MATHEMATICS COLLOQUIUM
2007 - 2008
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Address:  Mathematics | University of Alabama at Birmingham 
1300 University  Blvd | Birmingham, AL 35294-1170 
Phone:  943 2154 | Fax: 934 9025 
  • Friday, April 11, 2008
    James T. Rogers, Jr. Tulane University
    "Kuratowski's Cyclic Decomposition Theorem and Holomorphic Dynamics"
    2:00 pm / CH-458


    Abstract. Around 1925 Kuratowski proved that a plane continuum that is the common boundary of two of its complementary domains can be decomposed into "layers" in such a way that the resulting quotient space is a simple closed curve. We shall discuss this theorem and its proof; then we discuss applications of the theorem to complex dynamics.
  • Thursday, April 10, 2008
    Janusz Prajs California State University
    "Homogeneous continua and semi-indecomposable compact groups"
    2:30 pm / CH-458

    Abstract. The first part of this talk will be an introduction to homogeneous continua. In particular, motivation and history of this subject will be discussed. In the second part a new result will be presented showing there exists a large collection of semi-indecomposable homogeneous continua which admit compact group structure.
  • Friday, April 4, 2008
    Paul Yang Princeton University
    "On the Q-curvature equation"
    2:00 pm / CH-301

    Abstract. In this talk, I plan to explain the higher order Q-curvature equation that has come up in conformal geometry. I will give a couple of results exploiting the close connection of the Q-curvature equation to the Gauss-Bonnet integral. In addition, there is evidence that the corresponding equation in CR geometry has its own importance in the study of pseudo-convex domains.
  • Friday, March 28, 2008
    Hong Kun Zhang Northwestern University
    "Coupling measures in nonuniform billiards"
    2:00 pm / CH-301


    Abstract. Billiards are dynamical systems originating from the study of Boltzmann's Ergodic Hypothesis in statistical mechanics. For nonuniformly hyperbolic billiards, it is very important to determine the rate of convergence of sequence of distributions to equilibrium state. The general method is to introduce a one-step expansion condition, and show the reduced map of billiard map on a uniformly hyperbolic set enjoys exponential decay of correlations. The mixing rate is determined by the tail bound of return time function. In my PhD dissertation, there were several models failed this method. To overcome the difficulties, we introduce a much weaker one-step expansion condition. The main method we use is to couple measures supported on two families of unstable curves, and show the coupling rate is exponentially fast. This is a joint work with Nikolai Chernov.
  • Friday, March 21, 2008
    Mohameden Ould Ahmedou Eberhard-Karls-Universität Tübingen
    "On the prescribed scalar curvature problem on spheres: the zero degree case"
    2:00 pm / CH-301

    Abstract. In this talk we will report on some existence and multiplicity results for the prescribed scalar curvature problem on the sphere. Due to Kazdan-Warner obstructions, conditions on the function to be realized as a scalar curvature have to be given. From the analysis viewpoint, the existence of "critical points at infinity" for the associated Euler Lagrange functional, makes the existence results harder to prove. However it turns out that such noncompact orbits of the gradient can be treated as usual critical point once a "Morse Lemma at infinity" is performed. In particular their topological contribution to the level sets of the functional can be computed. Through a Morse theoretical approach to this noncompact variational problem, we prove some new existence as well as multiplicity results. Our methods give also an estimate on the Morse indices of the solution that we find and extend to other prescribed curvature problem, like the prescribed $Q$-curvature and the prescribed Webster curvature on $CR$-manifolds.
  • Friday, March 7, 2008
    Nikolaos Tzirakis University of Illinois at Urbana-Champaign
    "The scattering problem for the nonlinear Schroedinger equation"
    2:00 pm / CH-301


    Abstract. I will discuss recent progress on the scattering problem (mainly asymptotic completeness and existence of wave operators) for the semilinear Schroedinger equation with initial data in the energy space. In most cases these results are byproducts of global decay estimates that are true for all solutions. In particular after reviewing recent results I will show how one can derive new interaction Morawetz type (correlation) estimates in all dimensions. The idea behind these estimates is to view the evolution equation as describing the evolution of a compressible dispersive fluid.
  • Friday, February 15, 2008
    Imre Patyi Georgia State University
    Dolbeault and sheaf cohomology over complex Banach manifolds
    2:00 pm / CH-301


    Abstract. The Dolbeault and sheaf cohomology groups over finite dimensional complex manifolds are classical and fundamental objects. They can be defined also over complex Banach manifolds, and they are useful for some questions of global function theory. In this talk we survey some results on the Dolbeault and sheaf cohomology groups and on their interrelations over complex Banach manifolds. We show, e.g., that over closed complex Hilbert submanifolds of separable Hilbert space the real-analytic Dolbeault groups and the sheaf cohomology of the structure sheaf vanish, while there may be some unsolvable continuously differentiable $\overline{\partial}$-equations on the $(0,1)$-level over them.
  • Friday, February 8, 2008
    Lászl?Székely University of South Carolina
    Lovász Local Lemma. A new tool for asymptotic enumeration
    2:00 pm / CH-301


