Abstract:The study of groups of finite Morley rank generalizes the study of algebraic groups over algebraically closed fields with Morley rank generalizing the usual dimension function. In fact, these two classes of groups are suspected to have a deep connection as witnessed by the Cherlin-Zil'ber conjecture: an infinite simple group of finite Morley rank is isomorphic (as an abstract group) to an algebraic group over an algebraically closed field.

The goal of this talk is to give a brief introduction to groups of finite Morley rank and the Cherlin-Zil'ber conjecture. Ultimately, the talk will focus on recent work on how to recognize the "smallest" of the infinite simple algebraic groups: PSL_2.

This work if based on a joint project with Tamar Lando (UC Berkeley)

Abstract: The mid-Nineties of the last century saw a revival of interest in topological semantics for modal logic first introduced by A. Tarski and his student J.C.C. McKinsey in 1941. Spurred by the recent development of a variety of dynamic logical systems which were based on a mixture of topological view of space and a modal view of time, a small industry emerged working on simplifying Tarski and MacKinsey's classic completeness results. The ultimate goal was a set of techniques that will enable one to largely mechanize the construction, axiomatization, determination of computational complexity, and other important aspects of the logical systems based on the topo-semantics. One was after topological kin of Bisimulation, Correspondence Theory, Completeness via Sahlqvist Formulae, Tiling and related reduction techniques in complexity evaluation, and other sophisticated but user friendly techniques of the relational approach to modal logic.

It turns out that the success thus far is only partial. Although the work has produced a steady stream of results, papers and even some Ph.D. thesis on the topic, the progress has not been as smooth as initially envisaged. The field has yet to acquire an accessible set of techniques that is not only easily taught but also easily transferable to related results.

Our goal in this talk is to look closely at a variety of results in the topo-semantics over the last decade and extrapolate a set of techniques that go beyond a specific application to a specific logical systems.

It turns out that the concept best suited for analyzing such a varied and rich list of spaces is that of a self-similar fractal. As we will see in the talk, the conception of a fractal built in a series of self-similar stages enables one to understand techniques used in various semantic constructions in topological semantics for modal logic without much of a heavy technical investment in understanding the theories of fractals. A simple set of tools of the fractal approach seem curiously well crafted for the task of understanding topological spatial structures and reasoning.

Our work is inspired by two theorems of Sierpinski proved sometimes in the 1920s. He proved that two fractals, Sierpinski Carpet and Menger Sponge, are topologically universal. The former fractal embeds all planar Jordan curves; the latter embeds the unrestricted class of Jordan curves. As we will see, the two constructions are universal in an additional logical sense, and related constructions contain a substantial amount of topological structure.