December 28, 2020 - January 1, 2021
Public Access Category: All |
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- Time:
- 6:30pm
- Description:
- Talk 1*Speaker: *Anthony Iarrobino, Northeastern University, Boston, MA *
Date/Time: *29 December 2020, 6:30pm IST/ 1:00pm GMT/ 8:00am EST* (joining
time: 6:15 pm IST - 6:30 pm IST).
Google meet link: https://meet.google.com/tih-ghos-zzg
Title: *Jordan type and Lefschetz properties for Artinian algebras*
Abstract: The Jordan type of a pair (A,x), where x is in the maximum ideal
of a standard graded Artinian algebra A, is the partition P giving the
Jordan block decomposition of the multiplication map by x on A. When A is
Artinian Gorenstein, we say that (A,x) is weak Lefschetz if the number of
parts in the Jordan type P_x is the Sperner number of A – the highest value
of the Hilbert function H(A). We say that (A,x) is strong Lefschetz if P_x
is the conjugate of the Hilbert function.
Weak and strong Lefschetz properties of A for a generic choice of x have
been studied, due to the connection with topology and geometry, where A is
the cohomology ring of a topological space or a variety X. We discuss some
of the properties of Jordan type, and its use as an invariant of A, its
behavior for tensor products and free extensions (defined by T. Harima and
J. Watanabe).
If there is time, we will discuss an application to the study of local
Artinian Gorenstein algebras of fixed Hilbert function H; in recent work
with Pedro Macias Marques we show that in codimension three the properties
of Jordan type and of symmetric decompositions show that certain families
Gor(H) in codimension three or greater have several irreducible components.
The first part of the talk is based on work with Chris McDaniel and Pedro
Marques (arXiv:math.AC/1802.07383, to appear JCA).
- Time:
- 5:30pm
- Description:
- Speaker: *Suprajo Das, Chennai Mathematical Institute, India *.
Date/Time: *1 January 2021, 5:30pm IST/ 12:00 GMT/ 7:00am EST* (joining
time: 5:15 pm IST - 5:30 pm IST).
Google meet link: https://meet.google.com/kzx-jdjr-hyw
Title: *An inequality in mixed multiplicities of filtrations*.
Abstract: The theory of mixed multiplicities of (not necessarily
Noetherian) filtrations of $m_R$-primary ideals in a Noetherian local ring
$R$, has been recently developed by Cutkosky, Sarkar and Srinivasan. The
objective of this talk is to describe a generalisation of a Minkowski type
inequality given in their paper. We also recover a result of Cutkosky,
Srinivasan and Verma as a simple consequence of our inequality.
For more information and links to previous seminars, visit the website of
VCAS: https://sites.google.com/view/virtual-comm-algebra-seminar.