January  2023 semester

This semester's seminars:-

 

Title: High-degree vertices of random recursive trees


Abstract:

Random recursive trees are labeled trees constructed by sequentially adding a new vertex, with an associated vertex-weight, and connecting it to a previous vertex with probability proportional to its weight (the weights do not change throughout the process).


Contrary to the qualitative behavior of linear preferential attachment trees where high-degree vertices become established from the early stages of the process, high-degree vertices in random recursive trees keep changing throughout the process. 


We provide a description of both the degree and the height of uniformly chosen vertices and present some applications and open problems. This talk includes joint work with Sergio López-Ortega and Marco López-Ortiz.





Date-9th January 

Time-4 pm. 

Title:Vanishing of Tors of absolute integral closures in equicharacteristic zero.

Abstract: We show that vanishing of Tors of absolute integral closures characterizes regularity of a ring provided further that the ring is N-graded of dimension 2 finitely generated over an equicharacteristic zero field. This answers a question of Bhatt, Iyengar, and Ma. In spite of being a question in commutative algebra, the proof uses techniques from algebraic geometry and rational singularities.






Over zoom, time-2.30 pm 

Title: On the homological mirror symmetry of pair of pants

Abstract: A pair of pants and a polynomial xyz are related by homological mirror symmetry.

Namely, Abouzaid-Auroux-Efimov-Katzarkov-Orlov showed that geometric category of curves (Fukaya category) is derived equivalent to the matrix factorization category of xyz. We give a gentle introduction and the geometric idea behind the correspondences.

Recently, Burban-Drozd gave a complete classification of indecomposable Cohen-Macaulay modules of the singularity C[[x,y,z]]/(xyz),

and we find that these are mirror to the closed geodesics (and its iterated cones) of a hyperbolic pair of pants. This is a joint work with  K. Kim, K. Roh and W. Jeong.




Date-19th January 4 pm 

Chair: Dr. Praful Kumar Tale. 

Title : Algorithmic and structural results for path covering problems in graphs


Abstract: We discuss some recent developments around several related path-covering problems in graphs, where one wishes to cover the vertices of a graph by a minimum-size set of paths. We will discuss paths of several types : unrestricted, shortest paths, or induced paths. We will mostly focus on the shortest path version, and will present some hardness results, a constant-factor approximation algorithm for some graph classes (including chordal graphs), and an interesting relation between the solution size and the tree-width of the graph.

The talk is based on joint work from several recent papers:

https://hal.archives-ouvertes.fr/hal-03899912

https://arxiv.org/abs/2206.15088

https://arxiv.org/abs/2212.11653



Madhava hall, 3 pm. 

Title: Hilbert's 14 th problem, symbolic Rees algebras and set-theoretic complete intersections

Abstract: We shall discuss when the symbolic Rees algebra of an ideal is a Noetherian ring and its

relevance for the Hilbert's 14 th problem for the rings of invariants and  its generalization due to Zariski and Rees's solution.

Cowsik's conjecture and its relation to set theoretic complete intersections will be discussed.

If time permits, we will discuss a recent criterion for the Noetherian property

of the symbolic Rees algebra and its application to  set-theoretic complete intersection property of the ideal of certain

hyperplane arrangements. 


5. Dr. Soumen Sarkar, IIT, M  (27 January 2023).

Title: Equivariant cohomological rigidity of certain T-manifolds

Speaker: Soumen Sarkar

Abstract:

In this talk, I'll introduce the category of locally k-standard T-manifolds which includes well-known classes of manifolds such as toric and quasitoric manifolds, good contact toric manifolds and moment-angle manifolds. I'll compute fundamental groups and equivariant cohomology algebras of these spaces. Then, I'll discuss when the torus equivariant cohomology algebra distinguishes them up to weakly equivariant homeomorphism. This is a joint work with Jongbaek Song.


2nd February

Madhava Hall,

4 pm 

On a residual coordinate which is a non-trivial line


Let $R$ be a Noetherian integral domain. %The notation $A=R^{[n]}$ will

denote that $A$ is a polynomial ring in $n$ indeterminates over $R$.  A

polynomial $F$ in the polynomial ring $R[X,Y]$ is said to be a {\it

coordinate} if $R[X,Y]=R[F,G]$ for some $G \in R[X,Y]$.


