Seminar on Differential Geometry and Analysis
Spring Semester, 2006
Tuesdays, 4:00 - 4:50 pm, HH 2302
All interested faculty members and graduate students are welcome. For inquiry please contact the mathematics department office or Dr. Biao Ou.
| January 17 | Speaker: Professor Ivie Stein, University of Toledo Title: Atmospheric Refraction with Calculus of Variations Applied to Fermat's Principle Abstract: The problem is to determine the refracted ray path of electromagnetic radiation through the atmosphere given a variable index of refraction. We apply Fermat's Principle, which states that the radiation travels a path which is a stationary point of the time functional such as a path of least time. Using spherical coordinates with the origin at the center of the earth and by applying Euler's equations from the calculus of variations, we obtain a second order nonlinear system of ordinary differential equations. By introducing canonical variables, the second order system is converted to a first order nonlinear system which with appropriate initial conditions can be solved numerically. Examples are provided. |
| January 24 | Speaker: Professor Biao Ou, University of Toledo Title: A Uniqueness Theorem for Harmonic Functions on the Upper-Half Plane Abstract: Let R2+={(x,y)|y>0}. Consider the problem: -Δu = 0 in R2+, uy(x,0)=-exp(u(x,0)) on the boundary of R2+, and the integral of exp(2u) on R2+ is finite. I'll prove that any solution of the problem must be a fundamental solution of the Laplacian equation with a singularity on the lower-half plane. This problem arises from a context of conformal geometry. I'll turn the problem into a problem in complex analysis and use Picard's theorem to obtain the uniqueness. |
| January 31 | Speaker: Professor Mao-Pei Chui, University of Toledo Title: Constant Mean Curvature Hypersurfaces in Minkowski Space Abstract: In this talk, we will discuss the construction of spacelike constant mean curvature hypersurfaces in Minkowski space. Given a Lipschitz continuous function fon Sn-1, two families of hyperboloids are used as barriers to construct constant mean curvature hypersurfaces which satisfy the asymptotic boundary values defined by Λu(θ) =limr->∞ u(rθ)-r = f(θ), where θ is in Sn-1. Moreover, we use the barriers to estimate the gradient of the u. This enables us to show that the spacelike constant mean curvature hypersurfaces constructed are quasi-conformal to Hyperbolic space. |
| February 7 | Speaker: Dr. Zheng
Huang,
University
of
Michigan Title:The Canonical Metric on a Riemann Surface and Its Induced Metric on Teichmuller Space Abstract:We study the asymptotic behavior of the canonical metric on degenerating Riemann surface and explore some property of the induced metric on Teichmuller space, from a point view of geometric analysis. |
| February 14 | Speaker: Professor Mao-Pei Chui, University of Toledo Title:Contiuation on the topic of Tuesday, January 31 |
| February 21 MH 1003 |
Speaker: Professor Westcott Vayo, University of Toledo Title: Finding the Critical Shapes Abstract: In certain diseases shapes of cells are critical in diagnosis and treatment. One such situation is sickle cell anemia. One can describe sickled cells in mathematical terms and thereby provide insight into other cell properties. |
| February 28 | Speaker: Dr. Jean Cortissoz, University of Toledo Title: The Navier-Stokes Equations Abstract: We will give a survey on the existence and regularity theory for the 2D and 3D Navier-Stokes equations. Whenever possible, ideas for proofs of the presented results will be explained. |
| March 7 | Spring break, no seminar |
| March 14 | Speaker: Professor Bei Hu, University of Notre Dame Title: Liapounov-Schmidt and Hopf bifurcations for the geometry of the free boundary in a model from biology. Abstract:We consider the geometry of the free boundary for a system of partial differential equations that arises in a biological model. For any positive number R the system has a radially symmetric stationary solution with a spherical free boundary of radius R. The system also depends on a positive parameter μ, and for a sequence of the values there exist branches of symmetry-breaking stationary solutions. We prove that there exists a critical number μ* such that the stationary solution is asymptotically stable if 0<μ<μ* but linearly unstable if μ>μ*. Depending on the radius R, we can have either Liapounov-Schmidt or Hopf bifurcations. (This is a joint work with Professor Avner Friedman at the Ohio State University.) |
| March 21 | Speaker: Professor John McCuan, Georgia Institute of Technology Title: A Variational formula for floating bodies Abstract: We give a general necessary condition for criticality in capillary surface problems which involve rigid, fixed bounding structures (i.e., a container) and rigid floating objects as well. We also give applications to planar and minimal interfaces. |
| March 28 | Speaker: Professor Marianty Ionel, University of Toledo Title:Families of Calibrated 4-folds in C4 Abstract: In this talk I will present some families o f special Lagrangian submanifolds in C4 whose second fundamental form satisfy certain pointwise geometric conditions. |
| April 4 | Speaker: Professor Marianty Ionel, University of Toledo Title: Continuation on My Last Talk |
| April 11 | Speaker: Professor Rao Nagisetty Title: Zygmund on the Cantor_Lebesgue Theorem Abstract: Riemann, in his proof of the uniqueness of trigonometric series assumed that the coefficients an,bn tend to zero as n tends to infinity and also that the series converges to zero at all points. While generalizing Riemann's theorem, Cantor deduces Riemann's assumption on the coefficients by proving a lemma now famously known as Cantor's lemma. With the advent of Lebesgue measure, the theorem became Cantor-Lebesgue Theorem. Zygmund proved the analogue for the two dimensional torus. The proof is elementary and beautiful. |
| April 18 | Speaker: Professor Li-Zhen Ji, University of Michigan Title: Large Scale Geometry of S-arithmetic Groups and the Integral Novikov conjectures. Abstract: An important conjecture in topology is the Novikov conjecture on oriented homotopy invariance of the higher signatures, which can be formulated equivalently as the rational injectivity of the assembly map in algebraic surgery theory. The injectivity of the assembly map is called the integral Novikov conjecture in L-theory. There are also assembly maps in algebraic K- theory and C*-algebras, and the rational injectivity of each assembly map is called the Novikov conjecture, and the injectivity of the assembly map the integral Novikov conjecture in the corresponding theory. Various results on the integral Novikov conjectures have been obtained for discrete subgroups of Lie groups. In this talk, I will present some results for arithmetic groups such as SL(n ,Z) and S-arithmetic groups such as SL(n, Z[1/p1, ..., 1/pm]) of semisimple linear algebraic groups. The Novikov conjectures are closely related to the large scale geometry, in particular compactifications, of the universal covering of finite classifying spaces. |
| April 25 | Speaker: Professor Henry Wente, University of Toledo (Cancelled, no seminar.) |
| May 2 | Finals week, no seminar |