Pure Maths Colloquium Seminar - Invariants and singularities in prime characteristic - Moty Katzman (Sheffield) Seminar
- Time:
- 15:00 - 16:00
- Date:
- 27 April 2018
- Venue:
- Lecture Theatre 10C, Building 54, Mathematical Sciences, University of Southampton, Highfield Campus, SO17 1BJ
For more information regarding this seminar, please email Dr Jan Spakula at Jan.Spakula@southampton.ac.uk .
Event details
Abstract: The last few decades have seen the development of various powerful techniques to study algebraic varieties and their coordinate rings in prime characteristic. In this talk I will give a taste of these by exploring bounds relating classical invariants of local rings, namely, multiplicity, dimension and embedding dimension. I will first introduce the notion of F-purity, and bound multiplicity in terms of dimension and embedding dimension. This bound can the be extended to a new class of singularities: F-injective rings. We can then introduce a new invariant, the HSL number, which measures the failure to be F-injective, and use it to bound multiplicity. Finally, if time permits, we will see that some of these results yield bounds in characteristic zero, by reduction to prime characteristic. This is joint work with Wenliang Zhang.
Speaker information
Dr Moty Katzman , The University of Sheffield. Dr Katzman's research is in the area of commutative algebra. Specifically, he is interested in Characteristic p methods: Certain theorems in algebra can be proved by showing that they hold in positive characteristic, and in characteristic p one has extra structure given by the Frobenius map x↦xp. There are several tools, notably tight closure, which exploit this extra structure to prove some remarkable theorems. Also, Local cohomology modules: This modules derive their importance partly from the fact that they detect interesting properties of modules over commutative rings (e.g., depth.) Unfortunately, these objects tend to be very big are rather mysterious. It is very difficult to describe them in any detail even in seemingly easy cases. Dr. Katzman has recently been producing both examples showing that these objects are more complicated than previously conjectured but also instances where they can be understood fairly well; and Combinatorial aspects: One of the simplest family of modules imaginable are monomial ideals in polynomial rings and, perhaps surprisingly, these objects have a very rich structure, in some sense richer than the structure of graphs. Dr Katzman has recently been studying certain monomial ideals associated with graphs a discovering some surprising connections between the algebraic and combinatorial properties of these objects.