July 2021
Public Access Category: All |
5:30pm» Peter Schenzel, University of Halle, Leipzig, Germany.- Time:
- 5:30pm
- Description:
- Speaker: Peter Schenzel, University of Halle, Leipzig, Germany.
Date/Time: 2 July 2021, 5:30pm IST/ 12:00pm GMT / 8:00am EDT (joining time
5:15pm IST).
Google meet link: https://meet.google.com/hho-deai-mzm
Title: News about Koszul and \v{C}ech complexes: Another view at local
cohomology and completion.
Abstract: In the first part of the talk, we present some elementary new
facts about Koszul and \v{C}ech complexes with respect to a single
element. We construct free resolutions of the \v{C}ech complex for a
system of elements in a commutative ring. This is used in order to
construct quasi-isomorphisms between the \v{C}ech complexes and certain
Koszul complexes. The free resolution of the \v{Cech} complex is applied
in order to find relations to the left derived functors of the completion
as a certain Koszul homology. This material provides an elementary
introduction to some of the results of the speakers joint work with A.-M.
Simon (see "Completion, \v{C}ech and local homology and cohomology.
Interactions between them. Springer Monograph, 2018") as well as some
further developments. One focus is the study of weakly proregular
sequences and of proregular sequences which provides a new insight for the
local cohomology as well as the left derived functors of the completion.
Finally we shall present an application to prisms in the sense of Bhatt
and Scholze.
Chairperson: Le Tuan Hoa, Institute of Mathematics, Hanoi.
For more information and links to previous seminars, visit the website of
VCAS: https://sites.google.com/view/virtual-comm-algebra-seminar
- Time:
- 6:30pm
- Description:
- Speaker: Jonathan Montano, New Mexico State University, Las Cruces, NM, USA
Date/Time: 9 July 2021, 6:30pm IST/ 1:00pm GMT / 9:00am EDT (joining time
6:15pm IST).
Google meet link: https://meet.google.com/mda-vkyr-izm
Title: Mixed multiplicities of graded families of ideals
Abstract: We show the existence (and define) the mixed multiplicities of
arbitrary graded families of ideals under mild assumptions. In particular,
our methods and results are valid for the case of arbitrary m-primary
graded families. Furthermore, we provide a “Volume = Multiplicity formula”
for the mixed multiplicities of graded families of ideals. This is joint
work with Yairon Cid-Ruiz.
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: Yairon Cid-Ruiz, Ghent University, Krijgslaan, Belgium.
Date/Time: 16 July 2021, 5:30pm IST/ 12:00pm GMT / 8:00am EDT (joining
time 5:15pm IST).
Google meet link: https://meet.google.com/qca-qfgy-mpd
Title: Convex bodies and graded families of monomial ideals.
Abstract: We show that the mixed volumes of arbitrary convex bodies are
equal to mixed multiplicities of graded families of monomial ideals and to
normalized limits of mixed multiplicities of monomial ideals. This result
evinces the close relationship between the theories of mixed volumes from
convex geometry and mixed multiplicities from commutative algebra. This is
joint work with Jonathan Montaño. Time permitting, we will also speak
about some joint work with Fatemeh Mohammadi and Leonid Monin regarding
"multi-graded algebras and multi-graded linear series".
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: Kazuma Shimomoto, Nihon University, Tokyo, Japan.
Date/Time: 23 July 2021, 5:30pm IST/ 12:00pm GMT / 8:00am EDT (joining
time 5:15pm IST).
Google meet link: https://meet.google.com/fca-njsw-eox
Title: Perfectoid spaces - Part 1.
Abstract: In the first talk, I begin with a historical review of
perfectoid geometry. Then I talk about the definition of perfectoid rings
and tilting operations. Some basic examples are examined. I also show how
to use perfectoid rings by introducing recent results obtained by several
people. I end with a remark on a classical result on valuation rings.
For more information and links to previous seminars, visit the website of
VCAS: https://sites.google.com/view/virtual-comm-algebra-seminar
- Time:
- 2:30pm - 3:30pm
- Description:
- Title: On the Probabilistic Degree of an n-variate Boolean Function
Speaker: S. Venkitesh
Day-Date: Tuesday, July 27, 2021
Time: 14:30 - 15:30 hrs IST
Google Meet link: https://meet.google.com/qix-catf-kqy
Abstract: Consider the following question: Let f : {0,1}^n \to {0,1} be a
Boolean function that depends on all its input variables. (We call such a
function `truly n-variate'.) What is the least degree of a real-valued
multivariate polynomial P(X_1,...,X_n) that represents the function f ?
Nisan and Szegedy (1994) gave the lower bound log n - O(log log n), and
this was improved by Chiarelli, Hatami and Saks (2020) to a tight bound
log n - O(1).
We will explore a probabilistic analog of the above question: Given a
truly n-variate Boolean function f, what is the least degree of a
real-valued `random' polynomial that agrees with f, with high
probability, at every input? This least degree is called the
`probabilistic degree' of f, and is an important complexity measure for
Boolean functions. Our understanding of this complexity measure is
quite weak. For instance, we don't even know the probabilistic degree
of the OR function (which takes the value 0 at the all-0s input, and 1
at every other input). The best-known bounds put it between (log
n)^{1/2 - o(1)} and O(log n).
In this talk, we will give a near-optimal understanding of the
probabilistic degree of truly n-variate Boolean functions, modulo our
lack of understanding of the probabilistic degree of OR. We show that
if the probabilistic degree of OR is (log n)^c, then the minimum
possible probabilistic degree of a truly n-variate Boolean function is
at least (log n)^{c/(c+1) - o(1)}, and this bound is tight up to (log
n)^{o(1)} factors.
This is a joint work with Srikanth Srinivasan.
(https://eccc.weizmann.ac.il/report/2021/098/).