Heights of motives
Abstract
The height of a rational number a/b (a,b integers which are coprime) is defined as max(|a|, |b|). A rational number with small (resp. big) height is a simple (resp. complicated) number. Though the notion height is so naive, height has played a fundamental role in number theory. There are important variants of this notion. In 1983, when Faltings proved the Mordell conjecture (a conjecture formulated in 1921), he first proved the Tate conjecture for abelian varieties (it was also a great conjecture) by defining heights of abelian varieties, and then deducing Mordell conjecture from this. The height of an abelian variety tells how complicated are the numbers we need to define the abelian variety. In this talk, after these initial explanations, I will explain that this height is generalized to heights of motives. (A motive is a kind of generalisation of abelian variety.) This generalisation of height is related to open problems in number theory. If we can prove finiteness of the number of motives of bounded height, we can prove important conjectures in number theory such as general Tate conjecture and Mordell-Weil type conjectures in many cases.
16:30
A stochastic model of large-scale brain activity
Abstract
We have recently found a way to describe large-scale neural
activity in terms of non-equilibrium statistical mechanics.
This allows us to calculate perturbatively the effects of
fluctuations and correlations on neural activity. Major results
of this formulation include a role for critical branching, and
the demonstration that there exist non-equilibrium phase
transitions in neocortical activity which are in the same
universality class as directed percolation. This result leads
to explanations for the origin of many of the scaling laws
found in LFP, EEG, fMRI, and in ISI distributions, and
provides a possible explanation for the origin of various brain
waves. It also leads to ways of calculating how correlations
can affect neocortical activity, and therefore provides a new
tool for investigating the connections between neural
dynamics, cognition and behavior