11 problems

Let $L$ denote the set of all real random variables $X$ with a log-concave probability density function such that $EX=0$ and $EX^2=1$. Let $Y$ have the standard exponential distrib…

Concentration conjecture. Under mild conditions, the probability that the number of real zeros deviates from its expectation by at least $\varepsilon\log n$ is bounded above and be…

Let $G$ be a definable group in an o-minimal structure, let $S\subset G$ be a definable subset avoiding unboundedly large arithmetic progressions, and let $k=\dim(S)$. Consider the…

Concentration conjecture. There are positive constants $c_1$ and $c_2$ such that for every $n$, $k$, $P$, and $\varepsilon\geq 0$,

Leader–Radcliffe's conjecture.

The improved binomial MGF conjecture. The function

Let $P$ be a distribution on a finite alphabet of size $k$, let $\hat{P}_{n,k}$ be its empirical distribution from $n$ independent samples, and let $D(\hat{P}_{n,k} \| P)$ denote t…

Let $P$ be a distribution on a finite alphabet of size $k$, let $\hat{P}_{n,k}$ be its empirical distribution from $n$ independent samples, and let $D(\hat{P}_{n,k} \\| P)$ denote…

Let $\epsilon=\epsilon(n)$ be the radius of the geodesic balls, let $N_{c,\epsilon(n)}$ be the number of independently and uniformly selected centers used for the covering, and let…

Let $\{U_k,\mathcal{F}_k\}_{k=0}^h$ be the martingale sequence introduced in the subsection, and let $Z_2^{(m)}$ denote the corresponding refined exponential base and…

Uniform concentration inequality conjecture. Any symmetric log-concave probability measure satisfies $\mathrm{CI}(C)$ for some universal constant $C$.