Activities This Week

This will be an organizational meeting for this new seminar. Please bring or send your time constraints.

Time premitting I will also give a short survey talk: Uniformly recurrent subgroups and groups with a trivial amenable radical that fail to be $C^*$-simple. In this I will describe the new construction by Adrien Le-boudec of such groups, following the criterion for $C^*$-simplicity by Kalantar-Kennedy.

I will introduce the definitions of mutual and tight stationarity due to Foreman and Magidor. These notions generalize the property of stationarity from subsets of a regular cardinal to sequences of subsets of different regular cardinals (or, by some interpretations, to singular cardinals). Tight stationarity will then be related to pcf theory, and from a certain pcf-theoretic assumption we will define a ccc forcing which arranges a particularly nice structure in the tightly stationary sequences.

I will discuss how Hodge theory, and positivity phenomena from algebraic geometry in general, can be used to resolve fundamental conjectures in combinatorics, including Rotas conjecture for log-concavity of Whitney numbers and beyond. I will also discuss how combinatorics can in turn be used to explain and prove such phenomena, such as the Hodge-Riemann relations for matroids.

During the last two weeks, we discussed the definition of spectral flow and its connection to noncommutative geometry. This week, we will go over a proof of the integral formula for spectral flow which calculates the index pairing between (the equivalence classes of) a unitary and a p-summable semifinite Fredholm module.