    Abstract. Textbooks describe the Lovasz Local Lemma as the tool to find the proverbial needle in the haystack. We show that the lower bound part of a number of asymptotic enumeration results follow from the Lovasz Local Lemma. Our main tool is two generic constructions for negative dependency graphs. This is joint work with Lincoln Lu.
  • Friday, February 1, 2008
    Attila Maróti University of Southern California & MSRI
    Covering and pairwise generating finite groups
    2:00 pm / CH-301


    Abstract. Let G be a non-cyclic finite group that can be generated by two elements. Let \sigma(G) be the minimal number m so that G is the union of m proper subgroups, and let \mu(G) be the maximal number k such that there exists a subset X of G of size k such that any distinct pair of elements of X generates G. It is easy to see that \mu(G) \leq \sigma(G). In this talk we will investigate these numbers for various groups G. The motivation comes from a conjecture of Blackburn saying that \mu(G)/\sigma(G) tends to 1 as the sizes of the finite simple groups G tend to infinity.
  • Thursday, January 24, 2008
    Junfang Li McGill University (Montreal)
    Geometric evolution equations and Monotonicity formulas
    9:30--10:30 am / CH-458

    Abstract. In this talk, we will mainly focus on some classification results for special solutions of Hamilton's Ricci flow equation. We introduce various monotonicity formulas along the Ricci Flow and Normalized Ricci Flow equations which play crucial role in our results. If time permitted, we will also introduce an extrinsic evolution equation in Euclidean space and derive a monotonic geometric quantity along the flow equation which yields the isoperimetric inequality for Quermassintegrals on some non-convex domains.
  • Monday, January 14, 2008
    Iddo Ben-Ari University of California, Irvine
    Large deviations regimes for last passage percolation
    10:00 am / CH-458

    Abstract. The following model is known as directed last passage percolation on Z^d. For every nearest neighbor path X() on Z^d with initial position X(0)=0 and every time T=0,1,2,..., a random value V(0,X(0))+... + V (T-1,X(T-1)) is assigned, the V(t,x)'s being independent and identically distributed (IID) random variables. The last passage at time T is the maximal random value, the maximum taken over all such paths. The main object of study is its asymptotic behavior as T tends to infinity. The model can be viewed as a natural spatial extension of sums of IID random variables. It exhibits tight relations with some other models, including the totally asymmetric exclusion process, random growth models and random matrices. In this talk I'll discuss this model, focusing on its non-standard large deviations.
  • Friday, January 11, 2008
    Zoi Rapti University of Illinois at Urbana Champaign
    Predicting denaturation and functional sites in DNA
    10:00 am / CH-458


    Abstract. In this talk we'll give a brief introduction to DNA and describe those of its main features that are accounted for in the models that we use. In particular, we'll review the Peyrard--Bishop--Dauxois model, and a relatively recently developed Ising-type model originated by Frank--Kamenetskii and collaborators. Our main focus is the investigation of the correlation between the propensity of the DNA double helix to dissociate and the location of functional sites. We'll present some of our findings and conclude with the future directions of our research.
  • Monday, January 7, 2008
    Pavel Bachurin University of Toronto
    On ergodicity of multi-dimensional dispersing billiards
    9:30 -- 10:30 am / CH-458


    Abstract. Dispersing billiards represent an important class of dynamical systems, which includes many physical models, such as Hard Ball Systems. The study of their ergodic properties was initiated by Sinai in connection with Boltzman Ergodic Hypothesis. I shall survey the developments and present recent results on the ergodicity such billiards when the dimension of the billiard domain is at least 3.
  • Friday, November 16, 2007
    David G. Costa University of Nevada
    On existence of positive solution for a class of 2x2 semipositone systems
    2:00 -- 3:00 pm / CH-301


    Abstract. We consider the question of existence of positive solution for a class of 2x2 semipositone systems. Semipositone problems (where the nonlinearity satisfies f(0) < 0) were studied in the late 80's by Castro and Shivaji, who initially called them nonpositone, in contrast with the terminology 'positone' problems coined in the 60's by Cohen and Keller (when the nonlinearity f was positive and monotone). Typically such poblems arise in population models where there is 'harvesting'. Most cases considered in the literature involve one species. Here we present a class of 2x2 semipositone systems of ODEs. An application is given involving two competing species in an environment which is assumed to be one-dimensional. The main ingredients in our approach are phase-plane analysis and fixed point theory.
  • Wednesday(!!), October 31, 2007
    Norman Dancer University of Sydney
    Nonlinear elliptic equations with small diffusion
    3:00 -- 4:00 pm / CH-458 (big seminar room)


    Abstract. We show that in low dimensions we can obtain a very good understanding of stable solutions of nonlinear elliptic equations with small diffusion and also solutions which are not too unstable. We also discuss the relation of this problem with problems on all of R^N.
  • Friday, October 19, 2007
    Meijun Zhu University of Oklahoma
    Title: "Sharp Sobolev inequalities: from historic and geometric view points"
    2:00 -- 3:00 pm / CH 301


