2005-06 Colloquia talks

Abstract: Angiogenesis is the process by which new blood vessels develop from an existing vasculature through endothelial cell sprouting, proliferation and fusion. This process occurs during embryogenesis, would healing, arthritis and during the growth of solid tumors. In this talk I will review continuous and discrete mathematical models applied to tumor growth and angiogenesis. Most of these models are based on biochemical aspects of tumor growth and angiogenesis, however recent experimental results have shown that biomechanical aspects also play a pivotal role in the angiogenic process. This talk will elucidate some of these experimental findings.

Abstract: The reconstruction conjecture is one of the most seductive open problems in mathematics today. The question is simple to state and understand, and the conclusion seems inescapable. But beware - although significant progress has been made, this problem remains unsolved 65 years after it was originally posed. The conjecture centers around reconstructing a graph based on knowledge of all its subgraphs. (Recall that, loosely speaking, a graph is a network of dots with paths joining some pairs of dots to one another. In other words, we are not talking about the sorts of graphs produced by your calculator.) In this talk I will recount progress towards the resolution of the reconstruction conjecture, and introduce certain polynomials that I am cautiously optimistic will help in its solution.

Abstract: In the 1830s, John Scott Russell observed and studied a solitary surface water wave that he called the "great wave of translation." In 1895, Korteweg and de Vries (KdV) derived their equation that describes these solitary waves. Seventy years later, in 1965, Kruskal and Zabusky discovered that the solitary wave solutions of the KdV equation have the remarkable property of retaining their identities after collisions with other solitary waves. They gave these special waves the name "solitons." This discovery motivated a more detailed mathematical study of the KdV equation, including a search for conservation laws for the KdV equation that eventually led to devising the "inverse scattering method" for exact determination of the N-soliton solutions. In this talk, I will describe some of these discoveries. One feature I will demonstrate is that progress in science can be strongly influenced by non-scientific events and circumstances. These discoveries now have been extended and generalized to many different equations and applications, and a whole industry has developed pursuing these new directions.

Abstract: This talk will survey some aspects of subnormal operators that illustrate their connection with analytic function theory. Particular emphasis will be placed on the cyclic case and the problem of unitary equivalence for these operators will be shown to be equivalent to a function theory problem.

Abstract: Coxeter groups are generated by elements of order 2, subject only to the obvious relations. After recalling how a large class of Coxeter groups arise as "Weyl" groups in semisimple Lie theory, one classical (characteristic 0) and one less-classical (characteristic p) branch of representation theory will be discussed.

Abstract: Knot theory uses techniques from different fields of mathematics ranging from combinatorics to geometry. Thurston in the 1980's established the importance of geometric invariants like the volume in knot theory. A revolution in knot theory occurred with the discovery of the Jones polynomial in 1987. The relationship between these two ideas is still a mystery. In this talk I will discuss the geometric and combinatorial points of view in knot theory. I will describe some recent results and mention some conjectures about these invariants and their relationship to each other. This talk will be accessible to undergraduates and graduate students.

Abstract: The goal is to model relationship of Complex II activity through time and to determine activity rate of Succinate-UQ2 Reductase enzyme in isolated frozen liver mitochondria from broilers. The modeling sequence with various linear and non-linear models will be explored to obtain a model that is reasonably able to describe the relation. Mitochondria that are present in all eukaryotic cells except in the red blood cells are responsible for creating more than 90% of the energy needed by the body to sustain life and support growth. When they fail, less energy is generated within the cell resulting in cell injury and even cell death. Mitochondrial dysfunction has been implicated in degenerative disorders, ageing, and numerous human diseases. In this case study, we are interested in the activity of only one enzyme complex, namely Complex II or Succinate Ubiquinone Reductase (SQR) at a fixed concentration of substrate for 5-minute duration of the assay. The mean decrease in absorbance for 9 birds with two replicates per bird was recorded.

Abstract: Representations of quivers are natural generalizations of finite dimensional modules of polynomial algebras over a field and appear in many different directions of Mathematics, in particular in ring theory. However their deep relation to Lie algebras and quantum groups are only developed in the last 15 years due to the work of Ringel and Lusztig. In this talk I will outline the very basic theory of representations of quivers including Gabriel's Theorem on finite type and Kac's Theorem as well as Kac's Conjecture. Then I will outline Ringel's and Green's Hall algebra construction for quantum groups.

Abstract: Geometric group theory pursues the study of infinite groups via their actions on interesting geometric objects. At the start of this talk I will "remind" folks of how free products of groups act on trees. I will then introduce the McCullough-Miller complex, a space of actions of free products on trees. We will then explore how combinatorial properties of this space yield cohomological information about automorphism groups of free products.

