5:30pm» Matteo Varbaro, University of Genoa- Time:
- 5:30pm - 6:30pm
- Description:
- Date and Time: Tuesday 1st September 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/yqu-mvvy-jrs
Speaker: Matteo Varbaro, University of Genoa
Title: F-splittings of the polynomial ring and compatibly split
homogeneous ideals
Abstract: A polynomial ring R in n variables over a field K of positive
characteristic is F-split. It has many F-splittings. When K is a perfect
field every F-splitting is given by a polynomial g in R with the monomial
u^{p-1} in its support (where u is the product of all the variables)
occurring with coefficient 1, plus a further condition, which is not
needed if g is homogeneous (w.r.t. any positive grading). Fixed an
F-splitting s : R -> R, an ideal I of R such that s(I) is contained in I
is said compatibly split (w.r.t. the F-splittings). In this case R/I is
F-split. Furthermore, by Fedder’s criterion when I is a homogeneous ideal
of R, R/I is F-split if and only if I is compatibly split for some
F-splitting s : R -> R. If, moreover, u^{p-1} is the initial monomial of
the associated polynomial g of s w.r.t. some monomial order, then in(I) is
a square-free monomial ideal… In this talk I will survey these facts (some
of them classical, some not so classical), and make some examples,
focusing especially on determinantal ideals.
- Time:
- 7:00pm
- Description:
- The speaker is
Prof. Amritanshu Prasad from IMSc, Chennai. The following are the
details.
Title: Polynomials as Characters of Symmetric Groups.
Time: 7pm, Tuesday, September 1, 2020 (gate opens at 6:45pm).
Google meet link: meet.google.com/prm-feow-zwm.
Phone: (US) +1 740-239-3129 PIN: 706 683 026#
Abstract: Treating the variable $X_i$ as the number of $i$-cycles in a
permutation allows a polynomial in $X_1, X_2,\dotsc$ to be regarded as a
class function of the symmetric group $S_n$ for any positive integer $n$.
We present a simple formula for computing the average and signed average
of such a class function over the symmetric group. We use this formula to
investigate the dimension of $S_n$-invariant and $S_n$-sign-equivariant
vectors in polynomial representations of general linear groups.
This talk is based on joint work with Sridhar P Narayanan, Digjoy Paul,
and Shraddha Srivastava. Some of these results are available in the
preprint available at: http://arxiv.org/abs/2001.04112.
- Time:
- 5:30pm - 6:30pm
- Description:
- Date and Time: Friday 4th September 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/yqu-mvvy-jrs
Speaker: Mandira Mondal, Chennai Mathematical Institute.
Title: Density functions for the coefficients of the Hilbert-Kunz function
of polytopal monoid algebra
Abstract: We shall discuss Hilbert-Kunz density function of a Noetherian
standard graded ring over a perfect field of characteristic $p \geq 0$. We
will also talk about the second coefficient of the Hilbert-Kunz function
and the possibility of existence of a $\beta$-density function for this
coefficient.
Watanabe and Eto have shown that Hilbert-Kunz multiplicity of affine
monoid rings with respect to a monomial ideal of finite colength can be
expressed as relative volume of certain nice set arising from the convex
geometry associated to the ring. In this talk, we shall discuss similar
expression for the density functions of polytopal monoid algebra with
respect to the homogeneous maximal ideal in terms of the associated convex
geometric structure. This is a joint work with Prof. V. Trivedi. We shall
also discuss the existence of $\beta$-density function for monomial prime
ideals of height one of these rings in this context.
- Time:
- 3:00pm - 4:00pm
- Description:
Date and Time: Monday 07 September, 03.00pm - 04.00pm (joining time:
2.45pm - 3.00pm)
Meeting link: http://meet.google.com/aqe-hgbk-jpf
Speaker: Tamalika Koley, Indian Statistical Institute
Title: Current Status data with competing risks and missing failure types
Abstract:
Various studies on current status data with or without competing risks are present in the literature. In competing risks set up missing or uncertainty in failure types is a very common phenomenon. When observation on true failure type is not available, one observes a set of possible types containing the true cause. This gives rise to uncertainty in the true cause for the occurrence of the event of interest. As per our knowledge, the missing failure types in current status data with competing risks has not received much attention. The main purpose of this work is to focus and highlight this less-explored area of research. Throughout, for simplicity it is worked with two competing risks and both parametric and non-parametric analyses are carried out.
