Related papers: The Kannan-Lov\'asz-Simonovits Conjecture
An upper bound on the ergodic capacity of {\bf MIMO} channels was introduced recently in arXiv:0903.1952. This upper bound amounts to the maximization on the simplex of some multilinear polynomial $p(\lambda_1,...,\lambda_n)$ with…
The problem of efficiently generating random samples from high-dimensional and non-log-concave posterior measures arising from nonlinear regression problems is considered. Extending investigations from arXiv:2009.05298, local and global…
We establish the volume conjecture for (m,2)-cables of the figure 8 knot, when m is odd. For (m,2)-cables of general knots where m is even, we show that the limit in the volume conjecture depends on the parity of the color (of the Kashaev…
The Wigner-Dyson-Gaudin-Mehta conjecture asserts that the local eigenvalue statistics of large real and complex Hermitian matrices with independent, identically distributed entries are universal in a sense that they depend only on the…
The Weinstein conjecture, as the general existence problem for periodic orbits of Hamiltonian or Reeb flows, has been among the central questions in symplectic topology for over two decades and its investigation has led to understanding of…
The Conifold Gap Conjecture asserts that the polar part of the Gromov-Witten potential of a Calabi-Yau threefold near its conifold locus has a universal expression described by the logarithm of the Barnes $G$-function. In this paper, I…
We prove the generalized Franchetta conjecture for the locally complete family of hyper-K\"ahler eightfolds constructed by Lehn-Lehn-Sorger-van Straten (LLSS). As a corollary, we establish the Beauville-Voisin conjecture for very general…
Howard's conjecture, which states that in the linear instability problem of inviscid heterogeneous parallel shear flow growth rate of an arbitrary unstable wave must approach zero as the wave length decreases to zero, is established in a…
Estimates of densities of convolution semigroups of probability measures are given under specific assumptions on the corresponding L\'evy measure and the L\'evy--Khinchin exponent. The assumptions are satisfied, e.g., by tempered stable…
The purpose of this short note is to demonstrate uniform logarithmic Sobolev inequalities for the mean field gradient particle systems associated to an energy functional that is convex in the flat sense. A defective log-Sobolev inequality…
Kneser's 1955 conjecture -- proven by Lov\'asz in 1978 -- asserts that in any partition of the $k$-subsets of $\{1, 2, \dots, n\}$ into $n-2k-3$ parts, one part contains two disjoint sets. Schrijver showed that one can restrict to…
We study the log-concave measures, their characterization via the Pr\'ekopa-Leindler property and also define a subset of it whose elements are called super log-concave measures which have the property of satisfying a logarithmic Sobolev…
Kiefer and Wolfowitz [Z. Wahrsch. Verw. Gebiete 34 (1976) 73--85] showed that if $F$ is a strictly curved concave distribution function (corresponding to a strictly monotone density $f$), then the Maximum Likelihood Estimator $\hat{F}_n$,…
We prove the discrete analogue of Kakeya conjecture over $\mathbb{R}^n$. This result suggests that a (hypothetically) low dimensional Kakeya set cannot be constructed directly from discrete configurations. We also prove a generalization…
Recently, the authors proved [2] that the Maxwell-Stefan system with an incompressibility-like condition on the total flux can be rigorously derived from the multi-species Boltzmann equation. Similar cross-diffusion models have been widely…
The Wigner-Gaudin-Mehta-Dyson conjecture asserts that the local eigenvalue statistics of large random matrices exhibit universal behavior depending only on the symmetry class of the matrix ensemble. For invariant matrix models, the…
We prove that the planar hexagonal honeycomb is asymptotically optimal for a large class of optimal partition problems, in which the cells are assumed to be convex, and the criterion is to minimize either the sum or the maximum among the…
We study the local convergence of diffusive mean-field systems, including Wasserstein gradient flows, min-max dynamics, and multi-species games. We establish exponential local convergence in $\chi^2$-divergence with sharp rates, under two…
We extend to the framework of convergence in concentration virtually all the results concerning stability of Sobolev functions and differential operators known to be in place under the stronger measured-Gromov-Hausdorff convergence. These…
We prove Sarnak's density conjecture for the principal congruence subgroup of SL_n(Z) of squarefree level and discuss various arithmetic applications. The ingredients include new bounds for local Whittaker functions and Kloosterman sums.