"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)
"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)."
"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."