A zone of width $\omega$ on the unit sphere is defined as the set of points within spherical distance $\omega/2$ of a given great circle. Zones can be thought of as the spherical analogue of planks. In this talk we show that the total width of any (finite) collection of zones covering the unit sphere is at least $\pi$, answering a question of Fejes T\'oth from 1973.This is a joint work with Alexandr Polyanskii.
Markoff triples are integer solutions of the equation $x^2+y^2+z^2=3xyz$ which arose in Markoff's spectacular and fundamental work (1879) on diophantine approximation and has been henceforth ubiquitous in a tremendous variety of different fields in mathematics and beyond. After reviewing some of these, we will discuss joint work with Bourgain and Sarnak on the connectedness of the set of solutions of the Markoff equation modulo primes under the action of the group generated by Vieta involutions, showing, in particular, that for almost all primes the induced graph is connected. Similar results for composite moduli enable us to establish certain new arithmetical properties of Markoff numbers, for instance the fact that almost all of them are composite.
Time permitting, we will also discuss recent joint work with Magee and Ronan on the asymptotic formula for integer points on Markoff-Hurwitz surfaces $x_1^2+x_2^2 + \dots + x_n^2 = x_1 x_2 \dots x_n$, giving an interpretation for the exponent of growth in terms of certain conformal measure on the projective space.
The Euler--Poisson equations govern gas motion underself gravitational force. In this context the density is not strictly positive, it vanishes in the vacuum region, or falls off to zero at infinity. That causes a degeneration of the hyperbolic systems.The lecture will discuss local existence theorems under these circumstances and with a polytropic equation of state $p=K\rho^\gamma$, here $p$ is the pressure, $\rho $ the density and $\gamma>1$ is the adiabatic gas exponent. In particular, we shall discuss the question whether the initial data include the static spherical solutions for various values of the adiabatic constant $\gamma$. This is a joint work with U. Brauer, Universidad Complutense Madrid.
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About 15 years ago, Bourgain, Brezis and Mironescu proposed a new characterization of BV and W^(1,q) spaces (for q > 1) using a certain double integral functional involving radial mollifiers. We study what happens when one changes the power of |x-y| in the denominator of the integrand from q to 1. It turns out that for q > 1 the corresponding functionals "see" only the jumps of the BV-function. We further identify the function space relevant to the study of these functionals as an appropriate Besov space. We also present applications to the study of singular perturbation problems of Aviles-Giga type.
Using technical language, the Navier-Stokes equations with measure initial data (such as "point vortices"), in two (spatial) dimensions, have attracted much mathematical interest in the last twenty years. There are still many basic open problems, such as well-posedness in bounded domains, Hopf bifurcation into time-periodic solutions and many more. The talk will be non-technical, the only expected analytical background is advanced calculus. It will touch on the theoretical aspects as well as the indispensable accompanying numerical simulations. The general topic has fascinated many poets: "Waves, undulaing waves, liquid, uneven, emulous waves... laughing and buoyant" (Walt Whitman). However, the talk will be much more prosaic.
Spectral theory for general classes of first order systems has been less popular since 1990's. In this talk, I would like to propose a new class of first order systems which generalize both Maxwell and Dirac equations. In this new class, we can treat these two equations in a unified manner, although their physical backgrounds are very different from each other. The main point of my talk is space-time estimates for the new class of first order systems. The essential part of the idea is to derive uniform boundedness of the spectral densities. This talk is based on joint work with Matania Ben-Artzi.
Numerous optimization problems are solved using the tools of distributionally robust optimization. In this framework, the distribution of the problem's random parameter $z$ is assumed to be known only partially in the form of, for example, the values of its first moments. The aim is to minimize the expected value of a function of the decision variables $x$, assuming that Nature maximizes this expression using the worst-possible realization of the unknown probability measure of $z$. In the general moment problem approach, the worst-case distributions are atomic. We propose to model smooth uncertain density functions using sum-of-squares polynomials with known moments over a given domain. We show that in this setup, one can evaluate the worst-case expected values of the functions of the decision variables in a computationally tractable way. This is joint work with Etienne de Klerk (TU Delft) and Daniel Kuhn (EPFL Lausanne).