohomology is irreducible. In his work on categorified tangle invariants, Khovanov provides a topological model for a certain family of so-called two-row Springer varieties. We extend this model to all two-row Springer varieties. Using this, we are able to give a completely combinatorial construction of the Springer representation which can be expressed skein theoretically. If time permits, we will discuss our current work extending these results to three-row Springer varieties as well as connections to sl_3 link homology.

Abstract: This is based upon a summer research project of David Mullens that was supported by UCUR. We indicate that Heron's formula (which relates the square of the area of a triangle to a quartic function of its edge lengths) can be interpreted as a scissors congruence in 4-dimensional space.

Abstract: Business students have traditionally been challenged with gaining a working knowledge of quantitative applications. In an effort to improve student learning, Business Statistics and Operations Management courses were redesigned to incorporate technologies that students are familiar with and already using into a mechanism to deliver course content. Lectures have been removed from the live classroom, and are delivered digitally to the students via the iTunesU website. We developed podcasts incorporating both audio and video elements of the lectures, providing the students with a transportable, and repeatable lecture on each topic. Additionally, podcasts with example problems were developed and included on the iTunesU site, to support the transfer of learning from the lecture to the applied problem solutions. With the content delivered outside of the traditional classroom, student face-to-face time with the instructors in the classroom environment was reduced. Time together was used to focus on clarification of concepts, as well as repetition of working through and solving example problems. The immediate measure of student performance using this redesigned approach was that class test scores in the semester that introduced the redesign. Test scores on each of the exams during the semester had significantly higher averages than in preceding semesters. Further development of the blended course design and its technology based method of content delivery should continue to improve student test performance, and increase the overall performance of Business students in their quantitative abilities.

Abstract: The classical isoperimetric inequality for a plane curve has been generalized to a Minkowski inequality which is concerned with the Minkowski mixed area of two convex domains K1 and K2 in 2-dimensional Euclidean space. In this presentation, we introduce the symmetric mixed isoperimetric deficit of K1 and K2 that measures the homothety of K1 and K2, and establish some new results on Bonnesen-style symmetric mixed isohomothetic inequalities.

Abstract: The "uniform" distribution of many points on the unit sphere is a highly non-trivial problem with applications throughout the whole spectrum of modern science. Whether one studies electrons in equilibrium from Physics, large fullerene compounds from Chemistry, orifices of pollen grain from Biology, or data encoding from Computer Science, one arrives at spherical arrangements of points that minimize some energy functional. In this talk we shall make a short survey of the various problems in the literature and will focus on the separation properties of the extremal configurations and related minimal energy problems.

Abstract: This talk has two parts. Initially, I will present a survey of various minimal energy problems in potential theory having applications to different areas of research not only in mathematics, but in physics, biology, chemistry, etc. In addition to being a preview of a sequence of more detailed lectures to be given at subsequent colloquia and analysis seminars, the survey will introduce the necessary preliminaries to present the main result.

Let E be the union of finitely many intervals or arcs on the unit circle. In a joint work with David Benko we prove that the equilibrium measure of E has a convex density. This is true for both the classical logarithmic kernel, and the Riesz kernel. This seems to be a fundamental result in potential theory, maybe one for the books.

The electrostatic interpretation is the following: if we have a finite union of subintervals on the real line, or arcs on the unit circle, the electrostatic distribution of "many electrons" will have convex density on every subinterval.

Abstract: It is well-known that if a monic polynomial f(x) of degree n over a field F has n distinct roots, a_1,...,a_n, then f(x) = (x-a_n)...(x-a_1). This is not true if the field F is replaced by a division ring D, but there is a generalization, due to Gelfand and Retakh, which expresses f(x) as a product (x-b_n)...(x-b_1) where each b_i is a rational function of a_1,..., a_i. Here b_i is described by use of a certain rational function, called a quasideterminant, of an i by i matrix over the division ring D. We will discuss the definition and properties of quasideterminants, show how they can be used to generalize the theory of determinants of matrices over commutative rings to matrices over arbitrary rings, and show how they occur in the factorization of f(x) and related problems.

Abstract: Starting from the familiar factorization of integers into primes, we extend the concept of unique factorization to polynomials and beyond. In particular, we will discuss how unique factorization, or lack of it, influenced early attempts to prove Fermat's Last Theorem, and we will explore how it can be used to determine the probabilities associated with rolling a pair of dice.