- Time:
- 4:00pm - 5:00pm
- Description:
Speaker: Sarbeswar Pal, IISER Trivandrum
Time: Monday 7th September 4 to 5pm (joining time 3.50pm)
Google Meet Link: https://meet.google.com/wnf-ywcy-ozi
Title: Irreducibility of the moduli space of vector bundles over a very
general sextic surface.
Abstract: In this talk we will discuss the irreducibility question of the
moduli space of vector bundles over surfaces. More precisely we will
study the irreducibility question over a very general sextic surface. Our
technique is to use O’Grady’s method of deformation to the boundary as it
was exploited by Nijsse in the case of a very general quintic
hypersurface. We will discuss the main difficulties of using Nijsse's
method and how to overcome that.
- Time:
- 6:30pm
- Description:
- Date and Time: 8 September 2020, 6:30 pm IST/ 1:00 pm GMT/ 09:00 am EDT
(joining time : 6:15 pm IST - 6:30 pm IST)
Google Meet link: https://meet.google.com/tmh-yngo-ksk
Speaker: Irena Swanson, Purdue University
Title: Primary decomposition and powers of ideals
Abstract: This talk is about associated primes of powers of an ideal in
Noetherian commutative rings. Brodmann proved that the set of associated
primes stabilizes for large powers. In general, the number of associated
primes can go up or down as the exponent increases. This talk is about
sequences $\{ a_n \}$ for which there exists an ideal $I$ in a Noetherian
commutative ring $R$ such that the number of associated primes of $R/I^n$
is $a_n.$ This is a report on my work with Sarah Weinstein, with Jesse Kim
and ongoing work with Roswitha Rissner.
- Time:
- 3:00pm - 4:00pm
- Description:
- Speaker: Sarbeswar Pal, IISER Trivandrum
Time: Friday 11th September 3 to 4 pm (joining time 2.50pm)
Google Meet Link: https://meet.google.com/qvo-kduy-yco
Title: Irreducibility of the moduli space of vector bundles over a very
general sextic surface.
Abstract: In this talk we will discuss the irreducibility question of the
moduli space of vector bundles over surfaces. More precisely we will
study the irreducibility question over a very general sextic surface. Our
technique is to use O’Grady’s method of deformation to the boundary as it
was exploited by Nijsse in the case of a very general quintic
hypersurface. We will discuss the main difficulties of using Nijsse's
method and how to overcome that.
- Time:
- 3:00pm - 4:00pm
- Description:
- 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.
- Time:
- 4:00pm - 5:00pm
- Description:
- Speaker: Ramesh Sreekantan, ISI Bangalore
Time: Monday 21st September 4 to 5pm (joining time 3.50pm)
Google Meet Link: https://meet.google.com/qvo-kduy-yco
Title: Algebraic Cycles and Modular Forms
Abstract: There are many instances when special sub-varieties of Shimura
varieties give rise to modular forms. One such is the theorem of Gross and
Zagier linking Heegner divisors with coefficients of modular forms. We
discuss a generalisation of this theorem to higher codimensional cycles
which implies the existence of certain motivic cycles in the universal
families over these Shimura varieties. In special cases we construct some
examples which have applications to another conjecture of Gross and Zagier
on algebraicity of values of Greens functions.
