Talks

Speaker: D S Nagraj

Title: Vector Bundles

Abstract: Every mathematician is familiar with the concept of vector spaces and linear maps between vector spaces. In this talk we try to explain the concept of vector bundles and mapping between vector bundles, which is a natural generalisation of vector spaces and linear mapping between vector spaces.

Speaker: Mainak Poddar

Title: A Floer theory for toric orbifolds.

Abstract: will first give a general overview of Lagrangian Floer theory. Then I will outline the construction and some applications of such a theory for symplectic toric orbifolds.

This is based on a joint work with Cheol-Hyun Cho.

Speaker: Anupam Kumar Singh

Title: Asymptotics of powers in finite reductive groups

Abstract: Let G be a connected reductive group defined over a finite field F_q. Fix an integer M >1, and consider the power map x going to x^M on G. We denote the image of G(F_q) under this map by G(F_q)^M and estimate what proportion of regular semisimple, semisimple and regular elements of G(F_q) it contains. We prove that as q tends to infinity, all of these proportions are equal and provide a formula for the same. We also calculate this more explicitly for the groups GL(n, q) and U(n, q).

Speaker: Geetha Thangavelu

Title: On the determinant of Representations of Generalised Symmetric Groups

Abstract :The characterization of irreducible of representations of $S_n$ having non-trivial determinant was recently done by characterizing the corresponding integer partitions of n. And a closed formula to count the number of such representations is available. Also a similar characterization is available for the case of hyper-octahedral group and all irreducible finite coxeter groups. Motivated by this, in this talk I will talk about the case of generalized symmetric groups G(n,r) by giving a formula to compute the determinant of the irreducible representations of G(n,r) and various combinatorial results which will eventually count the number of non-trivial representations of G(n,r). This is a joint work with my student Amrutha P.

Speaker: Sugandha Maheshwary

Title: The upper and the lower central series of U (𝕫G)

Abstract: Given a group G, let U := U (𝕫G) be the group of units of the integral group ring 𝕫G. The elements of U, its center and their structures have been subject of investigation for a long time. Naturally, one also seeks understanding of the upper and the lower central series of U. For a finite group G, the upper central series is explored widely, whereas very basic questions remain unanswered for its lower central series. In this talk, I will discuss various aspects associated to the upper as well as the lower central series of U

Speaker: Jotsaroop Kaur

Title: Ramanujan's master theorem for Sturm Liouville operator

Abstract: Abstract: We prove an analogue of Ramanujan's master theorem in the setting of Sturm Liouville operator $$\mathcal L=\frac{d^2}{dt^2} +\frac{A'(t)}{A(t)} \frac{d}{dt}$$ on $(0,\infty)$, where $A(t)=(\sinh t)^{2\alpha+1}(\cosht)^{2\beta+1}B(t); \alpha,\beta> -\frac{1}{2}$ with suitable conditions on $B$. When $B\equiv 1$ we get back the Ramanujan's Master theorem for the Jacobi operator. This a joint work with Sanjoy Pusti.

Speaker: Sayan Bagchi

Title: The lacunary spherical maximal function on the Heisenberg group

Abstract: In this talk, we investigate the $L^p$ boundedness of the lacunary maximal function $A_rf$ associated to the spherical means on the Heisenberg group. By suitable adaptation of an approach of M. Lacey in the Euclidean case, we obtain sparse bounds for these maximal functions, which lead to new unweighted and weighted estimates. In order to prove the result, several properties of the spherical means have to be accomplished, namely, the $L^p$ improving property of the operator $A_rf$ and a continuity property of the difference $A_rf-\tau_y A_rf$, where $\tau_yf(x)=f(xy^{-1})$ is the right translation operator.

Speaker: Rahul Garg

Title: Weyl multipliers on Laguerre Sobolev Spaces

Abstract: In 1987, A. Bonami and S. Poornima showed that a non-constant function on the Euclidean space which is homogeneous of degree zero cannot be a Fourier multiplier on homogeneous Sobolev spaces. We define a notion of homogeneity of degree zero for bounded operators on $L^2(\mathbb{R}^n)$, and establish an analogue of the result of Bonami-Poornima for Weyl multipliers on Laguerre Sobolev Spaces.

This is joint work with Riju Basak and Sundaram Thangavelu.

Speaker : Amiya Kumar Mondal

Title: Integral representations of L-functions.

Abstract : L-functions are ubiquitous in number theory. Langlands' conjecture on L-functions describes the expected properties of the L-functions associated to automorphic representations. Integral representations of L-functions plays the central role in proving this conjecture. In this talk, we will describe a particular construction of integral representations, known as the 'Doubling Construction'. Towards the end, we will present recent progress in the doubling construction of integral representations for quasi-split special orthogonal groups.

