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We show that the Ryu-Takayanagi (RT) formula, originally introduced to compute the entropy of a holographic boundary CFT, can be interpreted as entropy of an algebra of bulk gravitational observables. In particular, we show that any Euclidean gravitational path integral satisfying a simple and familiar set of axioms defines type I von Neumann algebras of bulk observables acting on closed codimension-2 asymptotic boundaries. The entropies associated to these algebras, defined via the gravitational path integral, can be written in terms of standard density matrices and standard Hilbert space traces, and in appropriate semiclassical limits are computed by the RT formula with quantum corrections. Our work thus provides a bulk state-counting interpretation of the Ryu-Takayanagi entropy. Since our axioms do not severely constrain UV bulk structures, they may be expected to hold equally well for successful formulations of string field theory, spin-foam models, or any other approach to constructing a UV-complete theory of gravity.

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One way of constructing mirror partners to Fano varieties is via toric degenerations. The case in which this is best understood is the Grassmannian, using the well-known SAGBI basis of the Plucker coordinate ring indexed by semi-standard Young tableaux (SSYT). The mirror construction goes back to work of Eguchi—Hori—Xiong, however its geometry and combinatorics still plays an important role in current mirror constructions. In this talk, I will give an overview of this story, then turn to the question of what can be generalized for Kronecker moduli spaces. Like Grassmannians (which they generalize), Kronecker moduli spaces are high Fano index Picard rank 1 smooth Fano varieties. I will introduce linked SSYT pairs, which play the analogous role of SSYT for Grassmannians in understanding the coordinate ring of the Kronecker moduli space. This is joint work with Liana Heuberger.

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We investigate the recently found \cite{migdal2023exact} reduction of decaying turbulence in the Navier-Stokes equation in $3 + 1$ dimensions to a  Number Theory problem of finding the statistical limit of the Euler ensemble.
We reformulate the Euler ensemble as a Markov chain and show the equivalence of this formulation to the quantum statistical theory of free fermions on a ring, with an external field related to the random fractions of $\pi$. 

We find the solution of this system in the statistical limit $N\to \infty$ in terms of a complex trajectory (instanton) providing a saddle point to the path integral over the charge density of these fermions.

This results in an analytic formula for the observable correlation function of vorticity in wavevector space. This is a full solution of decaying turbulence from the first principle without assumptions, approximations, or fitted parameters.
We compute resulting integrals in \Mathematica{} and present effective indexes for the energy decay as a function of time Fig.\ref{fig::NPlot} and the energy spectrum as a function of the wavevector at fixed time Fig.\ref{fig::SPIndex}. 

In particular, the asymptotic value of the effective index in energy decay $n(\infty) = \frac{7}{4}$, but the universal function $n(t)$ is neither constant nor linear.

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