April 2021
Public Access Category: All |
6:30pm» Jason McCullough, Iowa State University, Ames, IA, USA- Time:
- 6:30pm
- Description:
- Speaker: *Jason McCullough, Iowa State University, Ames, IA, USA*
Date/Time: *2 April 2021, 6:30pm IST/ 1:00pm GMT / 9:00am EDT* (joining
time 6:15pm IST).
Google meet link: https://meet.google.com/iqh-jnea-omr
Title:* Rees-like algebras*
Abstract: Given their importance in constructing counterexamples to the
Eisenbud-Goto Conjecture, it is reasonable to study the algebra and
geometry of Rees-like algebras further. Given a graded ideal I of a
polynomial ring S, its Rees-like algebra is S[It, t^2], where t is a new
variable. Unlike the Rees algebra, whose defining equations are difficult
to compute in general, the Rees-like algebra has a concrete minimal
generating set in terms of the generators and first syzygies of I.
Moreover, the free resolution of this ideal is well understood. While it
is clear that the Rees-like algebra of an ideal is never normal and only
Cohen-Macaulay if the ideal is principle, I will explain that it is often
seminormal, weakly normal, or F-pure. I will also discuss the computation
of the singular locus, how the singular locus is affected by
homogenization, and the structure of the canonical module, class group, and
Picard group. This talk is joint work with Paolo Mantero and Lance E. Miller.
For more information and links to previous seminars, visit the website of
VCAS: https://sites.google.com/view/virtual-comm-algebra-seminar
- Time:
- 5:30pm
- Description:
- Speaker: Mitsuhiro Miyazaki, Kyoto University of Education, Kyoto, Japan
Date/Time: 9 April 2021, 5:30pm IST/ 12:00pm GMT / 8:00am EDT (joining
time 5:15pm IST).
Google meet link: https://meet.google.com/ikb-fxzc-vmd
Title: Hibi rings and the Ehrhart rings of chain polytopes - Part 1
Abstract: In 1985, Stanley submitted a paper titled "Two Poset Polytopes",
which was published in 1986, in which he defined the order and chain
polytopes of a finite partially ordered set (poset for short). On the
other hand, Hibi presented a notion of an algebra with straightening law
(ASL for short) on a finite distributive lattice, which nowadays is called
a Hibi ring, in a conference held in Kyoto 1985. This result was published
in 1987. It turned out that the Hibi ring on a distributive lattice D is
the Ehrhart ring of the order polytope of the poset consisting of
join-irreducible elements of D. In the first talk, we recall the
definition of Ehrhart rings, order and chain polytopes, and Hibi rings. We
recall some basic properties of Ehrhart rings and describe the canonical
module of them. Using these facts, we state some basic facts of Hibi
rings, i.e., the Ehrhart rings of the order polytopes of posets. We also
state some basic facts of the Ehrhart rings of chain polytopes of posets.
In the second talk, we focus on the structure of the canonical modules of
the Ehrhart rings of order and chain polytopes of a poset. We describe the
generators of the canonical modules in terms of the combinatorial
structure of the poset and characterize the level property. If time
permits, we describe the radical of the trace of the canonical module of
these rings and describe the non-Gorenstein locus. This final part is a
joint-work with Janet Page.
- Time:
- 5:30pm
- Description:
- Speaker: L. T. Hoa, Institute of Mathematics, Hanoi, Vietnam.
Date/Time: 23 April 2021, 5:30pm IST/ 12:00pm GMT / 8:00am EDT (joining
time 5:15pm IST).
Google meet link: https://meet.google.com/rev-pyja-byk
Title: Asymptotic behavior of Integer Programming and the stability of the
Castelnuovo-Mumford regularity.
Abstract: In the talk we explain a connection between Commutative Algebra
and Integer Programming and contains two parts. The first one is devoted
to the asymptotic behavior of integer programs with a fixed cost linear
functional and the constraint sets consisting of a finite system of linear
equations or inequalities with integer coefficients depending linearly on
$n$. An integer $N_*$ is determined such that the optima of these integer
programs are a quasi-linear function of $n$ for all $n\ge N_*$. Using
results in the first part, one can bound in the second part the indices of
stability of the Castelnuovo-Mumford regularities of integral closures of
powers of a monomial ideal and that of symbolic powers of a square-free
monomial ideal.
For more information and links to previous seminars, visit the website of
VCAS: https://sites.google.com/view/virtual-comm-algebra-seminar
- Time:
- 7:30pm
- Description:
- Speaker:* David Eisenbud, MSRI and University of California, Berkeley, CA,
USA*
Date/Time: *30 April 2021, 7:30pm IST/ 2:00pm GMT / 10:00am EDT* (joining
time 7:15pm IST). *(Please note the unusual time)*
Google meet link: https://meet.google.com/qei-hcip-jng
Title:* Layered resolutions and Cohen-Macaulay approximation*
Abstract: The representation theory of finite-dimensional algebras has an
important generalization to the study of maximal Cohen-Macaulay modules
(MCMs) over local Cohen-Macaulay rings. In the case of a hypersurface
ring,
this is the study of the matrix factorizations of the equation of the
hypersurface, and these come from minimal free resolutions of the MCMs. I
will talk about the "next" case---MCMs and their minimal free resolutions
over complete intersections. This is joint work with Irena Peeva.
For more information and links to previous seminars, visit the website of
VCAS: https://sites.google.com/view/virtual-comm-algebra-seminar