Abstract:

In your undergraduate mathematics education, one of the first algebraic objects that you were introduced to was the concept of a group or a ring. Rings are natural objects with an addition, multiplication and unit. Later on, in your graduate education, you probably took a course on commutative ring theory, where you learned that one can form a topological space via the prime ideals of the ring. In this setting the connections between algebra and geometry become more transparent.

In this talk, I will introduce the ideas of tensor triangular geometry and monoidal triangular geometry. Balmer introduced tensor triangular geometry, in 2005, as a way to study symmetric tensor categories using the additive and multiplicative structures on the objects. That is, one can view symmetric monoidal (triangulated) categories like a commutative ring, and study its spectrum. The presenter along with Vashaw and Yakimov have recently investigated monoidal triangular geometry where the multiplicative structure is not necessarly commutative. Our setting is more conducive to the study of arbitrary finite-dimensional Hopf algebras and finite tensor categories.

After introducing the basic concepts, I will demonstrate how to compute the Balmer spectrum using support data techniques. Examples will be shown that demonstrate how natural geometric objects can be realized via the Balmer spectra. At the end of the talk, I will discuss the Chinese remainder theorem for rings that dates back to the 5th century. A 21st century version of the Chinese remainder theorem will then be presented for monoidal triangulated categories that involves localization functors.

Many of the results in this talk involve joint work with Kent Vashaw and Milen Yakimov.