Speaker: Chandrakant Aribam

Title: Iwasawa invariants of some Artin representations

Abstract: We relate the Iwasawa invariants of elliptic curves with CM with those of some Artin Representations. We briefly indicate some applications of such a relation.

Speaker: Debargha Banerjee

Title: The Eisenstein cycles and Manin-Drinfeld properties

Abstract: Consider any subgroup of finite index inside the full modular group. We write down the Eisenstein cycles in the first homology group of modular curves associated to any cuspidal divisor for the subgroup. Our expression of these cycles are in terms of values at cusps of certain real valued function. This function is the harmonic conjugate of real analytic Eisenstein series of weight zero associated to the divisor. As an application, we give a criteria for a divisor to be torsion or not in the cuspidal group in terms of these Eisenstein cycles. As an application, we also compute these Eisenstein cycles for non-congruence subgroups associated to Fermat's curves and quotients of Fermat's curves and Heisenberg curves. In the process, we answered a question raised by KumarMurty-Ramakrishnan regarding the cuspidal subgroups associated to noncongruence subgroups corresponding to the Heisenberg curves. This is a joint work with Loic Merel.

Speaker : Jyoti Prakash Saha

Title : Families of Galois representations

Abstract : In 1916, Ramanujan predicted that the Fourier coefficients τ (n) of the ∆-function satisfy |τ (n)| ≤ d(n)n^11/2 , where d(n) is the number of positive divisors of n. This conjecture is established by Deligne and is equivalent to the Galois representations attached to ∆ being pure. Many Galois representations of arithmetic interest are expected to be pure (by the weight-monodromy conjecture and the generalized Ramanujan conjecture) and often known to be so. In the 1980s, Hida constructed p-adic families of ordinary cusp forms, he proved that the p-adic Galois representations attached to the p-ordinary cusp forms can be interpolated via big Galois representations. Further examples of p-adic families were constructed by Coleman–Mazur, Hida, Emerton, Ash–Stevens and others. This motivates to look for a notion of purity for families of Galois representations. In this talk, we will explain a formulation of this notion. We will also explain some results on ramification of big Galois representations.

Speaker: Soumya Bhattacharya

Title: Special factors of holomorphic eta quotients

Abstract: Holomorphic eta quotients supply examples of modular forms which are ubiquitous in Number Theory. We are particularly interested in detection of irreducibility of holomorphic eta quotients: A holomorphic eta quotient is called irreducible if it is not a product of two other holomorphic eta quotients. Like primality of natural numbers, irreducibility of holomorphic eta quotients have some nice applications and there already exist several results about them. Now, imagine a situation where we know the definition of prime numbers along with many striking results about them. However, given a natural number with more than 50 digits which has no obvious small prime factors, even our computers could not tell whether that integer is a prime or a composite. Wouldn't that be a plight? The practical situation for detection of primality of a integers was almost the same until Adleman, Pomerance and Rumely came up with their primality detection algorithm in 1983. Also in 2004, Agarwal, Kayal and Saxena provided a prime detection algorithm which runs in polynomial time. However, the situation with the detection of irreducibility of holomorphic eta quotients is worse than that. An eta quotient has two parameters, viz. weight and level. Though the weights of the factors of a irreducible holomorphic eta quotient is bounded above by the weight of it, the same does not hold for the levels of the factors. This is precisely the root cause of the difficulty of the detection of irreducibility of holomorphic eta quotients. However, we could show that when the level of the given holomorphic eta quotient is a prime power, then in a reasonable amount of time, we can tell whether is irreducible or not. More precisely, we take an arbitrary holomorphic eta quotient $f$ of a prime power level and we show that if $f$ is reducible,then we need not look beyond the level of $f$ for finding its factors. This in particular, implies that we require to consider the quotients of $f$ by only finitely many candidates which we could compute by any algorithm that lists the lattice points in a given polytope. The question that remains is: Does a similar result hold in general?

Speaker: Girija Shankar Tripathi

Title: WItt groups of real algebraic varieties

Abstract: In this talk the theory of triangular Witt groups for schemes is recalled and we discuss Witt theory for real algebraic varieties in motivic homotopy category.

Speaker: Sarbeshwar Pal

Title: Lazarsfeld-Mukai bundles on K3 surfaces associated to a pencil computing Clifford index

Abstract: Let $X$ be a smooth projective K3 surface over complex numbers and $C$ be an ample curve on $X$. Given a globally generated line bundle $A$ on $C$ one can associate a vector bundle on $X$, known as Lazarsfeld-Mukai bundle. In this talk we will discuss the semistability of the Lazarsfeld-Mukai bundle $E_{C, A}$ associated to a line bundle $A$ ion $C$ such that $|A|$ is a pencil on $C$ and computes the Clifford index of $C$. We will give a necessary and sufficient condition for $E_{C, A}$ being semistable.

