Related papers: The slicing conjecture via small ball estimates
A certain inequality conjectured by Vershynin is studied. It is proved that for any $n$-dimensional symmetric convex body $K$ with inradius $w$ and $\gamma_{n}(K) \leq 1/2$ there is $\gamma_{n}(sK) \leq (2s)^{w^{2}/4}\gamma_{n}(K)$ for any…
We settle the Path Decomposition Conjecture (P.D.C.) due to Tibor Gallai for minimally connected graphs, i.e. trees. We use this validity for trees and settle the P. D. C. using induction on the number of edges for all connected graphs. We…
We make two improvements upon Joyce's gluing theorems of for compact special Lagrangian submanifolds with isolated conical singularities. Firstly, we get rid of a few technical hypotheses of them. Secondly, we replace another hypothesis by…
Bayesian inference provides a framework to combine various model components with shared parameters, allowing joint uncertainty estimation and the use of all available data sources. Unfortunately, misspecification of any part of the model…
Brizolis asked the question: does every prime p have a pair (g,h) such that h is a fixed point for the discrete logarithm with base g? The first author previously extended this question to ask about not only fixed points but also…
The famous (3n + 1) or Collatz conjecture has admitted some progress over the last several decades towards the conclusion that the conjecture is true (i.e. that all Collatz sequences will eventually reach a value of one), but has stubbornly…
We prove Strichartz estimates for the semigroups associated to stratified and/or rotating inviscid geophysical fluids using Fourier restriction theory. We prove new results for rotating stratified fluids, and recover results from [16] for…
In this paper we study singularities in arbitrary characteristic. We propose Finite Determination Conjecture for Mather-Jacobian minimal log discrepancies in terms of jet schemes of a singularity. The conjecture is equivalent to the…
We give a simple, short and self-contained presentation of Bourgain's discretised projection theorem from 2010, which is a fundamental tool in many recent breakthroughs in geometric measure theory, harmonic analysis, and homogeneous…
The goal of this paper is to push forward the study of those properties of log-concave measures that help to estimate their Poincar{\'e} constant. First we revisit E. Milman's result [40] on the link between weak (Poincar{\'e} or…
We recently formulated important Modular Bourgain-Tzafriri Restricted Invertibility Conjectures and Modular Johnson-Lindenstrauss Flattening Conjecture in the Appendix of \textit{[arXiv: 2207.12799.v1]}. For the sake of wide accessibility…
We prove log-concavity of exit probabilities of lattice random walks in certain planar regions.
It is well known that axiom of choice implies the existence of non-measurable sets for Lebesgue's measure on R as well as the existence of "paradoxical" decompositions of the unit ball of R^3 (Banach-Tarski). This is generally interpreted…
Helmut Hofer introduced in '93 a novel technique based on holomorphic curves to prove the Weinstein conjecture. Among the cases where these methods apply are all contact 3--manifolds $(M,\xi)$ with $\pi_2(M) \ne 0$. We modify Hofer's…
We revisit an ingenious argument of K. Ball to provide sharp estimates for the volume of sections of a convex body in John's position. Our technique combines the geometric Brascamp-Lieb inequality with a generalised Parseval-type identity.…
Estimating copulas with discrete marginal distributions is challenging, especially in high dimensions, because computing the likelihood contribution of each observation requires evaluating $2^{J}$ terms, with $J$ the number of discrete…
About forty years ago it was realized by several researchers that the essential features of certain objects of Probability theory, notably Gaussian processes and limit theorems, may be better understood if they are considered in settings…
We prove an estimate for the number of lattice points lying in certain non-convex Euclidean domains of interest in Diophantine approximation. As an application, we generalise a result of Kruse (1964) concerning the almost sure order of…
We prove Strichartz estimates with a loss of derivatives for the Schr\"odinger equation on polygonal domains with either Dirichlet or Neumann homogeneous boundary conditions. Using a standard doubling procedure, estimates the on polygon…
Under a stronger genericity condition, we prove the local analogue of ghost conjecture of Bergdall and Pollack. As applications, we deduce in this case (a) a folklore conjecture of Breuil--Buzzard--Emerton on the crystalline slopes of…