- Time:
- 5:30pm
- Description:
- Date and Time: 22 September 2020, 5:30 pm IST (joining time : 5:15 pm IST
- 5:30 pm IST)
Google meet link: meet.google.com/sdz-bspz-uhu
Speaker: Shunsuke Takagi, University of Tokyo
Title: F-singularities and singularities in birational geometry - Part 1
Abstract: F-singularities are singularities in positive characteristic
defined using the Frobenius map and there are four basic classes of
F-singularities: F-regular, F-pure, F-rational and F-injective
singularities. They conjecturally correspond via reduction modulo $p$ to
singularities appearing in complex birational geometry. In the first talk,
I will survey basic properties of F-singularities. In the second talk, I
will explain what is known and what is not known about the correspondence
of F-singularities and singularities in birational geometry. If the time
permits, I will also discuss its geometric applications.
- Time:
- 10:30am
- Description:
- Speaker: Dr. Brett Parker, Monash University.
Time: 10:30 AM, IST, 24 September 2020 (gate open: 10:20 AM).
Google meet link: meet.google.com/qeh-paqu-yaq.
Phone: (US) +1 276-796-8106 PIN: 654 044 511#.
Title: Tropical counts of Gromov-Witten invariants in dimension 3.
Abstract: Tropical curves appear when we study holomorphic curves under
certain degenerations, or relative to normal-crossing divisors. In many
cases, there is a correspondence between counting tropical curves and
Gromov邑itten invariants. In complex dimension 3, this correspondence has
the wonderful feature that each tropical curve corresponds to
Gromov邑itten invariants counting curves in all genus. I will illustrate
some examples of this correspondence, including some interesting examples
counting Gromov-Witten invariants in log Calabi-Yau manifolds, where our
tropical curves live in a 3 dimensional integral affine space with
singularities along a 1-dimensional locus.
- Time:
- 4:00pm - 5:00pm
- Description:
- Date and Time: Friday 25 September, 04.00pm - 05.00pm
Google Meet link: https://meet.google.com/mvd-txng-kgf
Speaker: Sarjick Bakshi, CMI
Title: GIT quotients of Grassmannian and smooth quotients of Schubert
varieties
Abstract: The Geometric invariant theory (GIT) quotients of the
Grassmannian variety and its subvarieties lead to many interesting
geometric problems. Gelfand and Macpherson showed that the GIT quotient of
n-points in {\mathbb P}^{r-1} by the diagonal action of PGL(r,\mathbb{C})
is isomorphic to the GIT quotient of Gr_{r,n} with respect to the
T-linearized line bundle {\cal L}(n \omega_r). Howard, Milson, Snowden and
Vakil gave an explicit description of the generators of the ring of
invariants for n even and r=2 using graph theoretic methods. We give an
alternative approach where we study the generators using Standard monomial
theory and we will establish the projective normality of the quotient
variety for odd n and r=2.
Let r < n be positive integers and further suppose r and n are coprime. We
study the GIT quotient of Schubert varieties X(w) in the Gr_{r,n}
admitting semistable points for the action of T with respect to the
T-linearized line bundle {\cal L}(n \omega_r). We give necessary and
sufficient combinatorial conditions for w for which the GIT quotient of
the Schubert variety is smooth.
- Time:
- 5:30pm
- Description:
- Date and Time: 25 September 2020, 5:30 pm IST (joining time : 5:15 pm IST
- 5:30 pm IST)
Google meet link: meet.google.com/sdz-bspz-uhu
Speaker: Shunsuke Takagi, University of Tokyo
Title: F-singularities and singularities in birational geometry - Part 2
Abstract: F-singularities are singularities in positive characteristic
defined using the Frobenius map and there are four basic classes of
F-singularities: F-regular, F-pure, F-rational and F-injective
singularities. They conjecturally correspond via reduction modulo $p$ to
singularities appearing in complex birational geometry. In the first talk,
I will survey basic properties of F-singularities. In the second talk, I
will explain what is known and what is not known about the correspondence
of F-singularities and singularities in birational geometry. If the time
permits, I will also discuss its geometric applications.
