Trends in Contemporary Mathematics

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Emphasizing Algebraic Thinking, Problem Solving, and Communication

Connections and extensions are mentioned.

Current Trends in Arithmetical Algebraic Geometry

Geometric Numerical Integration is a subfield of the numerical treatment of differential equations. It deals with the design and analysis of algorithms that preserve the structure of the analytic flow. The present review discusses numerical integrators, which nearly preserve the energy of Hamiltonian systems over long times. Backward error analysis gives important insight in the situation, where the product of the step size with the highest frequency is small. Modulated Fourier expansions permit to treat nonlinearly perturbed fast oscillators.

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A big challenge that remains is to get insight into the long-time behavior of numerical integrators for fully nonlinear oscillatory problems, where the product of the step size with the highest frequency is not small. This paper is a short survey of classical and recent results on rationality or stable rationality for smooth projective varieties. We describe or reinterpret in a geometric form cohomological or Chow theoretic criteria for stable rationality, like the triviality of unramified cohomology or the universal triviality of the Chow group of zero cycles, and show that they are effective on examples.

This short paper surveys around themes related to a conjecture of R. Pink which, roughly speaking, generalises in particular the well-known Manin-Mumford conjecture a theorem of M. Raynaud to abelian varieties varying in families. In a series of works in collaboration with D. Masser we have established this for pencils of abelian surfaces, and in further work with D. Bertrand, Masser and A. Pillay we have considered other commutative algebraic groups of dimension 2.

This is analogue to the classical one for integers, and was studied already by Abel. We give a brief introduction to some of the results in the Minimal Model Programme using the elementary theory of directed graphs. In this paper we give a survey of some of the occurrences of Dynkin and extended Dynkin diagrams in algebra.

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It is based on my lecture at the INdAM day in , with some later developments included. We analyze a model tracking problem for a 1D scalar conservation law. It consists in optimizing the initial datum so to minimize a weighted distance to a given target during a given finite time horizon.

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Even if the optimal control problem under consideration is of classical nature, the presence of shocks is an impediment for classical methods, based on linearization, to be directly applied. We adapt the so-called alternating descent method that exploits the generalized linearization that takes into account both the sensitivity of the shock location and of the smooth components of solutions. This method was previously applied successfully on the inverse design problem and that of identifying the nonlinearity in the equation.

The efficiency of the method in comparison with more classical discrete methods is illustrated by several numerical experiments.

Future of mathematics

We propose here a rigorous derivation of this law and of some extensions of it starting from a microscopic model consisting of two species of charged particles interacting both via the self-consistent electromagnetic field, and via some collisional processes. The purpose of the present paper is to present, through historical perspectives, some of the recent developments on the Boltzmann equation in the kinetic theory and multi-dimensional shock waves in gas dynamics.

Recent developments in Nonlinear Potential Theory show the possibility of estimating solutions to nonlinear degenerate equations via potentials. We give a brief description of these results. Guerra: Interpolation and comparison methods in the mean field spin glass model.

Contemporary Math

De Sole, V. Kac, D. Valeri: Integrability of dirac reduced Bi-Hamiltonian equations. Luck: Some open problems about aspherical closed manifolds abstract.

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    Debarre, B. Lass: Monomial transformations of the projective space. Vazquez: Progress in the theory of nonlinear diffusion. Asymptotics via entropy methods.

    Bibliographic Information

    Hairer: Challenges in geometrical numerical integration. Voisin: Integral hodge classes, decompositions of the diagonal and rationality questions. Zannier: Unlikely intersections and Pell's equations inn polynomials. Cascini: Birational geometry of projective varieties and directed graphs.

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