TIFR CAM Alumni Meeting 2024
Date : Tuesday October 29, 20224
Venue : TIFR CAM, LH-111
Schedule
Sl | Time | Speaker |
---|---|---|
1 |
9:15 AM - 9:30 AM |
Introduction by the Dean, Prof. K. Sandeep |
2 |
9:30 AM - 10:30 AM |
Prof. Saugata Bandyopadhyay, IISER Kolkata |
3 |
11:00 AM - 12:00 Noon |
Prof. Mousomi Bhakta, IISER Pune |
4 |
12:00 Noon - 1:00 PM |
Prof. Sudarshan Kumar, IISER Thiruvananthapuram |
5 |
2:30 PM - 3:00 PM |
Mr. Ravi Jaiswal, TIFR CAM |
6 |
3:00 PM - 3:30 PM |
Dr. Subhankar Mondal, TIFR CAM |
7 |
4:00 PM - 4:30 PM |
Mr. Subhajit Ghosh, TIFR CAM |
8 |
4:30 PM - 5:00 PM |
Dr. Dipti Ranjan Parida, TIFR CAM |
9 |
5:00 PM - 5:30 PM |
Mr. Monideep Ghosh, TIFR CAM |
- Tea Breaks: 10:30 AM - 11:00 AM, 3:30 PM - 4:00 PM
- Lunch Break: 1:00 PM - 2:30 PM
- High Tea: 5:30 PM
Equivalence of flat coordinates for blinear forms & related problems
Speaker : Prof. Saugata Bandyopadhyay, IISER Kolkata
Multiplicity of solutions for a class of critical exponent problems in the hyperbolic space
Speaker : Prof. Mousomi Bhakta, IISER Pune
In this talk I will discuss the multiplicity of positive solutions to problems of the type
−∆BN u − λu = a(x)|u|2 ∗−2u + f(x) in B^N , u ∈ H1 (B^N ),
where B^N denotes the ball model of the hyperbolic space of dimension N ≥ 4, 2∗ = 2N/ N−2, N(N−2)/ 4 < λ < (N−1)^2/ 4 and f ∈ H^−1 (B^N ) (f != 0) is a non-negative functional in the dual space of H^1 (B^N ). The potential a ∈ L^∞ (B^N ) is assumed to be strictly positive, such that limd(x,0)→∞ a^(x) = 1, where d(x, 0) denotes the geodesic distance. In the profile decomposition of the functional associated with the above equation, concentration occurs along two different profiles, namely, hyperbolic bubbles and localized Aubin-Talenti bubbles. Using thedecomposition result, we derive various energy estimates involving the interacting hyperbolic bubbles and hyperbolic bubbles with localized Aubin-Talenti bubbles. Finally, combining these estimates with topological and variational arguments, we establish a multiplicity of positive solutions in the cases: a ≥ 1 and a < 1 separately. (This is a joint work with Debdip Ganguly, Diksha Gupta and Alok Kumar Sahoo.)
On a second-order numerical approximation for non-local conservation laws
Speaker : Prof. Sudarshan Kumar, IISER Thiruvananthapuram
In this talk, we present the convergence analysis for a second-order numerical scheme designed for traffic flow models governed by non-local conservation laws. While first-order methods in computational fluid dynamics are known for their robustness and reliability, particularly in proving well-posedness, higher-order methods can provide significantly more accurate solutions without additional computational cost, especially for two- or three-dimensional problems. Our focus is on deriving a second-order scheme and showing its convergence to the entropy solutions of the given problem. The analysis proceeds in two main steps. First, we establish that the scheme converges to a weak solution by proving key properties such as the maximum principle, bounded variation estimates, and L1-Lipschitz continuity in time. In the second step, leveraging the characteristics of the proposed method, we show that the solution converges to the entropy solution. Finally, we extend our discussion to a two-dimensional non-
local problem that models crowd dynamics, where we analyze a second-order approximation that maintains the essential properties of the physical model and explore its significance.
