Paris

The points in question can be found on any semi-abelian surface over an

elliptic curve with complex multiplication. We will show that they provide

counter-examples to natural expectations in a variety of fields : Galois

representations (following K. Ribet's initial study from the 80's),

Lehmer's problem on heights, and more recently, the relative analogue of

the Manin-Mumford conjecture. However, they do support Pink's general

conjecture on special subvarieties of mixed Shimura varieties.

The n-amalgamation property has recently been explored in connection with generalised imaginaries (groupoid imaginaries) by Hrushovski. This property is useful when studying models of a stable theory together with a generic automorphism, e.g.

elimination of imaginaries (e.i.) in ACFA may be seen as a consequence of 4-amalgamation (and e.i.) in ACF.

The talk is centered around 4-amalgamation of stably dominated types in algebraically closed valued fields. I will show that 4-amalgamation holds in equicharacteristic 0, even for systems with 1 vertex non stably dominated. This is proved using a reduction to the stable part, where 4-amalgamation holds by a result of Hrushovski. On the other hand, I will exhibit an NIP (even metastable) theory with 4-amalgamation for stable types but in which stably dominated types may not be 4-amalgamated.

A graph is $\chi$-bounded with a function $f$ is for all induced subgraph H of G, we have $\chi(H) \le f(\omega(H))$.  Here, $\chi(H)$ denotes the chromatic number of $H$, and $\omega(H)$ the size of a largest clique in $H$. We will survey several results saying that excluding various kinds of induced subgraphs implies $\chi$-boundedness. More precisely, let $L$ be a set of graphs. If a $C$ is the class of all graphs that do not any induced subgraph isomorphic to a member of $L$, is it true that there is a function $f$ that $\chi$-bounds all graphs from $C$? For some lists $L$, the answer is yes, for others, it is no.  

A decomposition theorems is a theorem saying that all graphs from a given class are either "basic" (very simple), or can be partitioned into parts with interesting relationship. We will discuss whether proving decomposition theorems is an efficient method to prove $\chi$-boundedness. 

TBA

Can investors with irrational beliefs be neglected as long as they are rational on average ? Does unbiased disagreement lead to trades that cancel out with no consequences on prices, as implicitly assumed by the traditional models ? We show in this paper that there is an important impact of unbiased disagreement on the behavior of financial markets, even though the pricing formulas are "on average" (over the states of the world) unchanged. In particular we obtain time varying, mean reverting and countercyclical (instead of constant in the standard model) market prices of risk, mean reverting and procyclical (instead of constant) risk free rates, decreasing (instead of flat) yield curves in the long run, possibly higher returns and higher risk premia in the long run (instead of a flat structure), momentum in stock returns in the short run, more variance on the state price density, time and state varying (instead of constant) risk sharing rules, as well as more important and procyclical trading volumes. These features seem consistent with the actual (or desirable) behavior of financial markets and only result from the introduction of unbiased disagreement.