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Another important interdisciplinary approach in ecology is the search for large-scale or even "universal" empirical patterns with simple explanations, such as distribution of energy and biomass across the world and across various types of organisms (plants, herbivores, carnivores...). This lecture will present some insights and limits of simple deterministic patterns constrained by biological mechanisms.

Understanding the semiclassical limit of the quantization of even the simplest classically chaotic Hamiltonian, proved to be problematic from the inception of quantum mechanics 100 years ago. Arithmetically defined such systems provide instances for which this limit can be analyzed mathematically using central tools from number theory and homogeneous dynamics.As such these can be thought of as "solvable models ". We review some of the many developments for these quantizations.

1. basics on RV (probability, moments, mgf, continuous RVs)
2. summing RVs and central limit theorem
3. sampling (binomial, poisson)
4. intro on stochastic processes
5. random walk
6. birth-death process (basics and exctintiion)
7. birth-death-migration process (stationary probability)

In 1924, Bose communicated a derivation of the Planck distribution, to Einstein, where Bose introduced a key notion of indistinguishability of photon quanta. This was a turning point in the history of quantum mechanics. Bose's article is considered the fourth most important article, in the development of quantum mechanics, following those of Planck, Einstein and Niels Bohr. It was Dirac, who coined the names Bosons and Fermions.  We present a brief historical account of Bose’s discovery, followed by a bird’s eye view of the impacts of boson bloom and Bose-Einstein condensation in modern science and technology.

[1] Boson Bloom, G. Baskaran and A. May, J. Physics B, 57, 142001 (2024)

Quantum mechanics is a cornerstone of modern physics. Just as the 19th century was called the Machine Age and the 20th century the Information Age, the 21st century promises to go down in history as the Quantum Age. Quantum Computing promises unprecedented speed in solving certain classes of problems while Quantum Cryptography promises information theoretically secure communications. In this talk, I will discuss the world of single and entangled photons and also discuss ongoing work towards secure quantum communications, quantum information and precision experimental tests of principles of quantum mechanics in our Quantum Information and Computing lab at the Raman Research Institute, Bengaluru, India. I will end with our broad vision for the future, which includes establishment of long distance secure quantum communications in India and beyond involving satellite based, fibre based as well as quantum memory and repeater network based approaches towards the global quantum internet.
 

One important interdisciplinary approach of ecological systems is the "complexity" perspective, which is represented by many lectures in this school. This lecture covers complementary ideas about networks of species interactions, their structures, and the use of randomness to understand their general features.

In this talk we will review the dynamics of warm inflation and the effect of dissipation on the primordial spectrum. When inflation occurs in the so-called strong dissipative regime, this effect will lead to the enhancement of the spectrum over all scales. An in particular to large enough fluctuations over the last 10-20 efolds of inflation which on re-entry may form Primordial Black Holes and act as a source for the tensors at second order. We will discuss different realisations consistent with CMB observations, i.e., different combinations of potentials and dissipative coefficients. Typically the enhancement is maximal near the end of inflation, which result in a spectral density of Gravitational Waves (GW) peaked at large frequencies today, in the range of MHz. Although the frequency range is outside the reach of present and planned GW detectors, it might be reached in future high-frequency gravitational waves detectors, designed to search for cosmological stochastic GW backgrounds above MHz frequencies.