An obvious necessary condition for a polynomial $F$ to be a coordinate

is that it should be a {\it line} in $R[X,Y]$, i.e.,

$R[X,Y]/(F)=R^{[1]}$. When $R$ is a field of characteristic zero,

the famous Epimorphism Theorem of Abhyankar-Moh shows that any line in

$R[X,Y]$ is necessarily a coordinate.

The Generalized Epimorphism Theorem of S.M. Bhatwadekar shows  that the

result holds when $R$ is a seminormal domain of characteristic zero

or if $R$ contains $\mathbb Q$.

Examples of B. Segre and M. Nagata

show that when $R$ is a field of positive characteristic, a line need

not be a coordinate in $R[X,Y]$.


Another necessary condition for a polynomial $F$ in $R[X,Y]$ to be a

{\it coordinate} is that $F$ should be

a {\it residual coordinate}, i.e., for every prime ideal $P$ of $R$,

the image of $F$ in $k(P)[X,Y]$ should be a coordinate in $k(P)[X,Y]$,

where $k(P)$ denotes the residue field $R_P/PR_P$.

Again, it has been shown by Bhatadekar-Dutta that if  $R$ contains

$\mathbb Q$ or if $R$ is seminormal,

then a residual coordinate in $R[X,Y]$ is necessarily a coordinate.

Bhatwadekar has also shown that if the characteristic of $R$ is zero,

then a line in $R[X,Y]$ is necessarily

a residual coordinate. However, an example of Bhatwadekar-Dutta shows

that, in general, a residual coordinate in $R[X,Y]$ need not be

a line; in particular, it need not be a coordinate.


To summarize, when $R$ is neither seminormal nor contains $\mathbb Q$,

then a line need  not be a coordinate and

a residual coordinate need not be a line. This naturally raises the

question whether

a residual coordinate in $R[X,Y]$ which is also a line in  $R[X,Y]$ is

necessarily a coordinate in $R[X,Y]$.


We shall demonstrate a negative answer to the above question by

presenting a

residual coordinate $F$ in $R[X,Y]$ over the one-dimensional Noetherian

local domain $R=k[[t^2, t^3]]$,

where $k$ is a field of characteristic $>2$, such that

$F$ is a line in $R[X,Y]$ but $F-1$ is not.


Let $\widetilde{R}$ denote the ring $k[[t]]$ and $\tilde{k}$ the ring

$\widetilde{R}/t^2\widetilde{R}$.

The main tool in the construction of the example is the technique of

lifting of a certain

$\tilde{k}$-automorphism of $\tilde{k}[X,Y]$ to an

$\tilde{R}$-automorphism of $\tilde{R}[X,Y]$ introduced by T. Asanuma.


  A modification of the above example shows that the Generalized

Epimorphism Theorem

of S.M. Bhatwadekar cannot be extended to an arbitrary Noetherian domain

of characteristic zero.


This is a joint work with Prof. T. Asanuma.



3rd February

Madhava Hall,

4 pm 

Title: A sequence of operator algebras converging to odd spheres in the quantum Gromov-Hausdorff distance

 

Abstract. Marc Rieffel had introduced the notion of the quantum Gromov-Hausdorff distance on compact quantum metric spaces and found a sequence of matrix algebras that converges to the space of continuous functions on 2-sphere in this distance. One finds applications of similar approximations in many places in the theoretical physics literature. We shall recall Rieffel’s work and define a compact quantum metric space structure on the sequence of Toeplitz algebras on generalized Bergman spaces and prove that the sequence converges to the space of continuous function on odd spheres in the quantum Gromov-Hausdorff distance.




16th  February

Madhava Hall,

4 pm 



Title: Chern-Weil theory and Hilbert-Samuel theorem for semi-positive

singular toroidal metrics on line bundles

Abstract: In this talk I will report on joint work with A. Botero, D.

Holmes and R. de Jong. Using the theory of b-divisors and

non-pluripolar products we show that Chen-Weil theory and a Hilbert

Samuel theorem can be extended to a wide class of singular

semi-positive metrics. We apply the techniques relating semipositive

metrics on line bundles to b-divisors to study the line bundle of

Siegel-Jacobi forms with the Peterson metric. On the one hand we prove

that the ring of Siegel-Jacobi forms of constant positive relative

index is never finitely generated, and we recover a formula of Tai

giving the asymptotic growth of the dimension of the spaces of

Siegel-Jacobi modular forms.


