Mathematics Colloquia, Spring 2004

Thursdays 4:10 pm in 1206 Stevenson Center, unless otherwise noted

Tea at 3:30 pm in 1425 Stevenson Center

February 5, 2004 Mary Ann Horn, Vanderbilt University Theoretical challenges arising from the questions of controllability for elastic structures Abstract: In the context of control of elastic systems, the challenge of obtaining rigorous theoretical results has given rise to the development of new mathematical methods and expanded applications of known tools. This talk will give an overview of the questions of controllability and stabilizability for plates, cylindrical shells and three-dimensional elasticity, as well as the issues arising when single components are coupled into a more complex system. The need for and the utility of high-powered mathematical tools such as microlocal analysis, Carleman estimates and Riemannian geometry will be illustrated and examples of some of the current challenges will be discussed. Contact person: Guoliang Yu Kim Ruane, Tufts University Groups with specified boundary Abstract: The result I will speak about is in the spirit of two well-known results. The first says if $G$ is a convergence group acting on the circle, then $G$ is virtually fuchsian (proved by Casson-Jungreis, Tukia-Gabai). The second result says if $G$ is a one-ended word hyperbolic group with one-dimensional boundary, then that boundary is a circle, a Sierpinski carpet, or a Menger curve (proved by Kapovich and Kleiner). If the boundary is circle, the first result applies to show $G$ is virtually fuchsian. If it’s a Sierpinski carpet, then they have a theorem about the structure of the group. In this case, it is conjectured that $G$ is a geometrically finite Kleinian group (obviously related to Cannon’s Conjecture concerning S^2 boundary). The Menger curve boundary is the “generic” case for a word hyperbolic group with one-dimensional boundary and a characterization of these groups is far from known. I wish to consider the following related general question. Suppose $G$ acts geometrically on a $\cat(0)$ space $X$ whose boundary is homeomorphic to some space $Y$. Given a particular space $Y$, what can you say about $X$ and $G$? For instance, if $Y$ is a circle, then I will show that $X$ is either the Euclidean plane and $G$ is a Bieberbach group or $X$ is the hyperbolic plane in which case the above result applies to say that $G$ is virtually fuchsian. The case where $Y$ is the circle is not difficult to prove, but the techniques already show something very interesting. Next, I consider $Y$ the suspension of the Cantor set and show that $X$ is in fact “almost” isometric to the metric product of a tree with the real line and $G$ is virtually the product of a free group and $\mathbb Z$. This case is already much more difficult than the circle and again, the techniques used should hopefully yield some powerful results. Contact person: Mike Mihalik Frank Quinn, Virginia Polytechnic Institute and State University What is a manifold? Abstract: Short names such as “manifold” or “group” tend to be attached to central objects in an area, as identified by tradeoffs between generality and properties. More general objects with significantly fewer properties, such as “homology manifolds” or “semigroup”, and more special objects with slightly more properties, such as “real analytic manifold” or “algebraic group” are thought of as perturbations on the central object. However what we see as the “central object” often changes as we learn more. In this lecture I’ll trace changes in the meaning of “manifold” as it tracked development of the subject over the last century. We may be on the verge of another major change of viewpoint, though whether the word “manifold” will follow it this time remains to be seen. Contact person: Bruce Hughes Mikhail Perepelitsa, Northwestern University Note: Wednesday, 4:10 pm in SC 1308 Existence of solutions of two dimensional compressible Navier-Stokes equations with uniform in time point-wise bounds on density Abstract: We consider the two dimensional Navier-Stokes equations that describe the motion of compressible, viscous, barotropic flow. The bulk viscosity coefficient is assumed to be a function of the density of the flow that grows a least as fast as the equilibrium pressure. We prove that given periodic initial data in which the density is bounded and does not contain vacuum, there exists global in time weak solution and the density is uniformly bounded and does not contain vacuum. Contact person: Gieri Simonett Jennifer Pietenpol, School of Medicine, Vanderbilt University p53 and Mechanisms of Cell Cycle Checkpoints Abstract: Although most cells in an adult human are quiescent or in a non-proliferative state, specialized cells such as those of the hematopoietic system, those that generate skin, or those that line the gastrointestinal tract, maintain proliferation. On average, about 2 trillion cell divisions occur in an adult human every 24 hours. It is critically important that various cell types divide at a rate sufficient to produce the needed cells for growth and replacement. However, if any given cell type divides more rapidly than is necessary, the normal organization and functions of the organism will be disrupted as specialized tissues are invaded and interfered with by the rapidly dividing cells. Such is the course of events in cancer. Over the past two decades, unraveling the basic molecular events controlling eukaryotic cell cycle transitions has been an area of intense research pursuit. Studies in a variety of organisms have identified an evolutionarily conserved signal transduction system for controling cell cycle transitions through regulation of the activity of key enzymes called cyclin-dependent kinases. Further, many investigations have