In recent years, the classification of fermionic symmetry protected topological phases has led to renewed interest in classical constructions of invariants in homotopy theory. In this talk, we focus on the description of Steenrod squares for triangulated spaces at the cochain level, introducing new formulas for the cup-i products and discussing their universality through an axiomatic approach. We also examine the interaction between Steenrod squares and the algebra structure in cohomology, providing a cochain level proof of the Cartan relation as requested by Kapustin. Time permitting, we will also study the Adem relation from this perspective.
In their seminal 1977 paper, Gibbons and Hawking (GH) audaciously applied concepts of quantum statistical mechanics to ensembles containing black holes, finding that a semiclassical saddle point approximation to the partition function recovers the laws of black hole thermodynamics. In the same paper they insouciantly applied the formalism to the case of boundary-less de Sitter space (dS), obtaining the expected temperature and entropy of the static patch. To what ensemble does the dS partition function apply? And why does the entropy of the dS static patch decrease upon addition of Killing energy? I’ll answer these questions, and then generalize the GH method to find the approximate partition function of a ball of space at any fixed proper volume. The result is the exponential of the Bekenstein-Hawking entropy of its boundary.
Fluid mechanics has proven to be remarkably successful in describing a wide variety of substances, both familiar and exotic. The latter category includes relativistic fluids, often arising in the most extreme regimes found anywhere in the universe. One such example is the quark-gluon plasma (QGP) formed in collisions of heavy ions, which exists at temperatures hot enough to “melt” hadrons; another is the matter composing neutron stars, whose density is comparable to that of an atomic nucleus. Beyond the surprising fact that the aforementioned substances act as fluids, they share an additional similarity in that they may both be measurably viscous, a feature accounted for in models of the QGP but almost never in neutron star simulations.
In this talk I will overview progress toward the incorporation of dissipative effects such as viscosity into relativistic fluid models of astrophysical systems. I will begin by reviewing the modern inter- pretation of fluid mechanics as a gradient expansion about thermodynamic equilibrium, and will discuss the nuances of constructing a theory compatible with beyond-equilibrium thermodynamics and general relativity. I will then define and motivate a promising new formulation of relativistic dissipative hydrodynamics known as BDNK theory before summarizing recent work toward its application in models of neutron stars.
In the context of irreversible dynamics, the meaning of the reverse of a physical evolution can be quite ambiguous. It is a standard choice to define the reverse process using Bayes' theorem, but, in general, this is not optimal with respect to the relative entropy of recovery. In this work we explore whether it is possible to characterise an optimal reverse map building from the concept of state retrieval maps. In doing so, we propose a set of principles that state retrieval maps should satisfy. We find out that the Bayes inspired reverse is just one case in a whole class of possible choices, which can be optimised to give a map retrieving the initial state more precisely than the Bayes rule. Our analysis has the advantage of naturally extending to the quantum regime. In fact, we find a class of reverse transformations containing the Petz recovery map as a particular case, corroborating its interpretation as a quantum analogue of the Bayes retrieval.
Finally, we present numerical evidence showing that by adding a single extra axiom one can isolate for classical dynamics the usual reverse process derived from Bayes' theorem.