September 11-18, Distinguished Visiting Professor: Martin Evans, University of Alabama, Department of Mathematics.
A group G is said to be simple if its only normal subgroups are 1 and G, whereas G is said to be linear if it is isomorphic to a group of matrices with entries in some field. In this talk we will consider simple linear groups and use geometric methods to show that there are very strong restrictions on their local structure. This talk will be suitable for faculty as well as undergraduates who are familiar with the definitions of groups and rings.
Let G be a (possibly infinite) group. A subgroup H of G is said to be inert if H ∩ Hg has finite index in H for all g in G. Thus, in a sense, inert subgroups of G are ‘almost normal’subgroups of G. We will discuss several results due to Belyaev concerning inert subgroups in simple groups and give some applications to locally (soluble-by-finite) groups.
The FT of algebra first rigorously proved by Gauss states that each complex polynomial of degree n has precisely n complex roots. In recent years various extensions of this celebrated result have been considered. In this talk we discuss the extension of the FT of algebra to harmonic polynomials of degree n. In particular, a recent theorem of D. Khavinson and G. Swiatek proves that the harmonic polynomial z-p(z), deg p=n>1 has at most 3n-2 roots as was conjectured in the early 90's by T. Sheil-Small and A. Wilmshurst.The case n=3 was settled by B. Crofoot and D. Sarason. Unexpectedly, the proof of the general result involves complex dynamical systems. Still nothing is known for harmonic polynomials with conjugate degree of z larger than 1. (Students welcome)
Is it possible to find a number r > 0 such that for any power series, if│∑ anzn│ < 1 whenever z is a complex number with |z| < 1, then ∑|an|rn < 1? Harald Bohr knew the answer to this question in 1913. He was also on the Olympic soccer team with his brother, Niels Bohr, but that is not the topic of this talk. We will discuss his ideas and other related questions in a more general context and see when we can find such a number r, called a Bohr radius, in a general normed space of analytic functions.
In this year paper G. Neumann and D. Khavinson showed that the maximal number of zeros of rational harmonic functions z-r(z), deg r =n>1 is 5n-5. It turns out that this result resolves the conjecture by several astrophysists dealing with the estimate on maximal number of images of a star one can see if the light is deflected by n co-planar masses. The first nontrivial case of one mass was already studied by Einstein in the 30s. Further applications and open problems will be discussed as well.
The 3x+1 function is a simple function usually defined on the positive integers whose iteration forms intriguing patterns. There is a simple conjecture about the function that is believed to be true, but which has confounded mathematicians for decades. Come see the fascinating behavior and get a glimpse of the chaos that appears when the function is generalized to the complex plane.
Hedayat, Rao and Stufken (1988) first introduced balanced sampling designs for the exclusion of contiguous units. Under the assumption that contiguous units of a finite, one-dimensional, population provided similiar information, sampling plans that excluded the selection of contiguous units within a given sample, while maintaining a constant second-order inclusion probability for non-contiguous units, were investigated. In this presentation, preliminary results in the areas of variance estimation and the extension of balanced sampling plans excluding adjacent units to two-dimensional populations will be discussed.
Using computer simulations, we will learn about regular and chaotic dynamical systems as they arise in billiard systems and then generalize these ideas to look at billiards on surfaces (i.e. geodesic flow).
In a 1991 article in the Mathematical Intelligencer, David Robbins made the following proclamation. ''These conjectures are of such compelling simplicity that it is hard to understand how any mathematician can bear the pain of living without understanding why they are true.'' The conjectures Robbins was referring to concern the number of alternating sign matrices of various types, and so are sometimes called the alternating sign matrix conjectures. In this talk I will explain what alternating sign matrices are and what the alternating sign matrix conjectures say about them. One of the most striking of these conjectures was proved by Greg Kuperberg in 1995. Kuperberg was not the first to prove this conjecture, but his proof was certainly the simplest. I will describe the surprising connection between alternating sign matrices and a model of ordinary ice which led to this simplicity. I will also discuss a seemingly unrelated set of objects, called totally symmetric self-complementary plane partitions, which turn out to have a close and still mysterious connection with alternating sign matrices.
Brief description: In this talk we will explore how the complex exponential function arises in a natural way from two famous conformal maps of the earth -- the stereographic projection and Mercator's map. Without using any complex analysis (okay, well, hardly any), we will be able to establish some of the most basic properties of this important function. A familiarity with Calculus is adequate background for prospective members of the audience.
Brief Description: This presentation will survey two methods for generating images in the plane that exhibit "self-similarity" and "noninteger dimension" (so called fractals). We will see that the methods are based on a theoretical photocopier, called a Multiple Copy Reduction Machine (MCRM). The mathematical formulation of an MRCM will result in an Iterated Function System (IFS). Some fundamental results from linear algebra, topology, and real analysis will be discussed. No previous experience with fractals is required.
Thursday, November 18, 4:00-5:00 p.m., Rooke 116 The University Lectureship Committee and Mathematics Department PRESENT Dmitry Khavinson, University of Arkansas (Fayetteville), Department of Mathematical Sciences.The Isoperimetric Inequality Revisited
Greeks already knew that among all domains of equal area the circle has the least perimeter. The first proof however was obtained almost 2000 years later by J. Steiner and later completed by Caratheodory. In this talk we shall discuss various proofs of this remarkable fact finishing with a relatively recent proof via elementary complex analysis. It led to a beautiful free boundary problem that remains unsolved.We shall also discuss applications to an interesting class of physical problems concerning possible shapes of electrified droplets of perfectly conducting fluid in the presence of an electrostatic field.
Platonism, the philosophical theory that the existence of mathematical objects is independent of our knowledge of them, forms the foundation of conventional mathematics. The Platonists both require and accept the principle that a given proposition is either true or false regardless of whether one determines which. Some mathematicians, called constructivists, hold a drastically different point of view. They assert that mathematical objects do not exist independent of our construction of them. In particular, constructivists believe that a given proposition is neither true nor false until such a time as one determines which. After exploring the rich history of this philosophical debate, we will construct a real number that is neither positive, negative, nor zero.
I will introduce the notion of iterated Ore extensions. Then I will consider the representation of some particularly complicated examples.