Astrophysical jets associated with supermassive black holes (BHs) are believed to derive their power from the rotational energy of the BH itself, akin to how the Crab Nebula is powered by its pulsar. The Blandford-Znajek (BZ) mechanism, an electromagnetic Penrose process, provides a framework for understanding the physics of jet energetics. Specifically, it predicts the jet efficiency—the ratio of outflowing jet power to inflowing accretion power—to scale quadratically with the magnetic flux at and angular velocity of the black hole horizon. For rapidly spinning Kerr BHs, numerical simulations reveal jet efficiencies exceeding unity, a clear indicator of energy extraction from the black hole. At moderate spins, confirmation of energy extraction relies on the alignment of measured jet efficiencies with the BZ prediction. Over the past decade, this prediction has been validated across Kerr BHs with varying spin values. We present new findings from a large suite of magnetohydrodynamics accretion simulations conducted in spinning non-Kerr spacetimes, demonstrating that the BZ mechanism operates universally, extending its applicability to arbitrary BHs.

I will talk about the statistical mechanics of classical hard-core dimers on the quasiperiodic Ammann-Beenker tiling. After briefly dwelling on the structure of the configuration space, I will discuss its statistical mechanics. By explicitly constructing an RG transformation and implementing it with a Monte-Carlo algorithm, we will show that the model is described by an interacting fixed point, albeit with discrete (instead of continuous) scale invariance. The fixed point theory is described by another hard-core dimer model, unlike dimer models on periodic bipartite lattices which have effective field theory descriptions in terms of unconstrained continuum fields. I will show that the discrete-scale symmetry shows up as log-periodic modulations of power-laws in observables of the overlap loops of the double dimer model.

Topological combinatorics is a very young and exciting field of mathematical research at the crossroads of algebraic topology and discrete mathematics. Over the last forty years, it has gained popularity due to growing applications in math, computer science, and other applied areas. To be precise, this field is concerned with solutions to combinatorial problems related to fair division, graph coloring, evasiveness of graph properties etc., by applying sophisticated topological tools such as the Borsuk-Ulam theorem, characteristic classes, Morse theory and spectral sequences. In this talk, I will focus mainly on the topological spaces that arise in the context of various graph properties and their applications. I will start from the origins of this subject, i.e., Lovasz's striking proof of Kneser's conjecture from 1978. Then, I will describe the neighborhood complexes, independence and matching complexes, also mention some of the important applications. I plan to end with some recent work on newly discovered complexes.