A Log-Sobolev Inequality for the Multislice, with Applications

Let kappa in N_+^l satisfy kappa_1 + *s + kappa_l = n, and let U_kappa denote the multislice of all strings u in [l]^n having exactly kappa_i coordinates equal to i, for all i in [l]. Consider the Markov chain on U_kappa where a step is a random transposition of two coordinates of u. We show that the log-Sobolev constant rho_kappa for the chain satisfies rho_kappa^{-1} <= n * sum_{i=1}^l 1/2 log_2(4n/kappa_i), which is sharp up to constants whenever l is constant. From this, we derive some consequences for small-set expansion and isoperimetry in the multislice, including a KKL Theorem, a Kruskal - Katona Theorem for the multislice, a Friedgut Junta Theorem, and a Nisan - Szegedy Theorem

A sharp log-Sobolev inequality for the multislice

We determine the log-Sobolev constant of the multi-urn Bernoulli-Laplace diffusion model with arbitrary parameters, up to a small universal multiplicative constant. Our result extends a classical estimate of Lee and Yau (1998) and confirms a conjecture of Filmus, O'Donnell and Wu (2018). Among other applications, we completely quantify the "small-set expansion" phenomenon on the multislice, and obtain sharp mixing-time estimates for the colored exclusion process on various graphs.Comment: 23 pages, comments welcom

Mathematical Challenges in Electron Microscopy

Development of electron microscopes first started nearly 100 years ago and they are now a mature imaging modality with many applications and vast potential for the future. The principal feature of electron microscopes is their resolution; they can be up to 1000 times more powerful than a visible light microscope and resolve even the smallest atoms. Furthermore, electron microscopes are also sensitive to many material properties due to the very rich interactions between electrons and other matter. Because of these capabilities, electron microscopy is used in applications as diverse as drug discovery, computer chip manufacture, and the development of solar cells. In parallel to this, the mathematical field of inverse problems has also evolved dramatically. Many new methods have been introduced to improve the recovery of unknown structures from indirect data, typically an ill-posed problem. In particular, sparsity promoting functionals such as the total variation and its extensions have been shown to be very powerful for recovering accurate physical quantities from very little and/or poor quality data. While sparsity-promoting reconstruction methods are powerful, they can also be slow, especially in a big-data setting. This trade-off forms an eternal cycle as new numerical tools are found and more powerful models are developed. The work presented in this thesis aims to marry the tools of inverse problems with the problems of electron microscopy: bringing state-of-the-art image processing techniques to bear on challenges specific to electron microscopy, developing new optimisation methods for these problems, and modelling new inverse problems to extend the capabilities of existing microscopes. One focus is the application of a directional total variation to overcome the limited angle problem in electron tomography, another is the proposal of a new inverse problem for the reconstruction of 3D strain tensor fields from electron microscopy diffraction data. The remaining contributions target numerical aspects of inverse problems, from new algorithms for non-convex problems to convex optimisation with adaptive meshes.Cantab Capital Institute for Mathematics of Informatio

Hypercontractivity on high dimensional expanders

We prove hypercontractive inequalities on high dimensional expanders. As in the settings of the p-biased hypercube, the symmetric group, and the Grassmann scheme, our inequalities are effective for global functions, which are functions that are not significantly affected by a restriction of a small set of coordinates. As applications, we obtain Fourier concentration, small-set expansion, and Kruskal–Katona theorems for high dimensional expanders. Our techniques rely on a new approximate Efron–Stein decomposition for high dimensional link expanders