arXiv subjects

David Narváez

Publications and source records attributed to David Narváez.

2 recordsLinked to original sources

arXiv2023

The Resolution of Keller's Conjecture

We consider three graphs, $G_{7,3}$, $G_{7,4}$, and $G_{7,6}$, related to Keller's conjecture in dimension 7. The conjecture is false for this dimension if and only if at least one of the graphs contains a clique of size $2^7 = 128$. We present an automated method to solve this conjecture by encoding the existence of such a clique as a propositional formula. We apply satisfiability solving combined with symmetry-breaking techniques to determine that no such clique exists. This result implies that every unit cube tiling of $\mathbb{R}^7$ contains a facesharing pair of cubes. Since a faceshare-free unit cube tiling of $\mathbb{R}^8$ exists (which we also verify), this completely resolves Keller's conjecture.

math.CO↗

arXiv2021

On Lev Gordeev's "On P Versus NP"

In the paper "On P versus NP," Lev Gordeev attempts to extend the method of approximation, which successfully proved exponential lower bounds for monotone circuits, to the case of De Morgan Normal (DMN) circuits. As in Razborov's proof of exponential lower bounds for monotone circuits, Gordeev's work is focused on the NP-complete problem CLIQUE. If successful in proving exponential DMN circuit lower bounds for CLIQUE, Gordeev would prove that P $\neq$ NP. However, we show that Gordeev makes a crucial mistake in Lemma 12. This mistake comes from only approximating operations over positive circuit inputs. Furthermore, we argue that efforts to extend the method of approximation to DMN circuits will need to approximate negated inputs as well.

cs.CC↗