RECASTING THE ELLIOTT CONJECTURE

Let A be a simple, unital, finite, and exact C*-algebra which absorbs the Jiang-Su algebra Z tensorially. We prove that the Cuntz semigroup of A admits a complete order embedding into an ordered semigroup which is obtained from the Elliott invariant in a functorial manner. We conjecture that this embedding is an isomor phism, and prove the conjecture in several cases. In these same cases - Z-stable algebras all - we prove that the Elliott conjecture in its strongest form is equivalent to a conjecture which appears much weaker. Outside the class of Z-stable C*-algebras, this weaker conjecture has no known counterexamples, and it is plausible that none exist. Thus, we reconcile the still intact principle of Elliott's classification conjecture -that K-theoretic invariants will classify separable and nuclear C*-algebras- with the recent appearance of counterexamples to its strongest concrete form.

Generalized property R and the Schoenflies Conjecture

There is a relation between the generalized Property R Conjecture and the Schoenflies Conjecture that suggests a new line of attack on the latter. The new approach gives a quick proof of the genus 2 Schoenflies Conjecture and suffices to prove the genus 3 case, even in the absence of new progress on the generalized Property R Conjecture.

Traverso's isogeny conjecture for p-divisible groups

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Petits points et conjecture de Bogomolov

This paper is devoted to the statement known as the Bogomolov conjecture on small points. We present the outline of Zhang’s proof of the generalized version of the conjecture. An explicit bound for the height of a non-torsion variety of an abelian variety is obtained in the frame of Arakelov theory. Some further developments are mentioned.

The Cuntz semigroup, the Elliott conjecture, and dimension functions on C*-Algebras

We prove that the Cuntz semigroup is recovered functorially from the Elliott invariant for a large class of C¤-algebras. In particular, our results apply to the largest class of simple C¤-algebras for which K-theoretic classification can be hoped for. This work has three significant consequences. First, it provides new conceptual insight into Elliott's classification program, proving that the usual form of the Elliott conjecture is equivalent, among Z-stable algebras, to a conjecture which is in general substantially weaker and for which there are no known counterexamples. Second and third, it resolves, for the class of algebras above, two conjectures of Blackadar and Handelman concerning the basic structure of dimension functions on C¤-algebras. We also prove in passing that the Kuntz-Pedersen semigroup is recovered functorially from the Elliott invariant for all simple unital C¤-algebras of interest.

The bogomolov conjecture for totally degenerate Abelian varieties

We prove the Bogomolov conjecture for a totally degenerate abelian variety A over a function field. We adapt Zhang's proof of the number field case replacing the complex analytic tools by tropical analytic geometry. A key step is the tropical equidistribution theorem for A at the totally degenerate place.

Maltsiniotis's first conjecture for K1

We show that K1(E) of an exact category E agrees with K1(DE) of the associated triangulated derivator DE. More generally we show that K1(W) of a Waldhausen category W with cylinders and a saturated class of weak equivalences agrees with K1(DW) of the associated right pointed derivator DW.

The generic Hanna Neumann conjecture and post correspondence problem

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The entropy conjecture for partially hyperbolic diffeomorphisms with 1-D center

We prove that if f is a partially hyperbolic diffeomorphism on the compact manifold M with one dimensional center bundle, then the logarithm of the spectral radius of the map induced by f on the real homology groups of M is smaller or equal to the topological entropy of f. This is a particular case of the Shub's entropy conjecture, which claims that the same conclusion should be true for any C1 map on any compact manifold.

A conjecture of Bavula on homomorphisms of Weyl algebra

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Pre-play communication and credibility: A test of Aumann's conjecture

The effectiveness of pre-play communication in achieving efficientoutcomes has long been a subject of controversy. In some environments,cheap talk may help to achieve coordination. However, Aumannconjectures that, in a variant of the Stag Hunt game, a signal forefficient play is not self-enforcing and concludes that an "agreementto play [the efficient outcome] conveys no information about what theplayers will do." Harsanyi and Selten (1988) cite this example as anillustration of risk-dominance vs. payoff-dominance. Farrell and Rabin(1996) agree with the logic, but suspect that cheap talk willnonetheless achieve efficiency. The conjecture is tested with one-waycommunication. When the sender first chooses a signal and then anaction, there is impressive coordination: a 94% probability for thepotentially efficient (but risky) play, given a signal for efficientplay. Without communication, efforts to achieve efficiency wereunsuccessful, as the proportion of B moves is only 35%. I also test ahypothesis that the order of the action and the signal affects theresults, finding that the decision order is indeed important. WhileAumann s conjecture is behaviorally disconfirmed when the signal isdetermined initially, the signal s credibility seems to be much moresuspect when the sender is known to have first chosen an action, andthe results are not statistically distinguishable from those whenthere is no signal. Some applications and issues in communication andcoordination are discussed.

Collusion in growing and shrinking markets: empirical evidence from experimental duopolies

We study collusive behaviour in experimental duopolies that compete in prices under dynamic demand conditions. In one treatment the demand grows at a constant rate. In the other treatment the demand declines at another constant rate. The rates are chosen so that the evolution of the demand in one case is just the reverse in time than the one for the other case. We use a box-design demand function so that there are no issues of finding and co-ordinating on the collusive price. Contrary to game-theoretic reasoning, our results show that collusion is significantly larger when the demand shrinks than when it grows. We conjecture that the prospect of rapidly declining profit opportunities exerts a disciplining effect on firms that facilitates collusion and discourages deviation.

Competition and well-being

We study the effects of competition in a context in which people's actions can not be contractually fixed. We find that in such an environment the very presence of competition does neither increase efficiency nor does it yield any payoff gains for the short side of the market. We also find that competition has a strong negative impact on social well-being, the disposition towards others, and individually experienced well-being, the emotional state, of those on the long side of the market. We conjecture that this limits the possibilities of satisfactory interaction in the future and, hence, has negative implications for efficiency in the longer-run

Semigroups of matrices of intermediate growth

Finitely generated linear semigroups over a field K that have intermediate growth are considered. New classes of such semigroups are found and a conjecture on the equivalence of the subexponential growth of a finitely generated linear semigroup S and the nonexistence of free noncommutative subsemigroups in S, or equivalently the existence of a nontrivial identity satisfied in S, is stated. This ‘growth alternative’ conjecture is proved for linear semigroups of degree 2, 3 or 4. Certain results supporting the general conjecture are obtained. As the main tool, a new combinatorial property of groups is introduced and studied.

On punishment and well-being

The existence of punishment opportunities has been shown to cause efficiency in public goods experiments to increase considerably. In this paper we ask whether punishment also has a downside in terms of process dissatisfaction. We conduct an experiment to study the conjecture that an environment with stronger punishment possibilities leads to higher material but lower subjective well-being. The more general motivation for our study stems from the notion that people??s subjective well-being may be affected by the institutional environment they find themselves in. Our findings show that harsher punishment possibilities lead to signficantly higher well-being, controlling for earnings and other relevant variables. People derive independent satisfaction from interacting under the protection of strong punishment possibilities. These results complement the evidence on the neural basis of altruistic punishment reported in de Quervain et al. (2004).