I will discuss Dye's Theorem and the uniqueness of hyperfinite equivalence relations following mostly [Kerr-Li] Section 4.9.

For more details and the exact reference, click here.  

Suppose $u_j$ is an orthonormal basis of eigenfunctions of the Laplacian on a compact Riemannian manifold, and consider the probability measures $|u_j|^2 dvol$. The quantum ergodicity theorem of Shnirelman, Zelditch and Colin de Verdiere, states that if the geodesic flow on the manifold is ergodic then there is a density one subsequence of these measures that equidistributes. What about the remaining eigenfunctions? The quantum unique ergodicity conjecture of Rudnick and Sarnak states that for hyperbolic manifolds (or more generally for negatively curved manifolds), any sequence of measures as above equidistributes.  In 2006 Lindenstrauss proved that this is true for Hecke—Maass forms on congruence surfaces. These are eigenfunctions of both the Laplacian and the Hecke operators, which are discrete averaging operators coming from the arithmetic structure of the manifold. 

We will discuss our extensions of Lindenstrauss’ results to the three and (partially) four dimensional cases. One of the challenges that arise in the higher dimensional case is to rule out concentration of measure on totally geodesic submanifolds. We will focus on this issue and our new methods for showing non-concentration. 

Based on joint works with Alexandre de Faveri and Lior Silberman.

How likely is $G(n;p)$ to have a less-than-typical number of triangles? This is a foundational question in non-linear large deviation theory. When $p \ll 1/\sqrt{n}$ or $p \gg 1/\sqrt{n}$ the answer is fairly well-understood, with Janson's inequality applying in the former case and regularity- or container-based methods applying in the latter. We study the regime $p=c/\sqrt{n}$, with $c$ fixed, with the large deviation event having at most $\eta$ times the expected number of triangles, for a fixed $\eta \in [0,1)$.

We prove explicit formula for the log-asymptotics of the event in question, for a wide range of pairs $(c,\eta)$. In particular, we show that for sufficiently small $\eta$ (including the triangle-free case $\eta=0$) there is a phase transition as $c$ increases, in the sense of a non-analytic point in the rate function. On the other hand, if $\eta > 1/2$, then there is no phase transition.

As corollaries, we obtain analogous results for the $G(n;m)$ model, when $m = Cn^{3/2}$. In contrast to the $G(n;p)$ case, we show that a phase transition occurs as $C$ increases for \textit{all} $\eta$.

Finally, we show that the probability of $G(n;m)$ being triangle free, where $m=Cn^{3/2}$ for a sufficiently small constant $C$, conforms to a Poisson heuristic.

Joint with Matthew Jenssen, Will Perkins, and Aditya Potukuchi. Based on arXiv:2410.22951 and arXiv:2411.18563.

Threshold cryptography has been fundamental to secure distributed protocols for over three decades. However, it often comes at the expense of accountability: when secret information is shared among multiple parties, it can be difficult to determine who is at
fault if this information is leaked or misused.
In this talk, I will present a recent line of works that demonstrate that this trade-off is not inherent—we can indeed build accountable threshold cryptosystems. Most of the talk will focus on accountability in secret sharing. Suppose Alice uses a t-out-of-n secret sharing scheme to store her secret key on n servers. This guarantees that the servers learn nothing about her secret key, even if t-1 of them collude. But what happens if some servers decide to sell their shares? In this case, Alice should be able to hold them accountable; otherwise, they have a risk-free incentive to sell her shares. A secret sharing scheme that allows Alice to trace the leak back to the corrupted servers is called a traceable secret sharing scheme. I will present new definitions and practical constructions for traceable secret sharing, based on the natural and widely-used schemes of Shamir and Blakley.