Abstract: Musical timbre is inherently multidimensional and extremely complex in structure. Humans have the "natural" ability to segregate, identify and recognize sounds in a variety of situations - separated by a wall from the sound source, in a concert hall, within a noisy traffic environment or at a cocktail party. In this talk I will address some of the issues involved with machine-based automatic musical instrument timbre recognition and present my own recent research in this area. The developed timbre recognition system follows a "bottom-up", "data-driven" model and includes a pre-processing module, a feature extraction module, and a RBF/EBF (Radial/Elliptical Basis Function) artificial neural network-based pattern recognition module. 829 monophonic samples from 12 instruments have been used for testing the system. Significant emphasis has been put on feature extraction development and testing to achieve robust and consistent feature vectors that are eventually passed to the neural network module.

Abstract: An edge-regular graph on n vertices is, for some d > 0, a d-regular simple graph for which there exists a non-negative integer l such that every pair of adjacent vertices has exactly l common neighbors. We use the notation ER(n,d,l) to denote the set of all edge-regular graphs with parameters n, d, l as above. If G is a graph in ER(n,d,l), then there necessarily exists a non-negative integer p = n ñ 2d + l that counts the number of non-neighbors that each pair of adjacent vertices has; it has turned out surprisingly fruitful to focus on this parameter p when studying edge-regular graphs. Given parameters l and p, we present some recent results involving the upper bound and "near upper bound" on the number n of vertices for an edge-regular graph with those parameters; such an upper bound is expressed nicely in terms of l and p. We also show some newer results involving the lower bound on n as well, and discuss future work.

Abstract: A trajectory of a flow on a 3-manifold is wild if the closure of at least one of the semi-trajectories is a wild arc. A trajectory is 2-wild if the closure of each semi-trajectory is a wild arc.

We describe a method of embedding wild trajectories in flows on 3-manifolds. This methods yields interesting examples of dynamical systems. In particular:

1. Every boundaryless 3-manifold admits a flow with a discrete set of fixed points and such that every non-trivial trajectory is 2-wild.

2. Every closed connected 3-manifold admits a flow with precisely one fixed point and such that the closure of every non-trivial trajectory is 2-wild.

Outside the set of fixed points, the above flows can be constructed in either of the two categories: C-infinity or piecewise linear.

Abstract: Motivated by the spectral analysis of the Google matrix for the computation of the PageRank of the Google web search engine, we present a recent result on a relation of the spectra of a matrix and its special perturbation. The spectrum of the Google matrix is easily obtained as a corollary. This talk can be understood by undergraduate students who have studied elementary linear algebra.

Abstract: Suppose, if you will, that G is a finitely generated group. In general there are, of course, very many generating sets for G. How can one classify these? Why would one bother?

We will discuss what is arguably the most natural classification system, a useful graphical method for visualizing this classification, and analogies with other "stable" phenomena in mathematics.

Abstract: Supersymmetry was introduced by physicists and has a lot of applications in conformal field theory, string theory and gravitation. But supersymmetry appears also quite naturally in mathematics.

In this talk I will give several examples how supersymmetry helps to solve problems in mathematics. In the second part of the talk I will discuss connections between quantum groups, supersymmetry and representation theory in finite characteristic.

Abstract: Among the developing countries, a time lag is usually found between the collection of information and the release of the results of the survey. The concept of concurrent assessment of health and family welfare programs is discussed. The methodology used is the multicentric approach of the sampling survey based on clustered sampling.

Abstract: The primary treatment of well-differentiated thyroid cancer is surgery. After surgery for differentiated thyroid carcinoma, it is common practice to ablate the residual thyroid tissue with radioiodine-131. A single optimum dose required to ablate residual thyroid tissue is still controversial. On this aspect, various studies have been published and reported as randomized controlled clinical trial (RCT), but most of these do not fulfill each and every aspects of RCT. In order to determine an optimum dose of radioiodine for remnant ablation of differentiated thyroid cancer an equivalence RCT being conducted (since 2001) at All India Institute of Medical Sciences (AIIMS). Patients were randomly allocated in to three different treatments groups of radioiodine-131, namely 25 mCi, 50 mCi and 100 mCi. An attempt was made to follow all the required steps (as given in CONSORT guidelines, 2001) to conduct and report well-planned RCT. We did overcome certain logistic problems giving emphasis on various design related issues. This presentation aims to address various issues related to study design (unequal allocation ratio, randomization method etc., statistical consideration of equivalence trial) and statistical methods used for the analysis.

