Date and Time: Monday 14 September, 3:00 p.m - 4.00 p.m.
Google Meet Link: http://meet.google.com/hqk-vobu-npc
Speaker: Sumit Mishra, Emory University
Title: Local-global principles for norms over semi-global fields.
Abstract: Let K be a complete discretely valued field with
the residue field \kappa. Let F be the function field of a smooth,
projective, geometrically integral curve over K
and \mathcal{X} be a regular proper model of F such that
the reduced special fibre X is a union of regular curves
with normal crossings. Suppose that the graph associated to
\mathcal{X} is a tree (e.g. F = K(t)).
Let L/F be a Galois extension of degree n such that
n is coprime to \text{char}(\kappa).
Suppose that \kappa is an algebraically closed field or
a finite field containing a primitive n^{\rm th} root of unity.
Then we show that the local-global principle holds for the
norm one torus associated to the extension L/F
with respect to discrete valuations on F, i.e.,
an element in F^{\times} is a norm
from the extension L/F if and only if
it is a norm from the
extensions L\otimes_F F_\nu/F_\nu
for all discrete valuations \nu of F.
Time:
4:00pm-5:00pm
Description:
Speaker: Manish Kumar, ISI Bangalore
Time: Monday 14th September 4 to 5pm (joining time 3.50pm)
Google Meet Link: https://meet.google.com/qvo-kduy-yco
Title: On various conjectures of Abhyankar on the fundamental group and
coverings of curves in positive characteristic.
Abstract: Abhyankar made many conjectures related to the fundamental group
which led to a lot of interesting mathematics. We will discuss some of
them and their status in this talk.
Time:
7:00pm
Description:
15 September 2020, 7:00 pm IST/ 1:30 pm GMT/ 09:30 am EDT (joining time :
6:45 pm IST - 7:00 pm IST) Please note the unusual time
Google meet link: https://meet.google.com/ada-tdgg-ryd
Speaker: Ben Briggs, University of Utah
Title: On a conjecture of Vasconcelos - Part 1
Abstract: These two talks are about the following theorem: If $I$ is an
ideal of finite projective dimension in a ring $R$, and the conormal
module $I/I^2$ has finite projective dimension over $R/I$, then $I$ is
locally generated by a regular sequence. This was conjectured by
Vasconcelos, after he and (separately) Ferrand established the case that
the conormal module is projective.
The key tool is the homotopy Lie algebra, an object sitting at the centre
of a bridge between commutative algebra and rational homotopy theory. In
the first part I will explain what the homotopy Lie algebra is, and how it
can be constructed by differential graded algebra techniques, following
the work of Avramov. In the second part I will bring all of the
ingredients together and, hopefully, present the proof of Vasconcelos'
conjecture.