Activities This Week

In the study of infinite discrete groups it is useful to consider imperfect approximations by finitary models (either permutations or matrices). I will talk about the stability of such approximations, i.e. can it always be corrected to a perfect approximation, focusing mostly on amenable groups. The involved techniques include ergodic theory and dynamics as well as character theory of infinite groups. Some directions and open problems will be presented.

מה צריך בשביל לבנות את המספרים הממשיים? אפסילונים? גבולות? לא בהכרח.

בהרצאה נבנה ביחד את המספרים הממשיים. נא להביא אתכם את המספרים השלמים ואת פעולת החיבור עליהם (מי שחושש שזה לא יספיק שיביא גם בקבוק ספרייט ומספריים)

We explain different kinds of Selmer groups, which are subgroups of Galois cohomology, including Bloch-Kato, strict, and Greenberg Selmer groups. We state part of the Bloch-Kato conjectures and describe a bound joint with A. Betts and M. Leonhardt on the number of rational points on a general higher genus curve, conditional on the Bloch-Kato conjectures. Finally, we explain how to use some Iwasawa theory, specifically Kato’s Euler system and a control theorem of Ochiai, to deduce specific cases of Bloch-Kato associated with elliptic curves.

A dynamical system is called universal if any system with lower entropy can be embedded into it. In this talk, we will discuss universality for $R^d$ flows $(d>1)$ both in ergodic and Borel contexts. We will discuss a specification property that implies universality for $R^d$ flows and provide an example of a tiling dynamical system with this specification property. This is ongoing work with Tom Meyerovitch. This talk is a preliminary report.