Abstract: The low frequency behaviour of the canonical switching cell A canonical switching cell can use pulse width modulation to transform an input signal u into a rectangular signal p whose duty cycle linearly depends on the input signal u. After low-pass filtering p to suppress the switching frequency and its harmonics, we obtain the output signal y that is a good approximation of u. This kind of converter is attractive due to the high efficiency of the energy transfer from the power supply to the output signal y. While the operating principle of the canonical switching cell is intuitively clear and it appears in countless books on power electronics, a rigorous mathematical analysis of this system is not trivial, as the system is time-varying and nonlinear. Using abstract results about sampling and approximation of signals, partly due to Yakar Kannai, we outline a rigorous proof of the fact that the signal p is a good approximation of u in the low-frequency range, and we provide precise error bounds for this approximation. We comment about applications.

Abstract: Discussions of different economic and social reforms are quite common for religious philosophers, including adherents of Bible or Talmud; for social philosophers, including Plato, Aristotle, Kant, Hegel, Proudhon, Marx, Schumpeter and many others; for constructors of ideal states like Thomas More and Tommaso Campanella. In modern times, different political parties took different social and economic doctrines as foundations of their programs. Marxists and liberals, social democrats and anarchists, national democrats and national socialists – all of them have their particular views on what the best social order is, and all are subjected to slashing criticism of each other as well as of independent social thinkers. Critique of the socialistic views by Mises and Hayek is the best example.
Some of the doctrines were checked by real-life experiments. Mankind went through hundreds of reforms, including great revolutions, attempts of building socialist societies and economic miracles. Hundreds of works were devoted to different aspects of these events. Therefore, one should be surprised by the fact that in 1990s, when about 30 socialist countries started their transition to the market, we knew almost nothing on how to conduct reforms.
The idea, that a methodology of purposeful institutional changes is necessary, turned out to be very difficult. I will try to discuss some important steps in the development of the Theory of Reform, and then describe its state-of-the art. Since the theory is new, its architecture is not set completely. In the final part of the paper, I will present my own visions of this architecture.
It is a typical case in the practice of reforms, when a reformer, who seeks to introduce an institution with desired properties, discovers that its immediate implementation is impossible because of resource, technological, cultural, political or institutional constraints. In this case, one has to construct a sequence of interim institutions that, for each moment of time, satisfy the existing constraints, and, in the end, provide the implementation of the desired institution. I describe some methods and constructions that can be used to create sequences of interim institutions; illustrations are extracted from the reform experience of China, Russia, and EU. Then I outline main difficulties, which have to be overcome to develop an applicable Theory of Reform, and discuss the natural role of mathematical instruments in this development.

Abstract: Spectral geometry is a relatively young branch of mathematics which is developing rapidly. It blends differential geometry, differential equations and functional analysis. "Can one hear the shape of a drum?" This simple question, popularized by Marc Kac in 1966, marked the beginning of a golden age for spectral geometry. This subject studies the links between the geometry of a space and the eigenvalues of a (pseudo)differential operator acting on functions of that space. In this talk, the spaces under study will be bounded Euclidean domains and their boundaries. I will be interested in two operators: the Laplace-Beltrami operator and the Dirichlet-to-Neumann map. My goal will be to overview the isoperimetric properties of their eigenvalues. We will see that despite sharing many common features, the two operators are also drastically different from the point of view of isoperimetric control.

Abstract: First-order systems of partial differential equations appear in many areas of physics, from the Maxwell equations to the Dirac operator.
The aim of the talk is to describe a general method for the study of the spectral density of all such systems, connecting it to traces on the (geometric-optical) "slowness surfaces" .

Abstract:  In addition  to his mathematical and economic prowess, Yakar Kannai is a Talmid Khakham  of considerable stature.  This talk examines the meeting between those two worlds. It reports on Talmudic discussions of topics such as the Invisible Hand,  Risk Aversion,  Moral Hazard, and others  that play important roles  in economic theory, finding that the Talmud's treatment  can be surprisingly modern.  We will  *not* discuss bankruptcy (the "Three Women"),  as it was  the subject of a talk at the Weizmann Institute several years ago.