Mathematics

Global well-posedness for the L^2-critical periodic nonlinear Schrodinger equation in 1d and 2d

Nikos Tzirakis (University of Toronto)

Abstract: In this talk I will consider the initial value problem for the $L^{2}$-critical semilinear Schr\"odinger equation, with periodic boundary conditions. I will study these equations in the case that the energy of the initial data is infinite and thus there are no a priori bounds that one can use to extend the local-in-time solutions to global ones. In other words I will show that the problem is globally well-posed in $H^{s}({\Bbb T^{d}})$, for any $s>4/9$ and any $s>2/3$ in 1d and 2d respectively. In 1d we match the result that is already known for initial data on $\Bbb R$. The periodic problem is well studied and the previous best known global well-posedness result in 1d is due to J. Bourgain. I will use the "I-method". This method allows one to define a modification of the energy functional that is well defined for initial data below the $H^{1}({\Bbb T^{d}})$ threshold. Apart from that, the main new ingredient is the proof of new bilinear refinements of Strichartz estimates that hold true for periodic solutions defined on $[0, \lambda]$, $\lambda>0$. This is joint work with D. De Silva, N. Pavlovic and G. Staffilani.

Approximation of the logarithmic integral

Harold Diamond (UIUC)

Abstract: The logarithmic integral function li(x) provides an approximation of the prime counting function pi(x). We recall some standard approximations of li(x) and then describe another method based on continued fractions.

The problem of angle bisectors

Prof. John E. Wetzel (UIUC Department of Mathematics)

Abstract: In 1875 H. Brocard asked whether there exists a triangle having angle bisectors of given lengths. In 1889 van den Berg produced a polynomial equation of degree 16 in the given data for the circumradius. A few years later Barbarin reduced the question to a generally irreducible equation of degree 14; and he showed that if such a triangle exists, it is not constructible with Euclidean tools. In 1937 Wolff found an equation of degree 10 for the reciprocal inradius. And there matters stood until 1994, when two Romanian mathematicians, Petru Mironescu and Laurentiu Panaitopol, settled the matter by showing that any three given positive numbers are the lengths of the angle bisectors of precisely one triangle.

It is my intention to sketch their argument, which involves an application of the Brouwer fixed point theorem.

Inversion Formulas and Their Applications

Tian-Xiao He (Illinois Wesleyan University (Math Dept.))

Abstract: We present several inversion formulas and their applications in expansion and interpolation problems.

First, the concept of a generalized Stirling number pair can be characterized by a pair of inverse relations, in which the problem of expansion of A(t)f(g(t)) (a composition of any given formal power series) can be constructively solved with the aid of Sheffer-type differential operators. Using the generalized Stirling number pair, we establish an inversion formula that can be applied to find an expansion of an analytic function f in terms of a sequence of Sheffer-type polynomials. The result can be readily extended to the higher-dimensional setting.

Secondly, multivariate rational exponential Lagrange interpolation formulas, Hermite interpolation formulas, and Hermite-Fejér interpolation formulas of the Newton type are established using Carlita's inversion formulas. By letting q -> 1, the obtained formulas reduce to the corresponding multivariate polynomial interpolation formulas with combinatorial form.

Finally, we provide a wide class of Möbius inversion formulas in terms of the generalized Möbius functions and its application to the setting of the Selberg multiplicative functions.

Pushing down infinite Loeb measures, cont.

David Ross (visiting UIUC from Hawaii 05-06)

Abstract: Sufficient conditions are given under which the standard part map on a locally compact Hausdorff space can be used to push down an infinite nonstandard measure. This makes it easier to construct standard infinite Borel measures using nonstandard techniques.

Genus zero Gopakumar-Vafa Invariants

Sheldon Katz   [email] (UIUC Math and Physics)

Abstract: The Gromov-Witten invariants of a Calabi-Yau threefold are conjectured to be expressible in terms of the integer-valued Gopakumar-Vafa invariants, which still lack a precise mathematical definition 8 years after their introduction. I give a proposed definition of the genus 0 GV invariants and prove the conjecture for the local invariants of a contractible P^1. This talk is based on math.AG/0601193.

Continuous first order logic

Sylvia Carlisle (UIUC Math)

Abstract: I will give an introduction to continuous first order logic, comparing it with usual first order logic. I will give some examples of metric structures for this setting. As time allows I will show how to prove some theorems analogous to some of the basic theorems of first order logic, for example, the compactness theorem or the downward Lowenheim-Skolem theorem.