This semester's seminars:-
Professor Laura Eslava (UNAM, Mexico City). 6th January
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.
Shravan Patankar (University of Illionis at Chicago)
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.
Dr. Cheol-Hyun Cho (Seoul National University). 13th January 2023)
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.
Florent Foucaud (University Clermont Auvergne, France)
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
Prof J. K. Verma, IIT Bombay (20th January 2023)
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.
Professor Amartya Kumar Dutta (ISI, Kolkata)
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.
Professor Tirthankar Bhattacharya (IISC).
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.
Professor José Ignacio Burgos Gil (ICMAT, Spain) .
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.
Speaker: Professor Barbara Fantechi, SISSA (International School for Advanced Studies), Italy
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.
Professor Sandip Singh (IIT Bombay).
24th February
Madhava Hall,
4 pm
Prof. Joachim Toft, Linnaeus University (LNU), Sweden)
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.
16th March 4pm
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.
Professor Mahan Mj (TIFR)
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.
Professor Eknath Ghate (TIFR, Mumbai)
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.
Professor Arnab Saha (IIT Gandinagar)
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.
Mr. Poornendu Kumar (IISC)
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.
Prof. Eric Urban (Columbia University)
12th May
Future speakers:
Professor Anupam Saikia (IIT, Guwhati).
C. S. Rajan (Ashoka University).
Professor Dipendra Prasad (IIT Bombay) August 2023 semester