Speaker: Ramya Dutta (TIFR-CAM)

Time: October 14, Friday, 11:30 am
Venue: Room 216

Title:  Apriori decay estimates for Hardy-Sobolev-Maz'ya equations and
application to a Brezis-Nirenberg problem.

Abstract: In this talk we will discuss some qualitative properties and
sharp decay estimates of solutions to the Euler-Lagrange equation
corresponding to Hardy-Sobolev-Mazya inequality with cylindrical weight.
Using these sharp asymptotics we will establish a Brezis-Nirenberg type
existence result for class of $C^1$ sublinear perturbations of the
p-Hardy-Sobolev equation with cylindrical weight in a bounded domain in
dimensions $n > p^2$ and an appropriate notion of positivity for these
perturbations.

Virtual Commutative Algebra Seminar

Speaker: Parnashree Ghosh, Indian Statistical Institute Kolkata, India
Date/Time: 14 October 2022, 5:30pm IST/ 12:00pm GMT /8:00am ET (joining 
time 5:20 pm IST)

Gmeet link: meet.google.com/eap-qswg-xvg [1]

Title: On the triviality of a family of linear hyperplanes
Abstract:  Let k be a field, m a positive integer, V an affine 
subvariety of $A^{m+3}$ defined by a linear relation of the form $x_1^{ 
r_1} · · · x_r^{r_m} y = F(x_1, . . . , x_m, z, t),$ A the coordinate 
ring of V and $G = X_1^{ r_1} · · · X_r^{r_m} Y - F(X_1, . . . , X_m, Z, 
T).$ We exhibit several necessary and sufficient conditions for V to be 
isomorphic $A^{m+2}$ and G to be a coordinate in $k[X_1, . . . , X_m, Y, 
Z, T],$ under a certain hypothesis on F. Our main result immediately 
yields a family of higher-dimensional linear hyperplanes for which the 
Abhyankar-Sathaye Conjecture holds.

We also describe the isomorphism classes and automorphisms of integral 
domains of type A under certain conditions. These results show that for 
each integer d ⩾ 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.

This is joint work with Neena Gupta.

For more information and links to previous seminars,

visit the website of VCAS: 
https://sites.google.com/view/virtual-comm-algebra-seminar [2]



Links:
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[1] http://meet.google.com/eap-qswg-xvg
[2] https://sites.google.com/view/virtual-comm-algebra-seminar