focused on how the signaling pathways that mediate the cell cycle transitions are regulated and modified after cellular stresses. Human cells are continuously exposed to external agents (e.g., reactive chemicals and UV light) as well as internal agents (e.g. byproducts of normal intracellular metabolism such as reactive oxygen intermediates) that can induce cell stress. Eukaryotic cells have evolved cell cycle machinery with a series of surveillance pathways termed cell cycle checkpoints to ensure that cells copy and divide their genomes with high fidelity during each replication cycle. Cell cycle arrest after DNA damage is critical for maintenance of genomic integrity and loss of normal cell cycle checkpoint signaling is a hallmark of tumor cells. The ability to manipulate cell cycle checkpoint signaling also has important clinical implications, as modulation of the checkpoints in human tumor cells may enhance cellular sensitivity to chemotherapeutic regimens that induce DNA damage. This presentation will focus on the mechanics of the cell cycle as well as checkpoint signaling pathways, with particular focus on the tumor suppressor p53, and how this knowledge may lead to more efficient use of current anticancer therapies and the development of novel agents. The hope is that from a detailed understanding of these processes, more incisive approaches to cancer treatment will evolve that exploit the molecular defects in cancer cells. Contact person: Guoliang Yu Ralph Mckenzie, Vanderbilt UniversityMatthew Valeriote, McMaster UniversityPawel M. Idziak, Jagiellonian University Generative complexity in algebra Abstract: The generative complexity of a class $K$ of algebras is the function $G_K$ defined for positive integers $n$ by setting $G_K(n)$ equal to the number of non-isomorphic $n$-generated algebras in $K$. We have studied $G_K$ mainly for $K$ a locally finite variety (equationally defined class of algebras). Besides making some easy observations about the ways in which the behaviour of $G_K$ imposes algebraic and structural constraints on $K$, we shall outline the proof of a three-year-old result of P. Idziak, R. McKenzie and M. Valeriote giving a precise determination of all the locally finite varieties $K$ for which $G_K(n)$ is bounded by some polynomial function of $n$. In the category of varieties, the directly indecomposable locally finite varieties with polynomially many models are the varieties of modules over finite rings of finite representation type, and the finite matrix powers of varieties of $M$-sets with (possibly) some constants, where $M$ is any finite group. Contact person: Ralph Mckenzie Stefan Richtner, University of Tennessee, Knoxiville Analytic contractions and nontangential limits Abstract: Let f be a complex-valued holomorphic function in the open unit disc D of the complex plane. f is said to have nontangential limit L at z, |z|=1, if there is a fixed angle A inside D with vertex at z such that the limit of f(z) from within A is L. A theorem of Fatou’s implies that every function f in the Hardy space H2 has nontangential limits at a.e. point of the unit circle, and a theorem of Beurling’s exploits these boundary values to give precise information about the invariant subspace structure of that the linear operator Mz, (Mz)f(z) = zf(z) on H2. If H is a Hilbert space of analytic functions that properly contains H2, then there may be a set of positive measure inside the unit circle such that all functions in H have nontangential limits a.e. on this set. In this talk I will discuss efforts to describe conditions for the existence of such a set in terms of the norm on H and the relevance of this for the invariant subspace structure of Mz when Mz acts on H. Contact person: Dechao Zheng Roger Smith, Texas A&M University Cohomology of von Neumann Algebras Abstract: In the 1940’s, Hochschild introduced cohomology groups for algebras, and these were adapted by Kadison and Ringrose in the 1970’s to the functional analytic setting of von Neumann algebras, the weakly closed self-adjoint subalgebras of bounded operators on a Hilbert space. These groups $H^n(M,X)$ are defined in terms of a module $X$ over the algebra $M$ and can be used as isomorphism invariants. When $X=M$, they also act as obstruction groups whose vanishing gives structural information about the algebras. For example, the statement that $H^1(M,M)=0$ is, in different language, a celebrated theorem of Kadison and Sakai that derivations of a von Neumann algebra are always implemented by elements of the algebra. In their original work, Kadison and Ringrose conjectured that $H^n(M,M)$ should always vanish, and proved this in a number of cases. Further progress had to await the recently developed theory of completely bounded maps. In this talk we will survey the current state of affairs and describe some of our own work on this topic. The presentation will be aimed at a general audience with very little background assumed. Contact person: Dietmar Bisch Burt Ovrut, University of Pennsyvania [Joint colloquium with Department of Physics] NOTE: Thursday, 4:00 pm in SC 4327Tea at 3:30 pm in SC 6333 Brane Worlds and Geometry Abstract: The fundamental concepts underlying superstrings and M-theory will be reviewed. The compactification of M-theory to the observed four-dimensional spacetime will be discussed, leading to the idea that our universe is a “brane” floating in a higher dimensional space. To obtain realistic particle physics on the brane, one must have the appropriate geometric and vector bundle properties on the background compact manifold. These provide an important link between modern mathematics and high energy physics. Contact person: Tom Kephart

Colloquium Chair (Spring 2004): Guoliang Yu

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