    Abstract. I shall review the history of the study of sharp Sobolev inequalities on $R^n$ (thus $S^n$) (back to the early work of Hardy and Littlewood in 1929, including the story of the Bliss's Lemma), and describe their relation to the Yamabe problem and sharp Sobolev inequalities on manifolds. These motivates us to obtain the local sharp inequalities, which yield, among other results, a much simpler proof of the Onofri inequality. I shall also explain that these sharp inequalities on $S^n$ are often encoded in certain geometric flow problems. In fact, the steady state metrics of those flows are usually described by the sharp inequalities. Our current project on the study of low dimensional geometric flows will be described.
  • Friday, October 12, 2007
    András Bezdek Auburn University
    Title: Short history of long cylinders
    2:00 -- 3:00 pm / CH 301


    Abstract. The talk will aim at a general audience, and will explain the status of several problems in the area of discrete geometry. It will also contain many examples and counterexamples in connection with the following problems on cylinders in 3-space: 1. Littlewood (1968) proposed the problem of determining the maximum number of congruent infinite cylinders that can be arranged in E^3 so that any two of them are touching. The speaker proved that 24 is an upper bound and recently with G. Ambrus disproved the existence of a promising arrangement of 8 cylinders. 2. L. Fejes Tóth (1964) conjectured that the densest packing and thinnest covering of E^3 by congruent infinite cylinders is attained in the arrangement of parallel cylinders such that their cross-sections with a plane perpendicular to the cylinders form the densest packing and the thinnest covering of the plane with congruent circles. The speaker with W. Kuperberg solved the problem concerning packings. 3. Finding reasonably good upper and lower bounds for the translational packing densities for the family of short cylinders, convex cones is a typical problem in discrete geometry. Based on recent constructions several conjectures were formulated.
  • Friday, October 5, 2007
    Michael Loss Giorgia Institute of Technology
    Approach to equilibrium for the Kac master equation
    2:00 -- 3:00 pm / CH 301


    Abstract. The Kac model describes the local evolution of a gas of $N$ particles with three dimensional velocities by a random walk in which the steps correspond to binary collisions that conserve energy as well as momentum. In this talk it will be indicated how this stochastic equation is related to the spatially homogeneous Boltzmann equation and how the exact spectral gap of the one step transition operator can be calculated, thereby solving Kac's conjecture for three dimensional collisions. This is joint work with Eric Carlen, Maria Carvalho and Jeff Geronimo.
  • Friday, September 28, 2007
    Lee M. Goswick, University of Alabama at Birmingham
    On the arithmetic of integral quaternions
    2:00 -- 3:00 pm / CH 301


    Abstract. This talk deals with, among other things, the characterization of twin integer vectors in three dimensions, i.e. orthogonal integer vectors $x$ and $y$ of the same length, by exploiting the properties of the real algebra of quaternions. Several elementary results from recent work done in collaboration with E. W. Kiss, G. Moussong, and N. Simanyi will be reviewed, with an emphasis placed on their geometric interpretation.
  • Friday, September 21, 2007
    Tomio Umeda University of Hyogo, Japan
    The asymptotic limits of zero modes of massless Dirac operators
    2:00 -- 3:00 pm / CH 301


    Abstract. The question whether a zero mode of a Weyl-Dirac operator exists or not was revealed to be very important in the study of stability of Coulomb systems with magnetic fields (Froehlich, Lieb and Loss 1986), and the first examples of zero modes of Weyl-Dirac operators were constructed by Loss and Yau in 1986. Since then, a lot of contributions to the study of the zero modes of the Weyl-Dirac operators have been made. On the other hand, it seems that properties of the zero modes themselves are not well understood. In this talk, I first make a quick review of the history of the study of zero modes of the Weyl-Dirac operators, and then talk about the asymptotic property of zero modes of massless Dirac operators which are generalizations of the Weyl-Dirac operators. One of our main results asserts that all zero modes behave in the same manner at infinity. (This talk is based on joint works with Yoshimi Saito.)
  • Friday, September 14, 2007
    Leonid Parnovski University College, London
    Lattice point distribution in Euclidean and hyperbolic spaces
    2:00 -- 3:00 pm / CH 301
    Abstract. We study the number of lattice points inside a ball (or an annulus) of large radius as a function of the centre of a ball. The average value of this function is the volume of the ball, and we obtain upper and lower bounds (and asymptotics in some cases) for the average deviation of this function from its average value as well as for its variance. The surprising feature of these results is that in some cases the bounds depend on the arithmetic properties of the dimension of the space (but not the arithmetic properties of the lattice!)
  • Friday, September 7, 2007
    Éva Czabarka University of South Carolina
    Modeling evaluation of database retrieval
    2:00 -- 3:00 pm / CH 301


    Abstract. Bootstrap resampling of databases for evaluation of retrieval algorithms is routinely done by practitioners of bioinformatics. However, it is frowned upon in the statistics community, as one application where bootstrapping has no model behind it. We provide such a model and show that the use of bootstrap on the truncated ROC, a retrieval measure used e.g. in sequence retrieval, is justified. A resulting testing method has been in use for the annual update of PSIBLAST.