- Time:
- 11:30am
- Description:
- Date and Time: Monday 28 September, 11.30 am - 12.30 pm
Google Meet link: meet.google.com/ahu-peka-sto
Speaker: Oorna Mitra, IMSc
Title: Twisted Conjugacy in Linear Groups over Polynomial and Laurent
Polynomial Algebras over Finite Fields
Abstract: Given an automorphism \phi : G \to G, one has the \phi-twisted
conjugacy action of G on itself, given by g.x = g x\phi(g^{-1}). The
orbits of this action are called the \phi-twisted conjugacy classes. In
this talk, we will talk about twisted conjugacy in general and special
linear groups over F[t] and F[t, t^{-1}] where F is any subfield of the
algebraic closure of \mathbb{F}_p. This is joint work with P. Sankaran.
Some new results by Shripad Garge and myself regarding twisted conjugacy
in other classical groups over \mathbb{F}_q[t] and \mathbb{F}_q[t,t^{-1}]
will also be mentioned.
- Time:
- 4:00pm - 5:00pm
- Description:
- Speaker: Frank Gounelas, TU Munich
Time: Monday 28th September 4 to 5pm (joining time 3.50pm IST)
Google Meet Link: https://meet.google.com/qvo-kduy-yco
Title: Curves on K3 surfaces
Abstract: I will survey the recent completion (joint with Chen-Liedtke) of
the remaining cases of the conjecture that a projective K3 surface
contains infinitely many rational curves. As a consequence of this along
with the Bogomolov-Miyaoka-Yau inequality and the deformation theory of
stable maps, I will explain (joint with Chen) how in characteristic zero
one can deduce the existence of infinitely many curves of any geometric
genus moving in maximal moduli on a K3 surface. In particular this leads
to an algebraic proof of a theorem of Kobayashi on vanishing of global
symmetric differentials and applications to 0-cycles.
- Time:
- 5:30pm
- Description:
- Date and Time: 29 September 2020, 5:30pm IST/ 12:00GMT / 08:00am EDT
(joining time: 5:15 pm IST - 5:30 pm IST)
Google meet link: meet.google.com/vog-pdxx-fdt
Speaker: K.N. Raghavan, The Institute of Mathematical Sciences
Title: Multiplicities of points on Schubert varieties in the Grassmannian
- Part 1
Abstract: Given an arbitrary point on a Schubert (sub)variety in a
Grassmannian, how to compute the Hilbert function (and, in particular, the
multiplicity) of the local ring at that point? A solution to this problem
based on "standard monomial theory" was conjectured by Kreiman-Lakshmibai
circa 2000 and the conjecture was proved about a year or two later by them
and independently also by Kodiyalam and the speaker. The two talks will
be an exposition of this material aimed at non-experts in the sense that
we will not presume familiarity with Grassmannians (let alone flag
varieties) or Schubert varieties.
There are two steps to the solution. The first translates the problem from
geometry to algebra and in turn to combinatorics. The second is a solution
of the resulting combinatorial problem, which involves establishing a
bijection between two combinatorially defined sets. The two talks will
roughly deal with these two steps respectively.
Three aspects of the combinatorial formulation of the problem (and its
solution) are noteworthy: (A) it shows that the natural determinantal
generators of the tangent cone (at the given point) form a Groebner basis
(in any "anti-diagonal" term order); (B) it leads to an interpretation of
the multiplicity as counting certain non-intersecting lattice paths; and
(C) as was observed by Kreiman some years later, the combinatorial
bijection is a kind of Robinson-Schensted-Knuth correspondence, which he
calls the "bounded RSK".
Determinantal varieties arise as tangent cones of Schubert varieties (in
the Grassmannian), and thus one recovers multiplicity formulas for these
obtained earlier by Abhyankar and Herzog-Trung. (The multiplicity part of
the Kreiman-Lakshmibai conjecture was also proved by Krattenthaler, but
by very different methods.)
What about Schubert varieties in other (full or partial) flag varieties
(G/Q with Q being a parabolic subgroup of a reductive algebraic group G)?
The problem remains open in general, even for the case of the full flag
variety GL(n)/B, although there are several papers over the last two
decades by various authors using various methods that solve the problem in
various special cases. Time permitting, we will give some indication of
these results, without however any attempt at comprehensiveness.