Commutative algebra seminar
IPDF talk
Friday, 2nd Feb, 3:30-4:30 pm
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Host: Manoj Keshari
Venue: https://meet.google.com/gyb-jfbn-oiu?authuser=0
Speaker: Parnashree Ghosh
Affiliation: ISI Kolkata
Title: Applications of exponential maps to the epimorphism and Zariski
cancellation problem
Abstract: In the first part, we will discuss the Epimorphism Problem and also discuss the famous Abhyankar-Sathaye Epimorphism Conjecture. We will introduce ``Generalised Asanuma varieties" (GAV) of higher dimensions \geq 3 and see some necessary and sufficient conditions for these varieties to be isomorphic to the affine space. We see that this characterization immediately yields a family of higher dimensional hyperplanes satisfying the Abhyankar-Sathaye Conjecture.
In the second part, we see some necessary conditions for two GAVs to be isomorphic and also describe automorphisms of a certain subfamily of GAV. These results show that for each d \geq 3, there is a family of infinitely many pairwise non-isomorphic rings which are counterexamples to the Zariski Cancellation Problem for dimension d in positive characteristic. We further give a complete description of two important invariants called Makar-Limanov and Derksen invariants of a certain subfamily of GAV.
This talk is based on a joint work with Neena Gupta.