In set theory one encounters statements which cannot be decided within a background theory one works in. Some of these statemetns can be shown to be consistent within standard background theories widely accepted in mathematics, but some of them require the use of very large sets, known under the term ``large cardinals". In this talk I will discuss how large cardinas naturally arise when studying relative consistencies and also give examples from mainstream mathematics which lead to relative consistencies and large cardinals.
This presentation will begin with an introduction to noncommutative rings by way of several examples. I will then discuss two research programs that share the common motivation of producing methods to view noncommutative pehnomena as an extension of commutative algebra, as follows:
The search for a functor that extends the Zariski spectrum from commutative to noncommutative rings.
Finding ways to infer good ring-theoretic properties from good homological properties.
We can think of a line as the shortest curve joining its endpoints and of a circle as the shortest closed curve enclosing a fixed area (isoperimetric problem). In this talk we will discuss what happens when curves are replaced by surfaces in such problems, and how the solutions of these problems can shed light on the differential geometry of curved spaces.