Time and place : 11 am on Monday, 20th February, in Madhava Hall


Title:  An informal introduction to log smooth schemes


Abstract: The notion of log cotangent and tangent bundle was initially developed for manifolds with a choice of normal crossing divisor. This was vastly extended to log schemes by Illusie and Kato, and later studied extensively by Olsson. Log structures arise naturally in many moduli problems. In particular there is a notion of smooth log structures which is in an appropriate sense locally modelled on toric varieties, which we will describe.


24th  February

Madhava Hall,

4 pm 

10th  March

Title:FRACTIONAL FOURIER TRANSFORM, HARMONIC

OSCILLATOR PROPAGATORS AND STRICHARTZ


ESTIMATES


Abstract:-


We show that harmonic oscillator propagators and fractional Fourier

transforms are essentially the same. We deduce continuity properties for

such operators on modulation spaces, and apply the results to prove Strichartz

estimates for the harmonic oscillator propagator when acting on modulation

spaces. Especially we extend some results in [1, 2, 3, 4]. We also show that

general forms of fractional harmonic oscillator propagators are continuous

on suitable Pilipovi ́c spaces. Especially we show that fractional Fourier

transforms of any complex order can be defined, and that these transforms

are continuous on any Pilipovi ́c space and corresponding distribution space,

which are not Gelfand-Shilov spaces.

The talk is based on a joint work with Divyang Bhimani and Ramesh

Manna.


Title: L^p-improving estimates for Fourier integral operators and maximal operators


Abstract: In this talk, I will discuss the L^p improving smoothing estimate of Fourier integral operator (FIOs), how restrictions on square function estimates have hindered the study of local smoothing estimates of FIOs and how one is able to overcome these restrictions to obtain local smoothing estimates for FIOs with general amplitude function. Finally, we give an application of these estimates to maximal operators. This is a joint work with Prof. P. K. Ratnakumar.


17th March

Title:

Combining Rational maps and Kleinian groups via orbit equivalence

Abstract:

We develop a new orbit equivalence framework for holomorphically combining the dynamics of complex polynomials with that of Kleinian surface groups. We show that the only torsion-free Fuchsian groups that can be thus combined are punctured sphere groups. We describe a new class of maps that are topologically orbit-equivalent to Fuchsian punctured sphere groups. We call these higher Bowen-Series maps. The existence of this class ensures that the Teichmüller space of matings is disconnected. Further, they also show that, unlike in higher dimensions, topological orbit equivalence rigidity fails for Fuchsian groups acting on the circle. We also classify the collection of Kleinian Bers' boundary groups that are mateable in our framework. This is joint work with Sabyasachi Mukherjee.


Title:- Galois Representations


Abstract:


  Galois representations have become fundamental tools for studying

problems in number theory. The lecture will give an overview of our

recent work on the explicit shape of the reductions of two-dimensional

local Galois representations.

Date:-21st April 

Title: Delta Characters and Crystalline Cohomology


Abstract: Given a commutative smooth finite dimensional group scheme  defined over a discrete valuation ring  with a fixed lift of Frobenius, using the theory of delta characters, we construct a canonical filtered isocrystal  associated to .


For an elliptic curve  defined over , we show that our canonical filtered isocrystal  is weakly admissible. In particular, if  does not admit a lift of Frobenius, we show that  is canonically isomorphic to the first crystalline cohomology  in the category of filtered isocrystals. On the other hand, if  admits a lift of Frobenius, then  is isomorphic to the sub-isocrystal  of .

The above result can be viewed as a character theoretic interpretation of the crystalline cohomology. This is a joint work with Sudip Pandit.

28th April 

Title:Rational Inner Functions

Abstract

In this talk, we shall discuss rational inner functions on certain domains. A classical rational inner function is a rational map f from the unit disc D to its closure D with the property that f maps the unit circle T into itself. It is well known that the rational inner functions on the open unit disc in the complex plane are the finite Blaschke products. Rudin gave a structure of rational inner functions in the polydisc. We shall see a canonical structure of rational inner functions on the bidisc and the symmetrized bidisc. We shall discuss the role of rational inner functions in approximation theory and operator theory. Specifically, we shall see Caratheodory-type approximation results on certain domains. The classical Caratheodory approximation result states that any matrix-valued holomorphic function on the open unit disc with sup-norm no greater than one, can be approximated by rational inner functions (precisely the Blaschke functions), uniformly on compact subsets of the open unit disc.



12th May 

Future speakers: