July 2020
Public Access Category: All |
5:30pm» Manoj Kummini, Chennai Mathematical Institute- Time:
- 5:30pm - 6:30pm
- Description:
- Date and Time: 3 July 2020, 5:30 pm IST - 6:30 pm IST (joining time :
5:15 pm IST - 5:30 pm IST)
Google Meet Link: https://meet.google.com/gkc-hydx-fkn
Speaker: Manoj Kummini, Chennai Mathematical Institute
Title: Big Cohen-Macaulay algebras - Part 2.
Abstract: In the first talk, we will look at some applications of big
Cohen-Macaulay algebras. In the second we will give an outline of the
proof by Huneke and Lyubeznik that the absolute integral closure of a
noetherian local domain that is a homomorphic image of a Gorenstein local
ring is a (big) Cohen-Macaulay algebra over it.
- Time:
- 4:00pm - 5:30pm
- Description:
- Lecture 1: Monday July 6th, 2020, from 4 pm to 5:30 pm (joining time 3:45pm)
Title: Functorial Geometry and Moduli Spaces
Google Meet Link: https://meet.google.com/wnv-gzos-zga
Abstract: This lecture will introduce some basic concepts of functorial
algebraic geometry following Grothendieck, including flat descent and
representability of functors. A quick introduction will be given to the
moduli problem for vector bundles on curves, and their moduli spaces.
Preparatory reading: An account of the basics of holomorphic vector
bundles on compact Riemann surfaces is available in the 2007 ATM School
lecture notes of Nitsure, which can be downloaded from
https://www.ncmath.org/lecture-notes
- Time:
- 7:00pm
- Description:
- Speaker: Nathan Ilten (https://www.sfu.ca/~nilten/).
Time: 7pm IST, Thursday, 9 July, 2020 (opening time 6:45 pm IST).
Google Meet Link: meet.google.com/qyw-drtx-uvb
Phone: (US) +1 402-744-0304 PIN: 737 931 641#
Title: Fano schemes for complete intersections in toric varieties.
Abstract: The study of the set of lines contained in a fixed hypersurface
is classical: Cayley and Salmon showed in 1849 that a smooth cubic surface
contains 27 lines, and Schubert showed in 1879 that a generic quintic
threefold contains 2875 lines. More generally, the set of k-dimensional
linear spaces contained in a fixed projective variety X itself is called
the k-th Fano scheme of X. These Fano schemes have been studied
extensively when X is a general hypersurface or complete intersection in
projective space.
In this talk, I will report on work with Tyler Kelly in which we study
Fano schemes for hypersurfaces and complete intersections in projective
toric varieties. In particular, I'll give criteria for the Fano schemes of
generic complete intersections in a projective toric variety to be
non-empty and of "expected dimension". Combined with some intersection
theory, this can be used for enumerative problems, for example, to show
that a general degree (3,3)-hypersurface in the Segre embedding of
P2×P2P2×P2 contains exactly 378 lines.
- Time:
- 4:00pm - 5:30pm
- Description:
- Lecture 2: Monday July 13th, 2020, from 4 pm to 5:30 pm
Title: The Narasimhan-Seshadri Theorem and Moduli spaces.
Google Meet Link: https://meet.google.com/xjq-pigk-bwo
Abstract: This will be an introduction to the Narasimhan-Seshadri Theorem
(1965) and its proof. I will make use of some later developments in
algebraic geometry (involving deformation theory, GIT-based moduli, and
algebraic spaces) not available in 1965, to simplify the exposition.
- Time:
- 6:30pm
- Description:
- Speaker: Dr. Tushar Bag, IIT Patna.
Time: 6:30 pm (gate opens 6:15 pm), 15 July, Wednesday.
Google Meet link: meet.google.com/aoz-gdsp-szk.
Phone: (US) +1 678-667-1924 PIN: 997 745 256#.
Title: Quantum Codes via Skew Cyclic Codes.
Abstract: In this presentation, we show a construction of quantum codes
from skew cyclic codes. Also, we discuss the advantages in the
construction of quantum codes from skew cyclic codes than from cyclic
codes over. Skew cyclic codes are equivalent to cyclic codes in the
commutative set up of linear codes. We derive a necessary and sufficient
condition for a skew cyclic code to contain its dual. Using this
dual-containing property, we show a construction of quantum codes from
them. We also present the usefulness of considering skew polynomial rings,
where we illustrate that, being non-unique factorization domains, skew
polynomial rings have numerous options for factors. As a result, we get
more options to construct codes with better parameters.
- Time:
- 5:30pm - 6:30pm
- Description:
- Date and Time: 17 July 2020, 5:30 pm IST - 6:30 pm IST (joining time :
5:15 pm IST - 5:30 pm IST)
Google Meet Link: https://meet.google.com/gkc-hydx-fkn
Speaker: Dale Cutkosky, University of Missouri
Title: Mixed multiplicities and the Minkowski inequality for filtrations -
Part 2.
Abstract: We discuss the theory of multiplicities and mixed multiplicities
of filtrations of m-primary ideals. We show that many classical formulas
are true in this setting. We also consider the case of equality in
Minkowski's inequality. We give some general theorems characterizing when
this condition hold, giving generalizations of classical theorems of Rees,
Sharp, Teissier, Katz and others.
