Second Prairie Analysis
Seminar

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First Prairie Analysis Seminar

Title: About singular approximation to the identity

Abstract: We will discuss some geometrical techniques to attack a weak-type problem for a maximal operator with a rough kernel

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Arpad Benyi

Title: Symbolic calculus for bilinear pseudodifferential operators

Abstract: We introduce several natural classes of bilinear pseudodifferential operators. We study their boundedness properties and discuss a symbolic calculus for the transposes of certain operators of order zero.

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Pietro Poggi-Corradini

Title: Harmonic functions on homogeneous domains

Abstract: We describe recent advances in the description of the cone of positive harmonic functions vanishing on the finite boundary of certain planar domains; its connections to the study of entire functions and to solutions of a certain partial differential equation on subdomains of the torus; and we mention an interesting open problem.

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Dmitry Ryabogin

Title: Volumes of projections of convex bodies via Fourier transform.

Abstract: In this talk we present the Fourier analytic approach to projections of convex bodies based on a formula expressing the volume of hyperplane projections in terms of the Fourier transform of the curvature function. (This is a joint work with A. Koldobsky and A. Zvavitch)

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Michael Scott

Title: An Exact, Lightlike, Shock-Wave Solution of the Einstein Equations.

Abstract: Using an extension of a shock-matching theory first developed by Joel Smoller and Blake Temple we construct a new exact solution of the Einstein equations, which can be interpreted as an outgoing spherical shock wave that propagates at the speed of light. The solution is constructed by matching a Friedman Robertson Walker (FRW) metric, which is a geometric model for the universe, to a Tolman Oppenheimer Volkoff (TOV) metric, which models a static isothermal spacetime. The pressure and density are finite on each side of the shock throughout the solution, and the sound speeds, on each side of the shock, are constant and sub-luminous. Moreover, the pressure and density are smaller at the leading edge of the shock, which is consistent with the Lax entropy condition in classical gas dynamics. However, the shock speed is greater than all the characteristic speeds. The solution also yields a surprising result in that the solution is not equal to the limit of previously known sub-luminous solutions as they tend to the speed of light.

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Robert Smits

Title: Conformal Mappings and Brownian Motion

Abstract: Much work has been done recently on heat kernels and Brownian Motion in unbounded domains especially related to domains of finite area, conical domains or convex domains of parabolic type. In this talk I would like to discuss recent work which shows how in two dimensions conformal mappings can be used to extend these results to more general domains showing that the essence of the estimates depend on the growth of the inradius along quasigeodesics.

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Alex Stokolos

Title: Tangential boundary behavior of bounded harmonic functions in the unit disc

Abstract: Bounded harmonic functions in the unit disc $D$ converge nontangentially almost everywhere (Fatou, 1906) and fail to converge along the rotates of any given tangential curve (Littlewood, 1927). We study their boundary behaviour along tangential curves whose shape may change from point to point (a problem posed by W. Rudin). Let $\tau$ be the assignment of a curve $\tau_\theta$ in $D$ ending at $\theta$ and tangential to the boundary $bD$ of $D$, for each $\theta\in bD$. The authors announce a proof that convergence along $\tau$ fails if $\tau_\theta$ depends on $\theta$ in a measurable way, and to show that there is a family $\tau$ of tangential curves such that each bounded harmonic function in $D$ converges along $\tau_\theta$ for a set of points $\theta$ whose outer measure is equal to $2\pi$.

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Caroline Sweezy

Titile: Weighted inequalities and Littlewood-Paley functions for parabolic solutions on non-smooth domains

Abstract: Sufficient conditions for weighted inequalities involving parabolic gradients and their boundary functions on non-smooth domains are established by first proving a discrete Littlewood-Paley inequality. This is joint work wih J. Michael Wilson.

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Nikos Tzirakis

Title: Global existence for the semilinear quintic NLS

Abstract: In this talk I'll show how,using the I-method of J.Colliander,M.Keel,G.Staffilani,H.Takaoka and T.Tao, we can get global well posednesss results for the quintic defocusing NLS on R with initial data that are below the energy norm.

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