Speaker: Amit Hogadi
[Delegated to Neeraj Deshmukh].

Title: The 1-Resolution Property for Algebraic Stacks

Abstract: An algebraic stack is said to have the 1-resolution property if every coherent sheaf is the quotient of (summands of) a single given vector bundle. In this talk, we will show that algebraic stacks (under some finiteness hypothesis) with the 1-resolution property admit finite flat covers by quasi-affine schemes. Further, we also show that any algebraic space with the 1-resolution property is necessarily a quasi-affine scheme. We also show how to remove some of the finiteness conditions on the stack using approximation techniques. This gives us further refinements in characteristic zero. This is joint work with Amit Hogadi and Siddharth Mathur.

Speaker: Vivek Sadhu

Title: Relative Brauer groups and etale cohomology

Abstract: The aim of this talk is to define the relative Brauer group Br(f) of a map of schemes $f: X -----> S$ and study its properties. We will also relate Br(f) with the first etale cohomology group of a certain sheaf.

Speaker : Yashonidhi Pandey

Title: On a "wonderful" Bruhat-Tits group scheme

Abstract: In this note, we make a universal construction of Bruhat-Tits group scheme on wonderful embeddings of adjoint groups and adjoint Kac-Moody groups. These have natural classifying space properties reflecting the orbit structure on the wonderful embeddings. This is joint work with Vikraman Balaji.

Speaker: Prahlad Vaidyanathan

Title: Stable Ranks for C*-algebras and Rokhlin Actions

Abstract: We will discuss certain dimension theories for a C*-algebra, together called stable ranks. We will then discuss a notion of free-ness of a group action, called the Rokhlin property, for an action of a finite group on a C*-algebra. We will show that, if an action has the Rokhlin property, then it is possible to estimate the stable ranks of the crossed product C*-algebra in terms of those of the underlying algebra. This result was jointly proved with Ms. Anshu.

Speaker: Sutanu Roy

Title: Quantum E(2) groups and beyond

Abstract: Locally compact quantum groups, abbreviated as LCQGs, generalise the notion of locally compact groups in the realm of Operator Algebras and Noncommutative Geometry. For real 0<|q|<1, the q-deformations E(2) group, constructed by S. L. Woronowicz in the early nineties, formed an important class of examples of LCQG in the C*-algebraic setting. However, for complex, 0<|q|<1, the respective deformations of E(2) fail to be LCQGs. In this talk, we shall first discuss Woronowicz's construction of quantum E(2) groups and its generalisation for complex deformation parameters. The later part is a joint work in progress with Atibur Rahaman.

Speaker: Lingaraj Sahu

Title: C*-convexity

Abstract: Here we will discuss a generalized notion of convexity, known as C*-convexity in the operator algebraic framework and will give some characterization of C*-extreme points of the set of completely positive maps.

Speaker : Shibananda Biswas

Title : On homogeneous operators via quotient modules

Abstract : We show that the quotient modules obtained from submodules, consisting of functions vanishing to order k along a linear variety $\Z\subset\O$ of codimension at least 2, of an analytic Hilbert module, is homogeneous with respect to suitable subgroup of the automorphism group of the domain. We are then able to show that these homogeneous operators are reducible and decomposes to generalized Wilkin's operators as irreducible components. This is a joint work with Prahllad Deb and Subrata Shyam Roy.

Speaker: Anup Biswas

Title: Recent developments in the theory of generalized principal eigenvalue in R^N.

Abstract: In this talk we shall give a brief survey on the (Dirichlet) generalized principal eigenvalue in unbounded domains for non-degenerate elliptic operators. Then we shall discuss few latest result in this direction for infinity Laplacian operators.

Speaker: Anupam Pal Choudhury

Title: The equivalent media generated by bubbles of high contrasts

Abstract: In this talk, we shall discuss about the point-interaction approximations for the acoustic wave fields generated by a cluster of highly contrasted bubbles for a wide range of densities and bulk moduli contrasts. As a consequence, we can derive the equivalent fields when the cluster of bubbles is appropriately distributed (but not necessarily periodically) in a bounded domain of R ^3.

Speaker: Manas Kar

Title: Recovery of coefficients for p-Laplacian perturbed by a second order term.

Abstract: In this talk we consider inverse problems for weighted p-Laplace equation perturbed by a linear second order term. We give a procedure to recover the coefficients involved in the equation from the corresponding Dirichlet to Neumann map. This is a joint work with Catalin I. Carstea.

