פעילויות השבוע

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.