- Time:
- 11:30am
- Description:
- Date and Time: Monday, 02 November, 11.30am

Speaker: Ketan Sutar

Title: Borsuk-Ulam Theorem

Abstract: The Borsuk-Ulam theorem is a nice result in Algebraic Topology. It states that every continuous function from a sphere to a plane acts as antipode preserving for a point on sphere. In this talk we will discuss some basic definitions and concepts in Algebraic Topology. Then we will prove some theorems which will be helpful for the proof the Borsuk-Ulam theorem. We will also discuss its generalisation to plane known as The Bisection theorem.

Google Meet link: https://meet.google.com/afe-nzqz-sgt

- Time:
- 4:00pm - 5:00pm
- Description:
- Speaker: V Balaji, CMI, Chennai

Time: Monday 2nd November 4 to 5pm (joining time 3.45 pm IST)

Google Meet Link: https://meet.google.com/qvo-kduy-yco

Title: Torsors on semistable curves and the problem of degenerations.

Abstract: This is part 2 of the previous talk. Let G be an almost simple,

simply connected algebraic group over the field of complex numbers. In

this talk I will discuss a basic question in the classification of

G-torsors on curves, which is to construct a flat degeneration of the

moduli stack G-torsors on a smooth projective curve when the curve

degenerates to an irreducible nodal curve. In this second part I will

discuss the problem addressed earlier in the setting of principal G

bundles for an almost simple algebraic group. We will recall the earlier

picture for the sake of continuity.

- Time:
- 6:30pm
- Description:
- Date and Time: 3 November 2020, 6:30pm IST/ 1:00pm GMT/ 09:00am EDT

(joining time: 6:15 pm IST - 6:30 pm IST)

Speaker: Claudia Polini, University of Notre Dame, IN, USA

Google meet link: meet.google.com/urk-vxwh-nri

Title: Core of ideals - Part 1

Abstract:

Let I be an ideal in a Noetherian commutative ring. Among all the closures

of I, the integral closure plays a central role. A reduction of I is a

subideal with the same integral closure. We can think of reductions as

simplifications of the given ideal, which carry most of the information

about I itself but, in general, with fewer generators. Minimal reductions,

reductions minimal with respect to inclusion, are loosely speaking the

counterpart of the integral closure. However, unlike the integral closure,

minimal reductions are not unique. For this reason we consider their

intersection, called the core of I. The core is related to adjoint and

multiplier ideals. A motivation for studying this object comes from the

Briancon-Skoda theorem. Furthermore a better understanding of the core

could lead to solving Kawamata's conjecture on the non-vanishing of

sections of certain line bundle. In this talk I will discuss the

importance of the core, its ubiquity in algebra and geometry, and some

effective formulas for its computation.

- Time:
- 7:00pm
- Description:
- Daet and Time: Wednesday, 4th Nov 2020 at 7 pm

Speaker:Parvez Rasul

Title: Bezout’s theorem for algebraic curves in plane

Abstract: Algebraic geometry is concerned with the study of the properties of certain geometric objects (which are mainly solution sets of systems of polynomial equations) using abstract algebra. One of the earliest results to this end is Bézout’s theorem, which relates the number of points at which two polynomial curves intersect to the degrees of the generating polynomials. Here we reproduce an elementary proof of Bézout’s theorem for algebraic curves in plane. It states that if we have two algebraic plane curves, defined over an algebraically closed field and given by zero sets of polynomials of degrees n and m, then the number of points where these curves intersect is exactly nm if we count ”multiple intersections” and ”intersections at infinity”. To formulate and prove the theorem rigorously we go through some concepts which lie at the heart of algebraic geometry like projective space and intersection multiplicities at a common point of two curves.

Google Meet Link: https://meet.google.com/hhk-ijhb-ivr

- Time:
- 4:00pm
- Description:
- Date and Time: Thursday, 05 November at 04.00pm

Speaker: Subhajit Ghosh (IISc)

Title: Total variation cutoff for random walks on some finite groups

Talk link: https://meet.google.com/jmz-wnfu-mwh

Abstract: see attached document

- Time:
- 6:30pm
- Description:
- Date and Time: 6 November 2020, 6:30pm IST/ 1:00pm GMT/ 08:00am EDT

(joining time: 6:15 pm IST - 6:30 pm IST)

