Southeast Geometry Conference 2005
Titles and Abstracts

Daniel Dix (University of South Carolina)
- Title: Sharp rates of decay of correlations in special flows over a hyperbolic toral automorphism.
- Abstract: When a Hamiltonian system with two degrees of freedom is restricted to a level set of the Hamiltonian it is possible to obtain a strongly chaotic flow on a compact smooth three dimensional manifold X. For example, the geodesic flow on a compact surface of constant negative curvature. Another example is the special flow over a hyperbolic toral automorphism. In this example there is a global Poincare section diffeomorphic to a two-dimensional torus and the return time is a smooth positive function defined on this torus. The Poincare return mapping is a hyperbolic toral automorphism. The mixing properties of the flow depend on the nature of the return time function. These are measured by the rate at which the correlation integral formed from two functions defined on X tends to its limiting behavior. When the return time function is constant, say the constant 1, then the limiting behavior is periodic in time, and the rate of decay of the correlation to its limiting oscillation depends sensitively on certain regularity properties of the two functions. We give two examples of sharp decay estimates on the correlation, one corresponding to Holder-type regularity and another corresponding to certain classes of smooth functions. When the return time function is nonconstant the flow will typically be mixing, i.e. the correlation will tend asymptotically to a constant as t tends to infinity. There is a natural smooth projection mapping from X onto a circle. When the two functions being correlated factor through this projection mapping there is a natural conjecture about the sharp exponential rate of decay of the correlation which is related to a form of the central limit theorem. This decay rate is independent of the regularity of the two functions. We will discuss some evidence for the validity of this conjecture.
Ronald Fulp (North Carolina State University)
- Title: The quantum master equation.
- Abstract: The quantum master equation is usually formulated in terms of functionals of the components of mappings (fields in physpeak) from a space-time manifold M into a finite-dimensional vector space. The master equation is the sum of two terms one of which is the anti-bracket (an odd Poisson bracket or Gerstenhaber bracket) of functionals and the other is the Laplacian of a functional. Both these terms seem to depend on the fact that the mappings on which the functionals act are vector-valued. It turns out that neither the Laplacian nor the anti-bracket is well-defined for sections of an arbitrary vector bundle. We show that if the functionals are permitted to have their values in appropriate graded tensor products of the dual of the space of smooth functions on M, then both the anti-bracket and the Laplace operator can be invariantly defined. This permits one to develop the Batalin-Vilkovisky approach to BRST cohomology for sections of an arbitrary vector bundle.
Eric Grinberg (University of New Hampshire)
- Title: Tomography by maximally curved spheres in compact symmetric spaces.
- Abstract: CAT Scanners reconstruct a spacial image by recovering a density function from its line integrals (measured by x-ray data). It is natural to consider the recovery of a function on a Riemannian manifold from its integrals along geodesics. There are many results about the x-ray transform, e.g., injectivity results and estimates in negatively curved manifolds, and inversion formulae in symmetric spaces. One variation on the x-ray problem considers a subclass of geodesic measurements: short geodesics. Thickening the geodesic measurement makes the problem harder (or no easier). We thicken short geodesics to maximally curved spheres, which exist in abundance on compact symmetric spaces (Helgason) and consider the Radon-like transform that integrates over these spheres. A number of familiar transforms can be presented in this way, and new ones arise as well. We describe injectivity and inversion results in a variety of cases and suggest generalities.
Haydee Herrera (Rutgers University - Camden)
- Title: Elliptic genus on non-spin manifolds with S^1 actions.
- Abstract: We start by defining the elliptic genus on an oriented manifold of dimension 4n. We prove the vanishing of various characteristic numbers, such as the signature and the A-hat-genus, on manifolds with finite second homotopy group and which admit smooth circle actions. More precisely, we prove the vanishing of various coefficients of the elliptic genus on non-spin manifolds, with finite second homotopy group, when the circle action either satisfies a "parity" condition or has isolated fixed points only.
Alexander Koldobsky (University of Missouri - Columbia)
- Title: Intersection bodies and $L_p$-spaces.
- Abstract: Intersection bodies have become one of the key objects in convex geometry. We discuss a recently discovered connection between this class of bodies and $L_p$-spaces with $p<0.$ This connection allows to get new geometric results by extending to negative values of $p$ different results from the classical theory of $L_p$-spaces.
Jason Parsley (University of Georgia)
- Title: Biot-Savart Operator, Helicity, and Linking on S^3
- Abstract: We extend the Biot-Savart law from physics to an operator BS acting on all vector fields on S^3, a geometric setting for electrodynamics in positive curvature. We show that Maxwell's equations hold and that BS acts as a right inverse to curl. We then discuss its application to energy-minimization problems in geometry and physics that depend on curl eigenvalues. Also, BS allows us to construct linking integrals on S^3. As one further application, we can express the helicity of a vector field, a measure of how much it coils around itself, as H(V) = . We find upper bounds for helicity on the three-sphere; our bounds are not sharp but within an order of magnitude.
Conrad Plaut (University of Tennessee, Knoxville)
- Title: Universal covers of uniform spaces.
- Abstract: I'll speak about joint work with Valera Berestovskii on a very general covering space theory. In earlier work we developed a theory for topological groups for a large category of "coverable" groups that includes all connected, locally connected metrizable groups and even some totally disconnected groups. We are currently in the process of generalizing those results to uniform spaces, including, of course, metric spaces. One motivation for this is recent work by C. Somari and G. Wei concerning universal covers of limits of Riemannian manifolds with a uniform lower bound on Ricci curvature. I conjecture that our theory will apply to all connected, uniformly locally connected metric spaces, which includes, in particular, all inner metric spaces and hence all Gromov-Hausdorff limits of Riemannian manifolds, whether or not they are semi-locally simply connected.
Jesse Ratzkin (University of Connecticut)
- Title: Nondegeneracy of constant mean curvature surfaces.
- Abstract: Constant mean curvature surface have a definite asymptotic structure, which naturally leads to the following question: how much does the asymptotic structure determine a constant mean curvature surface? To answer that question, one must study the linearized mean curvature operator. I will describe joint work with Nick Korevaar and Rob Kusner in which we find explicit bounds for the dimension of the L^2-kernel of this linearized operator, and show the kernel is trivial in many cases.
Nándor Simányi (University of Alabama at Birmingham)
- Title: Rotation numbers of billiards (A topological approach). Abstract: So far we have studied the rotation sets of two families of billiards:
  (A) The inertia motion of a point in the flat torus $\Bbb T^m$ minus a strictly convex, compact obstacle $O$ with a smooth boundary. Here, by definition, the rotation set $R$ consists of all limiting points of the average displacements along orbit segments, when the lengths of these segments tend to infinity.
  
