Abstract: In a very original and influencial contribution, Y.K. introduced - almost 50 years ago - a topology on the set of preference relations. This was a radically new concept for mathematical economists. I shall discuss Y.K.'s motivation and economic justification for the definition of such a topology, formulate some important results and discuss further developments in Mathematical Economics which are based on Y.K.'s contribution.

Abstract: Only recently, existence of equilibrium in a continuous-time securities market in which the securities are endogenously dynamically complete has been proved. Given the fundamental nature of this result various extensions have been proposed. In the present paper we prove all results under optimal (much more general and natural) conditions. Namely, we only assume quasi-analyticity rather than analyticity of the basic economic ingredients, and we prove everything based solely on this hypothesis.

Abstract: Trudinger-Moser inequality is the counterpart of Sobolev inequality for the spaces $H^{1,d}$ in the dimension $d$. While functions in $\dot H^{1,p}(\mathbb R}^d$ with $p<d$ are integrable when raised to the power $\frac{pd}{p-d}$, the optimal integrable nonlinearity in the case $p=d$ is $e^{a|u|^{d/(d-1)}}$. We discuss various recent improvements of the Trudinger-Moser inequality, which are similar to the classical Hardy-Sobolev and Hardy-Sobolev-Mazya inequalities for the Sobolev imbedding case $p<d$, as well as compactness properties of the Trudinger-Moser functional which are quite different from those in the Sobolev case.

Abstract: A cooperative game is represented by a non--additive set function, the "coalitional function''. The underlying space is called "the set of players'' and the domain of the set function reflects "the coalitions''. The standard versions are a finite set of players in which case the coalitionals are all subsets. Or else one has a measurable space (say, an interval of reals); then one chooses the coalitions to be the Borel measurable sets. The coalitional function is thought of to represent the "worth of the coalition'' whenever its members choose to cooperate.

A solution concept is a mapping that assigns to certain games an "imputation'', that is a measure distributing the total wealth to the players, somehow reflecting the strength players may exercise by participating in coalitions. The "von Neumann--Morgenstern Stable Set'' has been proposed by these authors already in their founding volume "Theory of Games and Economic Behavior'' but it has never found an overwhelming acceptance - partially due to the fact that it is lacking nice linearity or computational properties - that other solution concepts enjoy (the Core, the Shapley Value).

We consider a (cooperative) "linear production game'' with a continuum of players. The coalitional function is generated by $r+1$ "production factors'' that is, nonatomic measures defined on an interval. r of these are orthogonal probabilities. The r+1^{st} measure involved has positive mass "across the carriers'' of the orthogonal probabilities.
That is, there is a more ``central'' production factor available throughout the market and not just in the corners of the market.

We provide convex vNM--Stable Sets of this game. According to the size of the central measure, we observe cases in which a vNM--Stable Set is uniquely defined to be either the core or the convex hull of the core plus a unique additional imputation. There are further situations when the central measure/commodity is available in abundance. Then there is a family of vNM--Stable Sets each one being generated as the convex hull of the core and additional imputations.

Abstract: Starting with a discussion of the ð–Neumann problem I shall speculate on the solution of its nonelliptic analogue, the so-called ð_b–Neumann problem, and on the existence and meaning of a nonelliptic index. The talk will point to the necessity of a deeper understanding of CR-geometry than we have at present.

Abstract: In this paper we present a theory of social interactions where agents' payoffs are determined by three factors: (i) their intrinsic preferences over feasible alternatives, (ii) impact of local interaction conducted through existing peer groups and (iii) global interaction (often called conformity effect). We use the apparatus of potential games (Rosenthal (1973) and Shapley-Monderer (1996)) to examine equilibria of social interaction games. We then apply the obtained results to formation of international alliances during World War II and analysis of countries' behavior in other international conflicts.