Speaker: Claudia Polini, University of Notre Dame, IN, USA

Google meet link: meet.google.com/urk-vxwh-nri

Title: The core of monomial ideals

Abstract: Let $I$ be a monomial ideal. Even though there may not exist any

proper reduction of $I$ which is monomial (or even homogeneous), the

intersection of all reductions, the core, is again a monomial ideal. The

integral closure and the adjoint of a monomial ideal are again monomial

ideals and can be described in terms of the Newton polyhedron of $I$. Such

a description cannot exist for the core, since the Newton polyhedron only

recovers the integral closure of the ideal, whereas the core may change

when passing from $I$ to its integral closure. When attempting to derive

any kind of combinatorial description for the core of a monomial ideal

from the known colon formulas, one faces the problem that the colon

formula involves non-monomial ideals, unless $I$ has a reduction $J$

generated by a monomial regular sequence. Instead, in joint work with

Ulrich and Vitulli, we exploit the existence of such non-monomial

reductions to devise an interpretation of the core in terms of monomial

operations. This algorithm provides a new interpretation of the core as

the largest monomial ideal contained in a general locally minimal

reduction of $I$. In recent joint work with Fouli, Montano, and Ulrich, we

extend this formula to a large class of monomial ideals and we study the

core of lex-segment monomial ideals generated in one-degree.

- Time:
- 11:30am
- Description:
- Date and Time: Saturday, 07 November, 11.30am

Speaker: Utsav Dewan

Title: Discrete Hilbert transform and 𝐿% convergence of Fourier

series

Abstract: We investigate the convergence of Fourier series of various L^p (T) functions via the discrete Hilbert transform.

Google Meet link: https://meet.google.com/afe-nzqz-sgt

- Time:
- 4:00pm
- Description:
- Date and Time: Saturday, 07 November, 04.00pm

Speaker: Shubham Niphadkar

Title: Model selection consistency in linear models

Abstract: This short note studies the effects of model selection in a linear regression model with two covariates. We study the effect of including redundant covariate and effect of excluding useful covariate. We explore consistency and uniform consistency of model selection.

Google Meet link: https://meet.google.com/tbg-fghh-nmg

- Time:
- 11:30am
- Description:
- Date and Time: Monday, 09 November, 11.30am

Speaker: Lovy Jain

Title: Lax-Milgram lemma and its applications

Abstract: Lax-Milgram lemma is an effective tool in checking the well-posedness of a weak formulation.

Derived from basic theorems of functional analysis, it saves hectic calculations that serves the purpose

otherwise in differential equations' analysis.

Google Meet Link: https://meet.google.com/afe-nzqz-sgt

- Time:
- 4:00pm - 5:00pm
- Description:
- Speaker: Nihar Gargava, EPFL

Time: Monday 9th November 4 to 5pm (joining time 3.45 pm IST)

Google Meet Link: https://meet.google.com/qvo-kduy-yco

Title: Asymptotic Lower Bounds on Sphere Packing Efficiency of Lattices

Abstract: In 1945, Siegel showed that the expected value of the

lattice-sums of a function over all the lattices of unit covolume in an

n-dimensional real vector space is equal to the integral of the function.

In 2012, Venkatesh restricted the lattice-sum function to a collection of

lattices that had a cyclic group of symmetries and proved a similar mean

value theorem. Using this approach, new lower bounds on the most optimal

sphere packing density in n dimensions were established for infinitely

many n. We will discuss this result, and some surrounding literature.

The talk will only assume the knowledge of Haar measure.

- Time:
- 11:00am
- Description:
- Pre-synopsis seminar

Student: Hiranya Kishore Dey

Date and Time: Tuesday, 10 November 2020 at 11.00am

Title: Descents, Excedances and Alternating-runs in Positive elements of

Coxeter Groups

Google Meet Link: https://meet.google.com/cpj-fbho-apd

All interested are cordially invited.

- Time:
- 5:30pm
- Description:
- Date and Time: 10 November 2020, 5:30pm IST/ 12:00GMT/07:00am EDT (joining

time 5:15pm - 5:30pm IST)

Speaker: Amartya Kumar Datta, ISI Kolkata

Google meet link: https://meet.google.com/jom-etrz-bdd

Title: G_a-actions on Affine Varieties: Some Applications - Part 1

Abstract: One of the hardest problems that come up in affine algebraic geometry is to decide whether a certain d-dimensional factorial affine domain is ``trivial'', i.e., isomorphic to the polynomial ring in d variables. There are instances when the ring of invariants of a suitably chosen G_a-action has been able to distinguish between two rings (i.e., to prove they are non-isomorphic), when all other

known invariants failed to make the distinction. It was using one such

invariant that Makar-Limanov proved the non-triviality of the

Russell-Koras threefold, leading to the solution of the Linearization

Problem; and again, it was using an invariant of G_a-actions that Neena

Gupta proved the nontriviality of a large class of Asanuma threefolds

leading to her solution of the Zariski Cancellation Problem in positive

characteristic.

