Talk
Speaker: NAMAN KRISHNA PANDE, Indian Institute of Technology Ropar
Host: S Baskar
Title: Scientific Machine Learning for Complex Systems
Time, day and date: 11:30:00 AM – 12:30:00 PM, Monday, August 10
Venue: Online (https://meet.google.com/jxd-pimp-pox)
Abstract: Multi-agent systems (MAS) are ubiquitous to various natural phenomena such as flocks of birds, crowd motion, vehicular flow and insect swarms, etc. These systems involve a large number of interacting agents whose collective behaviour gives rise to complex emergent phenomena. Analysing and controlling such systems poses major problems in various scientific and engineering fields. Optimal control (OC) of MAS aims to learn strategies that enable the agents to achieve common or individual objectives while adhering to some dynamical constraints. We aim to develop and analyse these systems through the mean-field game theoretical approach that considers each agent as rational and utility-optimising and comprehends their dynamics at the macroscopic scale. Furthermore, we will utilise physics-informed neural networks (PINN) based models to learn the dynamics of such complex physical systems typically governed by a system of partial differential equations. However, despite enjoying great success, PINNs have some fundamental flaws when dealing with various complex physical problems. We aim to address various limitations of PINNs by developing robust and scalable physics-informed learning frameworks that incorporate improved neural network architectures, efficient sampling techniques and integration of discrete representations of the governing physics directly into the network architecture and training process.

PDE seminar
Speaker: Harsh Prasad, Universität Bielefeld
Host: Harsha Hutridurga
Title: Nonlocal Ground States Are Superharmonic
Time, day and date: 4:00:00 PM – 5:00:00 PM, Monday, August 10
Venue: Ramanujan Hall
Abstract: he first eigenfunction of the fractional Laplacian is expected to share many of the geometric properties of the classical ground state, yet even its superharmonicity has remained open in general. Previous results were limited to special exponents, low dimensions, or particular classes of domains.

Based on joint work with Florian Grube, I will present a new proof showing that every first Dirichlet eigenfunction $\phi$ satisfies
[
-\Delta\phi \ge \lambda_1^{1/s}\phi
]
for every $s\in(0,1)$. In one dimension, this yields strict log-concavity and resolves a long-standing conjecture of Bañuelos, Kulczycki and Méndez-Hernández.

The proof is remarkably short and is based on a new combination of Bernstein function theory and maximum principle arguments. Besides establishing the superharmonicity estimate, the method suggests a broader framework for comparing local and nonlocal elliptic operators and may extend to more general subordinate processes.

Finally, I will discuss the Brascamp–Lieb type log-concavity questions that originally motivated this line of research. I will present evidence indicating that the nonlocal picture may be considerably more subtle than in the classical setting, including indications that natural Brascamp–Lieb analogues may fail for symmetric $\alpha$-stable processes, and I will conclude with several open problems and possible future directions.
Combinatorics seminar

Number theory seminar
Speaker: Sabyasachi Dhar, IIT Bombay
Host: U.K. Anandavardhanan
Title: Tate cohomology and local base change for p-adic groups
Time, day and date: 5:00:00 PM - 6:00:00 PM, Monday, August 10
Venue: Room No .215
Abstract: In 2016, Treumann and Venkatesh proposed a remarkable conjecture on local Langlands functoriality for l-modular representations of p-adic groups, which predicts that the mod-l functoriality is realized by Tate cohomology. In this talk, we discuss this conjecture in the context of local base change lifting.