Speaker: Mousomi Bhakta

Title: Nonlocal scalar field equations

Abstract: In this talk we will give a brief survey on scalar field equations and discuss some recent results in nonlocal scalar field equations and open questions in that area.

Speaker: Rajib Datta

Title: Operator Splitting for the fractional Korteweg-de Vries Equation

Abstract: Our aim is to analyze operator splitting for the fractional Korteweg-de Vries equation, $u_t = u u_x + -(-\Delta)^{\alpha/2} u_x$, $\alpha\in [1,2]$. Under the appropriate regularity of the initial data, we demonstrate the convergence of approximate solutions obtained by the Godunov and Strang splitting. We show that for the Godunov splitting, first order convergence in $L^2$ is obtained for the initial data in $H^{1+\al}$ and in case of the Strang splitting, second order convergence in $L^2$ is obtained for initial data in $H^{1+2\al}$. The obtained rates are expected in comparison with the KdV $(\al=2)$ case. This is a joint work with Tanmay Sarkar.

Speaker: Saugata Bandyopadhay

Title: On the Equation (∇u)^t H(u)(∇u) = G

Abstract: Let n ∈ ℕ, n ⩾2 and let Ω ⊆ ℝⁿ be open. Let H, G : ℝⁿ → ℝ^nₓn be of appropriate regularity. We discuss the existence of an immersion u : Ω → ℝⁿ of appropriate regularity, satisfying

(∇u)^t H(u)(∇u) = G in Ω. (1)

We consider Cauchy, Dirichlet and Dirichlet-Neumann problems.

Equation (1) comes up in diverse contexts. When H (and hence G) is symmetric and positive definite, Equation (1) is connected to the problem of equivalence of Riemannian metrics. The symmetric case is also important in the non-linear elasticity theory because of its connection with the Cauchy-Green deformation tensor. When H (and hence G) is skew-symmetric, Equation (1) comes up in the context of the problem of equivalence of differential two-forms.

The aim of the talk is to present a survey of recent progress and advances in the context of Equation (1). We also discuss the general case when H, G are neither symmetric nor skew-symmetric.

Speaker: Atreyee Bhattacharya

Title: On certain Riemannian functionals

Abstract: Riemannian functionals and the geometry of their critical points are classical tools for analysing the existence of standard Riemannian metrics (e.g. Einstein metrics) and the rigidity of existing Riemannian structures. Given a compact Riemannian manifold, one can define several Riemannian functionals (on the space of Riemannian metrics of the manifold) by taking L^p p norms of curvature quantities such as scalar curvature, Ricci curvature, full curvature tensor, Weyl curvature, etc. After making sense of differentiability of such functionals, the stability of a critical point (/metric) can be determined by the second order infinitesimal behaviour of the functional at that point. Irreducible symmetric spaces are known to be critical points of all Riemannian functionals normalized by volume. More generally, Einstein metrics turn out to be critical points for certain functionals. However, in general, the moduli space of critical points or the stability of crit-ical points are not so well understood. In this talk, we will discusssome well known curvature functionals and the stability of their critical points corresponding to locally symmetric spaces. This is a jointwork with Soma Maity.

Speaker: Shane D' Mello

Title: Real algebraic knots

Abstract: We will discuss some problems and developments in the field of real algebraic knots.

Speaker: Somnath Basu

Title: A story of hypocycloids and special unitary matrices

Abstract: We discuss the classical non-self intersecting hypocycloids and how it relates to the set of mean eigenvalues of special unitary matrices. Working with general hypocycloids lead to symmetric functions of eigenvalues of the above matrices. This talk will be an expository account of how these ideas from geometry of curves connect with coefficients of characteristic polynomials.

Speaker: Tejas Kalelkar

Title: Bounds on Pachner moves in cusped hyperbolic manifolds

Abstract: A Pachner move is a local combinatorial change to the triangulation of a manifold. Any two geometric ideal triangulations of a one-cusped complete hyperbolic 3-manifold M are related by a sequence of Pachner moves through topological triangulations. We give a bound on the length of this sequence in terms of the total number of tetrahedra and a lower bound on dihedral angles. This leads to a naive but effective algorithm to check the equivalence of hyperbolic knots (given geometric ideal triangulations of their complement). Given a geometric ideal triangulation of M, we also give a lower bound on the systole length of M in terms of the number of tetrahedra and a lower bound on dihedral angles. This allows us to show that repeatedly twisting a pair of strands of a knot in S^3 gives knots whose complements do not admit any thick geometric ideal triangulation.