G_a actions are also involved in the algebraic characterisation of the

affine plane by M. Miyanishi and the algebraic characterisation of the

affine 3-space.by Nikhilesh Dasgupta and Neena Gupta. Miyanishi's

characterisation had led to the solution of Zariski's Cancellation Problem

for the affine plane. Using G_a-actions, a simple algebraic proof for

this cancellation theorem was obtained three decades later by

Makar-Limanov.

In this talk (in two parts), we will discuss the concept of G_a-actions

along with the above applications, and the closely related theme of

Invariant Theory. The concept of G_a-action can be reformulated in the

convenient ring-theoretic language of ``locally nilpotent derivation'' (in

characteristic zero) and ``exponential map'' (in arbitrary

characteristic). The ring of invariants of a G_a- action corresponds to

the kernel of the corresponding locally nilpotent derivation (in

characteristic zero) and the ring of invariants of an exponential map. We

will recall these concepts. We will also mention a theorem on G_a actions

on affine spaces (or polynomial rings) due to C.S. Seshadri.

We will also discuss the close alignment of the kernel of a locally

nilpotent derivation on a polynomial ring over a field of characteristic

zero with Hilbert's fourteenth problem. While Hilbert Basis Theorem had

its genesis in a problem on Invariant Theory, Hilbert's fourteenth

problem seeks a further generalisation: Zariski generalises it still

further. The connection with locally nilpotent derivations has helped

construct some low-dimensional counterexamples to Hilbert's problem. We

will also mention an open problem about the kernel of a locally nilpotent

derivation on the polynomial ring in four variables; and some partial

results on it due to Daigle-Freudenburg, Bhatwadekar-Daigle,

Bhatwadekar-Gupta-Lokhande and Dasgupta-Gupta. Finally, we will state a

few technical results on the ring of invariants of a G_a action on the

polynomial ring over a Noetherian normal domain, obtained by

Bhatwadekar-Dutta and Chakrabarty-Dasgupta-Dutta-Gupta.

- Time:
- 5:30pm
- Description:
- Date and Time: 13 November 2020, 5:30pm IST/ 12:00GMT/07:00am EDT (joining

time 5:15pm - 5:30pm IST)

Speaker: Amartya Kumar Datta, ISI Kolkata

Google meet link: https://meet.google.com/jom-etrz-bdd

Title: G_a-actions on Affine Varieties: Some Applications - Part 2

--------------------------

Abstract for both the talks: One of the hardest problems that come up in

affine algebraic geometry is to decide whether a certain d-dimensional

factorial affine domain is ``trivial'', i.e., isomorphic to the

polynomial ring in d variables. There are instances when the ring of

invariants of a suitably chosen G_a-action has been able to distinguish

between two rings (i.e., to prove they are non-isomorphic), when all other

known invariants failed to make the distinction. It was using one such

invariant that Makar-Limanov proved the non-triviality of the

Russell-Koras threefold, leading to the solution of the Linearization

Problem; and again, it was using an invariant of G_a-actions that Neena

Gupta proved the nontriviality of a large class of Asanuma threefolds

leading to her solution of the Zariski Cancellation Problem in positive

characteristic.

G_a actions are also involved in the algebraic characterisation of the

affine plane by M. Miyanishi and the algebraic characterisation of the

affine 3-space.by Nikhilesh Dasgupta and Neena Gupta. Miyanishi's

characterisation had led to the solution of Zariski's Cancellation Problem

for the affine plane. Using G_a-actions, a simple algebraic proof for

this cancellation theorem was obtained three decades later by

Makar-Limanov.

In this talk (in two parts), we will discuss the concept of G_a-actions

along with the above applications, and the closely related theme of

Invariant Theory. The concept of G_a-action can be reformulated in the

convenient ring-theoretic language of ``locally nilpotent derivation'' (in

characteristic zero) and ``exponential map'' (in arbitrary

characteristic). The ring of invariants of a G_a- action corresponds to

the kernel of the corresponding locally nilpotent derivation (in

characteristic zero) and the ring of invariants of an exponential map. We

will recall these concepts. We will also mention a theorem on G_a actions

on affine spaces (or polynomial rings) due to C.S. Seshadri.