Boundary behaviour of the Bergman and Szego kernels on generalized decoupled domains
Speaker : Mr. Ravi Jaiswal, TIFR CAM
In this talk, we will introduce the generalized decoupled domains. Within the class of domains, complex tangential directions are not necessarily decoupled individually, and boundary points may possess both finite and infinite type directions. Here’s an example of such a domain:
{ (z_1, z_2, z_3, z_4) ∈ C^4 : Re z_4 > e^−1/(|z_1|^2+|z_2|^2) + |z_3|^2 }
We will prove optimal lower and upper bounds of the Bergman and Szeg ̋o kernels near the boundary of bounded smooth generalized decoupled pseudoconvex domains in C^N
On inverse initial value identification for a non-local in time evolution equation
Speaker : Dr. Subhankar Mondal, TIFR CAM
We consider the inverse problem of retrieving the initial value of a time-fractional fourth order parabolic equation from source and final time observation. Since the considered problem is ill-posed, we obtain regularized approximations for the sought initial value by employing the quasi-boundary value method and its modification. We provide both the apriori and aposteriori parameter choice strategies and obtain the error estimates of H ̈older type under some typical source condition. If time permits, we shall also discuss about the optimality (in the sense of worst case error) of the obtained rates.
Number of connected components of polynomial lemniscates
Speaker : Mr. Subhajit Ghosh, TIFR CAM
For a monic polynomial p with complex coefficients, we define its unit lemniscate by
Λp := {z : |p(z)| < 1}
We are interested in the maximum number of connected components of Λp, when the roots ofp lie within a compact set K ⊂ C with positive logarithmic capacity c(K). Our main result concerns the asymptotic behavior of the maximum number of components. Let Cn(K) denote the maximum number of connected components of Λp for a monic polynomial p of degree n whose roots are contained in K. We prove that the quantity M(K) = lim supn→∞ Cn(K)n satisfies M(K) < 1 if c(K) < 1, and M(K) = 1 if c(K) ≥ 1 . This result resolves a question posed by Erd ̈os and colleagues in 1958.
Resonant and non-resonant energy transfers in wave turbulence
Speaker : Dr. Dipti Ranjan Parida, TIFR CAM
Wave dynamics are important in many fields, such as oceanography, plasma physics, and astrophysics. They are often described by dispersive, non-linear partial differential equations. These equations explain how energy spreads from excited waves through nonlinear interactions consisting of resonant and non-resonant waves. When wave amplitudes are small, resonant interactions typically dominate energy transfer, a study area known as wave turbulence. However, the effectiveness of these interactions is still unclear, and quasi- resonant interactions are not well understood. Moreover, full-scale, three dimensional simulations of hydrodynamics are complex and time-consuming, and they often make it hard to see how energy transfers happen through resonant interactions. To address these challenges, this study derives new one-dimensional mathematical models with quadratic non-linear terms, which result in three-wave resonant interactions. These models span a spectrum of dispersion relations: 1) sinusoidal and algebraic relations for exact resonance, and 2) Rossby and internal-gravity waves for quasi-resonance. Also, each model is characterized by distinct non-linear terms for varying degrees of nonlinearity. By exploring these simplified models, we found out that the resonant energy transfer is subdominant even for lower degrees of nonlinearity.
Log-perturbed Br ́ezis Nirenberg problem in hyperbolic space
Speaker : Mr. Monideep Ghosh, TIFR CAM
The well-known Br ́ezis-Nirenberg problem in a bounded domain Ω ⊂ R n gained popularity from their seminal work in 1983. This problem investigates the existence and non-existence of positive solutions
−∆u(x) = |u| (n+2) n−2 −1 u(x) + f(x, u) in Ω,
f is a lower-order perturbation. The article studied various examples of perturbations, the simplest example being f(x, u) = λu, λ ∈ R. Later, Mancini and Sandeep studied the
hyperbolic space analog of this problem with the perturbation λu. In recent times, taking motivation from the logarithmic-Schrodinger equation in quantum mechanics, attention has been paid to the log-perturbed Brezis-Nirenberg problem, more specifically perturbations of the form λu + θu ln u, where λ, θ ∈ R.
While the existence of solution for θ ≥ 0 and partial non-existence for θ < 0 has been obtained in the Euclidean space, by and large, the understanding of the threshold existence vs non-existence remains incomplete. In our talk, we will present our results in the hyperbolic case for the mentioned perturbation, which provides a complete picture of the existence vs non-existence dichotomy of positive solutions. This work is a joint effort with my advisor, Dr. Debabrata Karmakar, and his postdoc, Dr. Anumol Joseph.