### Article

## Which quartic double solids are rational?

We study the rationality problem for nodal quartic double solids. In

particular, we prove that nodal quartic double solids with at most six

singular points are irrational and nodal quartic double solids with at

least eleven singular points are rational.

Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k- rational functions on G, respectively, g. The conjugation action of G on itself induces the adjoint action of G on g. We investigate the question whether or not the field extensions k(G)/k(G)^G and k(g)/k(g)^G are purely transcendental. We show that the answer is the same for k(G)/k(G)^G and k(g)/k(g)^G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_n or C_n, and negative for groups of other types, except possibly G_2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.

Proceedings of Philisophy of Law International Symposium "Rationality in Law" (Buenos Aires, 5-7 May, 2014)

The proceedings of the conference "Rationality in Action: Intentions, Interpretations and Interactions". The project has been carried out as part of the HSE Program of Fundamental Studies.

The paper focuses on the concept of ‘financial strategies’ and addresses two problems: first, how to define the concepts of financial strategy and strategizing, and second, how to operationalize them into indicators for empirical research. The introduction to this new concept is based on the conviction that strategizing (which is understood as a specific attitude to life held by people who do not live for the moment, think about their future even if it is rather uncertain, set long-term financial goals and act towards achieving them), is an intrinsic factor in the financial behavior of people. It is argued that it is not possible to define financial strategy or to operationalize it objectively and universally since people operate in very different circumstances; i.e. in different institutional environments or at different stages of life, etc. The solution must be found in the interactionist sociological perspective with the emphasis on the construction of the interpretation of a situation: how individuals themselves make sense of financial strategizing in their own environment, the options they perceive and the constraints they feel.

Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a cross-section exists if and only if the universal covering isogeny Ĝ → G is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg's theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck's questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational cross-section in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational W-equivariant map T- - - >G/T where T is a maximal torus of G and W the Weyl group.