20 problems

The paper constructs an explicit family of $2$-query locally testable codes with linear distance. Constant-rate conjecture. For an appropriate choice of parameters, this family has…

Let $obreakforall obreaktext{?}$ For every $obreaktext{?}$

Good-distance conjecture. The code described in this section nevertheless has good distance.

Parallel LTC amplification conjecture. There exists such a family for which the construction gives a parallel variant of the theorem on LTC amplification:

Let $\binom{[n]}{k}$ denote the family of $k$-element subsets of $[n]=\{1,\ldots,n\}$. For a family $X_n\subseteq\binom{[n]}{k}$, let the local agreement property be the condition…

A quantum locally testable code is a quantum error-correcting code whose validity can be tested through local checks. Quantum-LTC conjecture. High-dimensional expansion will be a k…

A locally testable code (LTC) is an error-correcting code for which membership can be tested by querying only a small number of coordinates. A genuine high-dimensional expander is…

Locally testable codes (LTCs) are error-correcting codes whose membership can be tested by inspecting only a small number of coordinates; a code has high rate when its rate is boun…

Let $X$ be a $2$-dimensional simplicial complex, let ${\mathcal{F}}$ be a sheaf on $X$, and use the notation of the modification process, including the subspaces $E_r$, $E'_r$,…

Let $d,q\in\mathbb{N}$ with $d\geq 2$, and let $f:\mathbb{N}\cup\{0\}\to\mathbb{N}$. Let $X$ be covered by a $q$-thick affine building of dimension $d$, and let ${\mathcal{F}}$ be…

Let $X$ be a simplicial complex and let ${\mathcal{F}}$ be a sheaf on $X$. Consider the linear map … Assume that this map is injective and that…

Let $X$ be a sheaved high-dimensional expander and let the modification process alter its sheaf by successively eliminating excess cohomology classes. Say that the process terminat…

Consider expanding posets, including the Grassmann poset, and simplicial complexes equipped with local-spectral expansion. Special-class conjecture. Simplicial complexes and struct…

Let $X$ be a high-dimensional expander and let $f$ be a function subject to the paper's pseudorandomness condition, except that the condition is weakened by dropping the assumption…

Let $X$ be a sufficiently strong two-sided locally spectral expander, and let $f$ be a Boolean function on $X$. Write $I[f]$ for its total influence, and let $\tau$ be a face whose…

Let the underlying object be a locally spectral expanding structure, such as a simplicial complex or a more general ranked poset, and consider hypercontractivity beyond the second…

Let $F$ and $F'$ be non-archimedean local fields with the same residue order, and let $\mathbb{P}_{\mathrm{fr}}^{2}(\mathcal{O}_{r})$ and…

Let $q$ be a prime power, let $d\in\mathbb{N}$, and let $0<\gamma<1$. A $d$-dimensional measured poset $X$ is sparse when $|X(0)|=n$ and $|X(d)|=O(n)$, where the implied constant m…

Let $K$) be a non-Archimedean local field, let $\utilde G$ be a $K$-simple simply connected group of $K$-rank $r$, set $G=\utilde G(K)$, and let $\Gamma$ be a torsion-free cocompac…

Let $\mathcal{B}$ be the Bruhat–Tits building associated with $\mathrm{PGL}_{d}(F)$, where $F$ is a local field and $d\geq3$. For a finite quotient $X$ of $\mathcal{B}$, let…