We will also discuss the close alignment of the kernel of a locally

nilpotent derivation on a polynomial ring over a field of characteristic

zero with Hilbert's fourteenth problem. While Hilbert Basis Theorem had

its genesis in a problem on Invariant Theory, Hilbert's fourteenth

problem seeks a further generalisation: Zariski generalises it still

further. The connection with locally nilpotent derivations has helped

construct some low-dimensional counterexamples to Hilbert's problem. We

will also mention an open problem about the kernel of a locally nilpotent

derivation on the polynomial ring in four variables; and some partial

results on it due to Daigle-Freudenburg, Bhatwadekar-Daigle,

Bhatwadekar-Gupta-Lokhande and Dasgupta-Gupta. Finally, we will state a

few technical results on the ring of invariants of a G_a action on the

polynomial ring over a Noetherian normal domain, obtained by

Bhatwadekar-Dutta and Chakrabarty-Dasgupta-Dutta-Gupta.

- Time:
- 4:00pm - 5:00pm
- Description:
- Speaker: Roy Skjelnes, KTH, Stockholm

Time: Monday 16th November 4 to 5pm (joining time 3.45 pm IST)

Google Meet Link: https://meet.google.com/qvo-kduy-yco

Title: Classification of smooth Hilbert schemes.

Abstract: The Hilbert scheme parametrizing closed subschemes in a fixed

projective n-space having Hilbert polynomial p is a projective scheme.

Each such polynomial p can be described in terms of an integer partition,

and this can then be used to classify which Hilbert schemes are smooth.

The corresponding subschemes parametrized are described by a

generalization of partial flags. I will try to explain the classification

result and the underlying geometry. These new results are based on a joint

work with Greg Smith.

- Time:
- 6:30pm
- Description:
- Date/Time: 17 November 2020, 6:30pm IST/ 1:00pm GMT/ 8:00am EDT (joining

time: 6:15 pm IST - 6:30 pm IST)

Speaker: Giulio Caviglia, Purdue University, USA

Google meet link: meet.google.com/gyc-baih-xas

Title: The Eisenbud-Green-Harris Conjecture

Abstract: The $f$-vector of a simplicial complex is a finite sequence of

integers defined by the number of $i$-dimensional faces of the complex.

All possible such vectors are completely characterized thanks to a

classical theorem by Kruskal and Katona. This result, when rephrased in

terms of Hilbert functions of certain quotients of polynomial rings by

monomial ideals, extends the celebrated theorem of Macaulay on

lexicographic ideals.

The Eisenbud-Green-Harris conjecture is a further generalization of both

the Kruskal-Katona theorem and the well-known Cayley–Bacharach theorem for

plane curves. I will survey the known results on this conjecture including

a recent joint work with Alessandro De Stefani.

- Time:
- 4:00pm
- Description:
- Date and Time: Wednesday, 18 November, 04.00pm

Speaker: Anik Roy

Title: Testing Independence among random vectors based on Univariate Test

Abstract: The problem of testing independence of random vectors has

received increased attention in recent years. There are lots of method for

testing independence among univariate random variables and also random

vectors. In this presentation we carry out a test of Independence for

random vectors

based on univariate test.

Google Meet Link: https://meet.google.com/rxi-ebqz-qhy

- Time:
- 5:30pm
- Description:
- Date/Time: 20 November 2020, 5:30pm IST/ 12:00 GMT/ 7:00am EDT (joining

time: 5:15 pm IST - 5:30 pm IST)

Speaker: Parangama Sarkar, IIT Palakkad, India

Google meet link: meet.google.com/gyc-baih-xas

Title: Frobenius Betti numbers of finite length modules

Abstract: Let $(R, m)$ be a Noetherian local ring of dimension $d > 0$ and

$M$ be a finitely generated $R$-module of finite length. Suppose char R =

$p > 0$ and $d = 1.$ De Stefani, Huneke and Núńez-Betancourt explored the

question: what vanishing conditions on the Frobenius Betti numbers force

projective dimension of $M$ to be finite. In this talk we will discuss the

question for $d ≥ 1.$ This is joint work with Ian Aberbach.

- Time:
- 2:30pm
- Description:
- Date and Time: Saturday, 21 November, 02.30pm

Speaker: Sakshi

Title: Lambert's proof of irrationality of pi

Abstract: In the 1760s, Johann Heinrich Lambert proved that the number (pi) is irrational. It was the

very first proof of irrationality of pi and was proved using the continued fraction expression of tan X. This

talk includes a brief introduction to continued fractions and focuses on a simplified version of Lambert's proof given by M. Laczkovich in 1997.

