Tootfinder

Quiver varieties and root multiplicities in rank 3
Patrick Chan, Peter Tingley
https://arxiv.org/abs/2508.04804 https://arxiv.org/pdf/2508.04804

Quiver varieties and root multiplicities in rank 3
Building on our previous work in rank two, we use quiver varieties to give a combinatorial upper bound on dimensions of certain imaginary root spaces for rank 3 symmetric Kac-Moody algebras. We describe an explicit method for extracting combinatorics when the Dynkin diagram is bipartite (i.e. two of the nodes are not connected). As in rank two we believe these bounds are quite tight and we give computational evidence to this effect, although there is more error in rank 3 than in rank 2.

Asynchronous Collective Tree Exploration: a Distributed Algorithm, and a new Lower Bound
Romain Cosson, Laurent Massouli\'e
https://arxiv.org/abs/2507.15658

Asynchronous Collective Tree Exploration: a Distributed Algorithm, and a new Lower Bound
We study the problem of collective tree exploration in which a team of $k$ mobile agents must collectively visit all nodes of an unknown tree in as few moves as possible. The agents all start from the root and discover adjacent edges as they progress in the tree. Communication is distributed in the sense that agents share information by reading and writing on whiteboards located at all nodes. Movements are asynchronous, in the sense that the speeds of all agents are controlled by an adversary a…

$TIME[t] \subseteq SPACE[O(\sqrt{t})]$ via Tree Height Compression
Logan Nye
https://arxiv.org/abs/2508.14831 https://arxiv.org/pdf/2508.14831

$TIME[t] \subseteq SPACE[O(\sqrt{t})]$ via Tree Height Compression
We prove a square-root space simulation for deterministic multitape Turing machines, showing $\TIME[t] \subseteq \SPACE[O(\sqrt{t})]$. The key step is a Height Compression Theorem that uniformly (and in logspace) reshapes the canonical left-deep succinct computation tree for a block-respecting run into a binary tree whose evaluation-stack depth along any DFS path is $O(\log T)$ for $T = \lceil t/b \rceil$, while preserving $O(b)$ work at leaves, $O(1)$ at internal nodes, and edges that are logs…