- 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.