Google Meet Link: https://meet.google.com/hsf-yhwu-eaf

- Time:
- 4:00pm
- Description:
- Date and Time: Saturday, 21 November, 04.00pm

Speaker: Anik Roy

Title: Testing Independence among random vectors based on Univariate Test

Abstract: The problem of testing independence of random vectors has received increased attention in recent years. There are lots of method for testing independence among univariate random variables and also random vectors. In this presentation we carry out a test of Independence for random vectors based on univariate test.

Google Meet Link:

https://meet.google.com/cyq-wbid-zcx

- Time:
- 4:00pm - 5:00pm
- Description:
- Speaker: Gianfranco Casnati, Politecnico di Torino

Time: Monday 23rd November 4 to 5pm (joining time 3.45 pm IST)

Google Meet Link: https://meet.google.com/qvo-kduy-yco

Title: Ulrich bundles on surfaces

Abstract: An Ulrich bundle on a variety embedded in the projective space

is a vector bundle that admits a linear resolution as a sheaf on the

projective space.

Ulrich bundles have many interesting properties. E.g., the existence of

Ulrich bundles of low rank on a hypersurface $X$ is related to the problem

of finding a linear determinantal or a linear pfaffian description of the

equation of $X$.

Ulrich bundles on curves can be easily described. This is no longer true

for Ulrich bundles on a surface. In the talk we focus our attention on

this latter case. In particular we deal with surfaces $S$ such that

$q(S)=0$ and the hyperplane linear system is non-special. In this case, we

discuss some recent existence results, discussing also the case of

surfaces of degree up to $8$.

- Time:
- 6:30pm
- Description:
- Date/Time: 24 November 2020, 6:30pm IST/ 1:00pm GMT/ 8:00am EDT (joining

time: 6:15 pm IST - 6:30 pm IST)

Google meet link: meet.google.com/dhe-jsbw-jem

Speaker: Tai Huy Ha, University of Tulane

Title: The ideal containment problem and vanishing loci of homogeneous

polynomials

Abstract: We shall discuss Chudnovsky’s and Demailly’s conjectures which

provide lower bounds for the answer to the following fundamental question:

given a set of points in a projective space and a positive integer m, what

is the least degree of a homogeneous polynomial vanishing at these points

of order at least m? Particularly, we shall present main ideas of the

proofs of these conjectures for sufficiently many general points.

- Time:
- 7:30pm
- Description:
- Speaker: Alexey Garber.

Time: 7:30 pm (IST) gate opens 7:15 pm IST, 26 November, 2020

Google meet link: meet.google.com/zis-ovwc-tjy.

Title: Voronoi conjecture for five-dimensional parallelohedra.

Abstract: In this talk I am going to discuss a well-known connection

between lattices in $\mathbb{R}^d$ and convex polytopes that tile

$\mathbdd{R}^d$ with translations only.

My main topic will be the Voronoi conjecture, a century old conjecture

which is, while stated in very simple terms, is still open in general.

The conjecture states that every convex polytope that tiles

$\mathbb{R}^d$ with translations can be obtained as an affine image of

the Voronoi domain for some lattice.

I plan to survey several known results on the Voronoi conjecture and give

an insight on a recent proof of the Voronoi conjecture in the

five-dimensional case. The talk is based on a joint work with Alexander

Magazinov.

- Time:
- 5:30pm
- Description:
- Date/Time: 27 November 2020, 5:30pm IST/ 12:00 GMT/ 7:00am EDT (joining

time: 5:15 pm IST - 5:30 pm IST)

Google meet link: meet.google.com/gfz-iazq-xcc

Speaker: Ryo Takahashi, Nagoya University

Title: Getting a module from another and classifying resolving subcategories

Abstract: Let $R$ be a commutative noetherian ring. Let $M$ and $N$ be

finitely generated $R$-modules. When can we get $M$ from $N$ by taking

direct summands, extensions and syzygies? This question is closely related

to classification of resolving subcategories of finitely generated

$R$-modules. In this talk, I will explain what I have got so far on this

topic.

- Time:
- 4:00pm - 5:00pm
- Description:
- Speaker: Pranav Pandit, ICTS

Time: Monday 30th November 4 to 5pm (joining time 3.45 pm IST)

Google Meet Link: https://meet.google.com/qvo-kduy-yco

Title: Noncommutative geometry of Landau-Ginzburg models

Abstract: I will outline a categorical framework for studying the

symplectic geometry of Landau-Ginzburg models and the algebraic geometry

of Tyurin degenerations. The main ingredients in this story are sheaves of

categories and spherical functors. I will explain how this framework can

be used to construct Calabi-Yau structures on categories of branes, and

(shifted) symplectic structures on certain moduli spaces of branes. This

talk is based on joint work with L. Katzarkov and T. Spaide.