Opening

Karl Rubin
University of California Irvine, USA

Euler systems in number theory

Abstract: Euler systems were introduced about 20 years ago by Kolyvagin, and have become one of the best tools available for relating special values of zeta functions to arithmetic. Euler systems have been used successfully to prove analytic class number formulas, and to make progress on the Birch and Swinnerton-Dyer conjecture for elliptic curves. In this talk we will describe the basic examples and applications of Euler systems, and explain why an Euler system should be viewed as the algebraic incarnation of a zeta function.

Pierre Deligne
Institute for Advanced Study, Princeton, USA

Multizeta values, from the 1740's to now

Abstract: Euler introduced multizeta values, and proved identities relating them to zeta values, possibly with the hope of shedding light on the values of the zeta fucntions at odd integers. The modern interpretation of multizeta values as "periods" gives a framework to understand which algebraic relations they should satisfy, but still lacks effectiveness.

Eberhard Zeidler
Max-Planck-Institut, Leipzig, Germany

Euler and the Mathematical Principles of Modern Natural Philosophy

Abstract: In 1744 Euler founded the calculus of variations. This is a basic tool for describing the four fundamental forces in nature: strong, weak, electromagnetic, and gravitative interaction. In the 20th century it was discovered that the principle of local symmetry is basic for formulating both the standard model in cosmology and the standard model in elementary particle physics. It is the challenge for the mathematics and physics of the 21th century to create a rigorous unified theory for the fundamental forces acting in nature.

Stefan Müller
MPI for Mathematics in the Sciences, Leipzig

Rigidity, Geometry and Elastica

Abstract: One of Euler's beautiful contributions to mechanics and analysis is his theory of elastica, one-dimensional objects which are unstretchable but bendable. A fundamental problem is to relate Euler's and other theories of thin elastic objects to three-dimensional nonlinear elasticity. While since Euler there has been an enormous amount of heuristic ansatz-based derivations of large variety of theories, a mathematical understanding has been suprisingly difficult. In this talk I will discuss a new approach to the rigorous derivation of lower-dimensional theories. The key ingredient is a new geometric rigidity estimate which generalizes Fritz John's work on rotation and strain and can be seen as a quantative version of the classical fact a map from n-dimensional Euclidean space to itself whose differential is a rotation (almost) everywhere is already a rigid motion.
   This talk is based on joint work with Gero Friesecke and Richard James.

Craig Fraser
University of Toronto, Kanada

Leonhard Euler and the History of Mathematics: Changing Perspectives

Abstract: Euler has always been recognized as the pre-eminent figure of eighteenth-century analysis, rivaled only by Lagrange in his place in the history of the subject. In traditional history this view of Euler's greatness has been combined with a more critical assessment of his understanding of fundamental principles. Euler is seen as someone who was an enormously energetic and resourceful pragmatist with a fairly limited grasp of foundational conceptions. In 1947 Rudolph Langer wrote of Euler's "naive faith in the infallibility of formulas and the results of manipulations upon them." In 1970 Imre Lakatos characterized Euler as a "naive inductivist" who possessed a somewhat uncritical view of foundational matters. In the view of these historians basic conceptions remained undeveloped by him and would only receive a proper formulation in the writings of such nineteenth-century mathematicians as Cauchy and Weierstrass.
   Over the past thirty years, historical research has significantly altered our understanding of Euler's contributions to analysis. Recent scholarship has developed an account of his work that is more consistent with how he and other eighteenth-century figures understood the subject. Euler possessed a coherent mathematical philosophy that was reflected in his advocacy at the middle of the century of the separation of calculus from geometry, and was also expressed in his conception of the relationship of mathematics and physical science. Drawing on these historical studies, the present paper presents a survey of Euler's research in calculus, infinite series, differential equations and calculus of variations, showing how his perspective on mathematical foundations was articulated in his treatment of these subjects.

Ronald J. Stern
University of California, Irvine

Euler, Polyhedron, and Smooth 4 dimensional Manifolds

Abstract: We will discuss Euler's interest in polyhedra, the open problem of whether every closed manifold of dimension greater than 4 is a polyhedron, and how this leads us to the study of smooth 4-dimensional manifolds. We will then discuss our current (lack of) understanding of smooth 4-manifolds with an emphasis on those with small Euler characteristic.

Günter M. Ziegler
Technische Universität Berlin, Deutschland

Euler's Polyhedron Formula - at the starting point of today's Polytope Theory

Abstract: The Euler Polyhedron Formula, known as e - k + f = 2 (as shown on the poster), or in modern scientific notation as
           f₀ - f₁+ f₂ = 2,
is a crucial starting point for f-vector theory. While regular polyhedra (the "Platonic Solids") were studied since antiquity, it puts "general polyhedra" into the center of attention.
In further major steps we will meet

• the Euler-Poincaré formula for d-dimensional polytopes (due to Schläfli 1852),
• the characterization of the f-vectors of 3-polytopes (due to Steinitz 1906),
• the characterization of the f-vectors of simplicial d-polytopes (McMullen's ''g-Conjecture, proved by Billera-Lee 1980 and Stanley 1980), and
• the characterization of the f-vectors of 4-polytopes (work in progress, still incomplete).

Alfio Quarteroni
Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland and MOX, Politecnico di Milano, Milan, Italy

Mathematical Modelling for Environment, Medicine, and Sport: Euler's Legacy

Abstract: In this talk I will review some classical mathematical models to describe complex phenomena in continuum mechanics. Applications will concern environmental problems, cardiovascular flow problems, and problems arising from sport competition.
   For environmental problems on large geographical scale, as well as on the whole of the circulatory system, multiscale geometrical models allow an effective treatment of the interaction between large, 3D components, and small branches that can be modelled by dimensional reduction techniques. Besides, I will analyse the complex fluid-structure interaction problem that models the artery wall deformation.
   In sport, mathematical models are often used to improve shape design. In particular I will mention recent results achieved in the design of America's cup yachts , through the simulation of the fluid dynamics around boat appendages and of the fluid-sails interaction.
   A common denominator of the different approaches is the problem reduction technique at interfaces in terms of Steklov-Poicaré operators.
   Another one is the pervasivity of Euler's equations and modelling paradigms in modern large-scale scientific computing.

Anthony Tromba
University of California Santa Cruz, USA

VARIATIONS AND SINGULARITIES

Abstract: Euler and Lagrange initiated the study of the first variation of non-linear integral functionals and Legendre, Jacobi, Weierstrass and H.A. Schwarz the second variation. Schwarz also studied the second variation associated to The Plateau Problem. We will discuss how one can now calculate N-th order variations associated to Plateau's Problem and how these variations are related to the existence or non existence of singularities of minimal immersions.

Roger Penrose
University of Oxford, United Kingdom

Euler's Profound Influence on Twistor Theory

Abstract: Twistor theory provides a scheme whereby space-time notions are reformulated in terms of complex analysis and geometry. Space-time points are regarded as secondary entities, to be constructed from spining massless elements (like photons). It owes a great debt to Euler in a number of basic respects. Some of the most striking have to do with twistor diagram theory (the twistor formulation of quantum field theory), where Euler's summing of a divergent series plays a crucial role.