Abstract: Balance is an important issue affecting the health of people of all ages. Veterans, people with chronic fatigue, people who have had strokes, amputations or head injuries have balance problems and are at a greater risk for falls. Diagnostic approaches are needed to find those most at risk and to analyze whether treatments for such persons are effective. In this presentation, we discuss the diagnostic equipment commonly used to study balance and how it determines postural sway and arrives at an ìEquilibrium Scoreî. We then develop our model, which employs an inverted pendulum approach. The model is used to examine postural sway and computes ankle moment and ankle stiffness. Finally, we develop a new metric of balance health (called the Postural Stability Index) and discuss its advantages over ìEquilibrium Scoreî.

Abstract: Dehn surgery removes a solid torus neighborhood of a knot in a 3-manifold and reattaches it in another way to produce another 3-manifold. It remains an open question which knots in the 3-sphere admit a Dehn surgery yielding a lens space though Berge has conjectured a classification. We will discuss Berge's knots, some of the differences among them, and the related questions of when a knot in a lens space admits a Dehn surgery yielding a 3-sphere or another lens space.

Abstract: Somewhere on Earth at this very moment there are two antipodal points (that is, points directly opposite from one another through the Earth) where the temperatures are identical and the pressures are also identical. This meteorological fact follows immediately from the theorem in topology known as the Borsuk-Ulam Theorem. We'll see a neat proof of this fact whose primary technical tool is a wrapping rope.

Abstract: The problem of uniformly distributing points on spheres (more generally, on compact sets in Euclidean n-space) is an interesting, long standing, and difficult problem with numerous applications in diverse areas (approximation theory, numerical integration, spherical t-designs, crystallography, molecular structure, electrostatics, etc.). In this talk, we will discuss classical and recent results concerning the asymptotics, separation radius, and mesh norm of optimal arrangements of N points on d-dimensional compact sets A embedded in Euclidean n-space, which interact through a power law (Riesz) potential V=1/r^s, where s>0. In particular, we will see that, for a large class of sets, such optimal points are "well distributed" on A and have local energies asymptotically of the same order, as N approaches infinity.

Abstract: Many proofs are logically sound, in that each step follows from the previous one, yet the proof as a whole sheds very little light on whatever was proven. I call such proofs "Hows," in contrast to "Whys," which are proofs that dramatically get to the heart of the matter. In this talk, we look at some well-known Hows, and then look at some (surprisingly less well known) Whys. We will also look at some Hows that are crying out for as-yet-unknown Whys. This talk is designed to be accessible to advanced undergraduates.

Abstract: The LU decomposition and the QR decomposition in matrix theory have the corresponding versions in the complex classical groups, called the Bruhat decomposition and the Iwasawa decomposition. Both decompositions can be viewed as certain group orbits over flag manifolds. In this talk, we extend both decompositions to B_K\G/P_G with finite cardinalities, where K is a symmetric subgroup of G, B_K is a Borel subgroup of K, and P_G is a parabolic subgroup of G. They share many analogous properties with the Bruhat decomposition B_G\G/B_G and the Iwasawa decomposition K\G/B_G. We will use some examples on Grassmannians to illustrate orbits and invariants for B_K\G/P_G, and compare them with the Bruhat decomposition and the Iwasawa decomposition. Implications of these decompositions in representation theory will be presented.

Abstract: In this talk we will discover that Hilbert did cohomology in the 1890's, he just didn't realize it. Starting with Lie superalgebra cohomology one is quickly led to invariant theory. Thus, what is usually perceived as an abstract and technical subject instead becomes a problem of polynomials. We will introduce Lie superalgebras, cohomology, and see how invariant theory allows us to calculate cohomology explicitly and easily. The talk is intended to be accessible to a wide audience.

Abstract: We introduce the use of "regular production systems" - a certain generalization of symbolic substitution systems - as a tool for analyzing tilings in general. These systems precisely capture the combinatorial structure of any set of tiles residing on a two-dimensional surface, though in this talk we are particularly interested in tilings of the hyperbolic plane.

We briefly discuss a number of applications, such as the construction of the first known "strongly aperiodic" set of tiles in the hyperbolic plane. As another application, we conjecture necessary and sufficient conditions under which we may tile the sphere, hyperbolic or Euclidean plane by copies of a given triangle, and prove the conjecture on all but a measure-zero set in the space of all triangles. We give a new proof of Poincare's triangle theorem as an aside.

We also show most triangles that do tile are "weakly aperiodic"; that is, they admit tilings, and admit tilings that are invariant under some infinite cyclic symmetry, but do not admit tilings with a compact fundamental domain. Decidability and rigidity play an interesting role.

The talk will use only elementary methods and is intended to be suitable for a wide audience.

Abstract: In this talk we investigate the structure of loops in which the associativity relation between pairs of nontrivial elements is transitive. We also look at other properties that come out of the study of this class of loops.