  (B) The inertia motion of a point in a rectangle minus a strictly convex, compact obstacle $O$ with a smooth boundary inside the interior of the rectangle. Here the rotation set $R$ is, by definition, the set of all limiting values of the average wrapping around the obstacle by orbit segments whose lengths tend to infinity.
  
  We will present a detailed description of the above rotation sets. The proofs use ideas from geometry, topology, and a bit of combinatorics.
  
  (These are joint results with A. Blokh and M. Misiurewicz.)
Laszlo Szekely (University of South Carolina)
- Title: Crossing numbers of graphs
- Abstract: I will discuss why the crossing number is a successful concept to measure non-planarity of graphs. Although we do not even know exactly the crossing number of complete graphs, estimates on crossing numbers give invaluable tools and have a number of applications. I will also show a number of open problems, some old and some more recent.
Roman Vershynin (University of California, Davis,)
- Title: Probabilistic approach to asymptotic convex geometry: global and local.
- Abstract: Global convex geometry views a convex body as a whole; local convex geometry studies its lower-dimensional sections. The "complexity" of a convex body is measured by the number of transformations (whether global or local) needed to bring the body to a "uniform" state. The uniform state is typically an ellipsoid; the transformations are usually chosen at random. I will review a recent progress of the probabilistic methods in global and local convex geometries.
Shirley L. Yap (University of Pennsylvania)
- Title: Prescribing Curvature Forms.
Artem Zvavitch (Kent State University)
- Title: The Busemann-Petty problem for arbitrary measures.
- Abstract The Busemann-Petty problem asks whether symmetric convex bodies in $R^n$ with smaller (n-1)-dimensional volume of central hyperplane sections necessarily have smaller n-dimensional volume. Clearly, the Busemann-Petty problem is a triviality for $n=2$ and the answer is ``yes''. Minkowski's theorem shows that an origin-symmetric star-shaped body is uniquely determined by the volume of its hyperplane sections. In view of this fact it is quite surprising that the answer to the original Busemann Petty problem can be negative. Indeed, it is affirmative if $n\le 4$ and negative if $n\ge 5$. In this talk we will present a generalization of the Busemann-Petty problem to essentially arbitrary measure in place of the volume. We also present applications of the latter result by proving several inequalities concerning the measure of sections of convex symmetric bodies in $R^n$.
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Southeast Geometry Conference 2005 Titles and Abstracts

Southeast Geometry Conference 2005
Titles and Abstracts