- Time:
- 4:00pm - 5:00pm
- Description:
- Speaker: Sujoy Chakraborty
Time: Monday 20th July 4 to 5pm (joining time 3.50pm)
Google Meet Link: https://meet.google.com/ubd-taca-nio
Title: Moduli of Parabolic bundles over a curve and its first rational
Chow group
Abstract: In this talk, we will aim to study the rational Chow group of
1-cycles for the moduli of semistable Parabolic bundles of fixed rank,
determinant and weight over a curve. Chow groups are interesting and
important objects to study for various reasons. Unfortunately, not much is
known about the Chow groups for various moduli space of semistable vector
bundles of a fixed rank and determinant over a curve. We will first
discuss the notions related to Parabolic bundles and their moduli, and
study the effect on Chow group of 1-cycles as we vary the generic weight.
As a consequence, we can get an explicit description of the Chow group of
1-cycles in a particular case of rank 2 Parabolic bundles, extending an
earlier result of I. Choe and J. Hwang.
- Time:
- 6:30pm - 7:30pm
- Description:
- Date and Time: Tuesday 21 July 6.30 pm-7.30 pm
Google Meet Link: https://meet.google.com/gkc-hydx-fkn
Speaker: Melvin Hochster, University of Michigan.
Title: Tight Closure, lim Cohen-Macaulay sequences, the content of local
cohomology, and related open questions - Part 1.
Abstract: The talks will give multiple characterizations of tight closure,
discuss some of its applications, indicate connections with the existence
of big and small Cohen-Macaulay algebras and modules, as well as variant
notions, and also explain connections with the theory of content. There
will be some discussion of the many open questions in the area, including
the very long-standing problem of proving that Serre intersection
multiplicities have the behavior one expects.
- Time:
- 11:00am - 12:00pm
- Description:
- Title of the Thesis: Operator theory on two domains related to
$\mu$-synthesis
Abstract: We shall discuss Nagy-Foias type operator theory for operators
associated with two domains
related to $\mu$-synthesis, namely the tetrablock and the symmetrized
polydisc.
Date and Time: July 24, 2020, Friday from 11 am - 12 noon
WebEx Link Details:
Meeting link:
https://iitbombay.webex.com/iitbombay/j.php?MTID=m99f0f4c8b20a85a7ed79974d06f07269
Meeting number:
166 634 3026
Password:
CWvrJia7E24
All are cordially invited to participate.
- Time:
- 6:30pm - 7:30pm
- Description:
- Date and Time: Friday 24 July 6.30pm - 7.30pm.
Google meet link: https://meet.google.com/epm-ddze-asm
Speaker: Hai Long Dao, The University of Kansas.
Title: Reflexive modules over curve singularities
Abstract: A finitely generated module $M$ over a commutative ring $R$ is
called reflexive if the natural map from $M$ to $M^{**} = Hom(Hom(M,R),
R)$ is an isomorphism. In understanding reflexive modules, the case of
dimension one is crucial. If $R$ is Gorenstein, then any maximal
Cohen-Macaulay module is reflexive, but in general, it is quite hard to
understand reflexive modules even over well-studied one-dimensional
singularities. In this work, joint with Sarasij Maitra and Prashanth
Sridhar, we will address this problem and give some partial answers.
- Time:
- 6:30pm - 7:30pm
- Description:
- Date and Time: Tuesday 28 July, 6:30 pm IST - 7:30 pm IST (joining time :
6:15 pm IST - 6:30 pm IST)
Google Meet Link: https://meet.google.com/oku-xudb-imy
Speaker: Melvin Hochster, University of Michigan.
Title: Tight Closure, lim Cohen-Maculay sequences, content of local
cohomology, and related open questions - Part 2.
Abstract: The talks will give multiple characterizations of tight closure,
discuss some of its applications, indicate connections with the existence
of big and small Cohen-Macaulay algebras and modules, as well as variant
notions, and also explain connections with the theory of content. There
will be some discussion of the many open questions in the area, including
the very long standing problem of proving that Serre intersection
multiplicities have the behavior one expects.
- Time:
- 5:30pm
- Description:
- Date and Time: Friday 31 July 2020, 5:30 pm IST / 12:00 GMT / 08:00am EDT
(joining time : 5:15 pm IST - 5:30 pm IST)
Google Meet link: https://meet.google.com/xxm-cidr-yqa
Speaker: Neena Gupta, ISI Kolkata.
Title: On the triviality of the affine threefold $x^my = F(x, z, t)$ -
Part 2.
Abstract: In this talk we will discuss a theory for affine threefolds of
the form $x^my = F(x, z, t)$ which will yield several necessary and
sufficient conditions for the coordinate ring of such a threefold to be a
polynomial ring. For instance, we will see that this problem of four
variables reduces to the equivalent but simpler two-variable question as
to whether F(0, z, t) defines an embedded line in the affine plane. As one
immediate consequence, one readily sees the non-triviality of the famous
Russell-Koras threefold x^2y+x+z^2+t^3=0 (which was an exciting open
problem till the mid 1990s) from the obvious fact that z^2+t^3 is not a
coordinate. The theory on the above threefolds connects several central
problems on Affine Algebraic Geometry. It links the study of these
threefolds with the famous Abhyankar-Moh “Epimorphism Theorem” in
characteristic zero and the Segre-Nagata lines in positive characteristic.
We will also see a simplified proof of the triviality of most of the
Asanuma threefolds (to be defined in the talk) and an affirmative solution
to a special case of the Abhyankar-Sathaye Conjecture. Using the theory,
we will also give a recipe for constructing infinitely many counterexample
to the Zariski Cancellation Problem (ZCP) in positive characteristic. This
will give a simplified proof of the speaker's earlier result on the
negative solution for the ZCP.