"Visual proofs of identities were common at the Greek time, such as the Pythagoras theorem. In the same spirit, with the renaissance of combinatorics, visual proofs of much deeper identities become possible. Some identities can be interpreted at the combinatorial level, and the identity is a consequence of the construction a weight preserving bijection between the objects interpreting both sides of the identity. In this lecture, I will give an example involving the famous and classical Ramanujan continued fraction. The construction is based on the concept of "heaps of pieces", which gives a spatial interpretation of the commutation monoids introduced by Cartier and Foata in 1969. For more informations  go to website of the combinatorial course "The Art of Bijective Combinatroics"  I am giving at IMSc (2016-2019) https://www.imsc.res.in/~viennot/abjc-course.html"

"The intriguing question, why the present scale of the universe is free from any perceptible footprints of rank-2 antisymmetric tensor fields (generally known as Kalb-Ramond fields) is addressed. A quite natural explanation of this issue is given from the angle of higher-curvature gravity, both in four- and in five-dimensional spacetime. The results here obtained reveal that the amplitude of the Kalb-Ramond field may be actually large and play a significant role during the early universe, while the presence of higher-order gravity suppresses this field during the cosmological evolution, so that it eventually becomes negligible in the current universe. Besides the suppression of the Kalb-Ramond field, the extra degree of freedom in F(R) gravity, usually known as scalaron, also turns out to be responsible for inflation. Such F(R) gravity with Kalb-Ramond fields may govern the early universe to undergo an inflationary stage at early times (with the subsequent graceful exit) for wider range of F(R) gravity than without antisymmetric fields. Furthermore, the models—in four and five dimensional spacetimes are linked to observational constraints, with the conclusion that the corresponding values of the spectral index and tensor-to-scalar ratio closely match the values provided by the Planck survey 2018 data."

"ery strong magnetic fields can arise in non-central heavy-ion collisions at ultra-relativistic energies, which may not decay quickly in a conducting plasma. We carry out relativistic magneto-hydrodynamics (RMHD) simulations to study the effects of this magnetic field on the evolution of the fluid and on resulting flow fluctuations in the ideal RMHD limit. Our results show that the magnetic field leads to enhancement in elliptic flow for small impact parameters while it suppresses it for large impact parameters. Interestingly, we find that magnetic field in localized regions can temporarily increase in time as evolving fluid energy density fluctuations lead to reorganization of magnetic flux. This can have important effects on the chiral magnetic effect. We discuss the situation of nontrivial magnetic field configurations arising from collision of deformed nuclei and show that it can lead to anomalous elliptic flow. Special (crossed body-body) configurations of deformed nuclei collisions can lead to the presence of a quadrupolar magnetic field, which can have very important effects on the rapidity dependence of transverse expansion (similar to beam focusing from quadrupole fields in accelerators)."

"We will discuss various aspects of the Ramanujan-Petersson conjecture about the growth of the Fourier coefficients of cusp forms of half-integral weight. This is recent joint work with S. Gun"

"Blazars are active galactic nuclei with strong jets. They tend to exhibit dramatic and unpredictable flux variations, namely outbursts. Certain observed outbursts from an exceptional Blazar OJ287 can be explained by invoking a massive black hole binary as its central engine. Detailed General Relativistic modeling allowed us to predict a major optical outburst during November 2015. The outburst did occur within the expected time range, peaking on 5/12/2015. A multi-wavelength observational campaign confirmed the occurrence of certain impact flare and the presence of a major thermal component in the flare, as predicted. These observations and subsequent analysis allowed us to establish the presence of a spinning supermassive black hole binary that spirals in due to the emission of nano-Hertz gravitational waves in the central engine of OJ287. I will briefly list our on-going efforts that should be interesting to the Event Horizon Telescope consortium and the International Pulsar Timing Array."