Results 91 - 100 of 24638
We present here the first part of Jorge Garza-Vargas’s three-part mini-course on the strong convergence phenomenon, from our ICM Satellite Conference on Spectral Theory, High-Dimensional Expansion, and Pseudorandomness. The course introduces and motivates the notion of strong convergence, discusses recent proof techniques, and showcases concrete applications.
High-dimensional expansion (HDX) is a generalization of expansion in graphs to higher dimensions (i.e., hypergraphs). In this first of three lectures from our ICM satellite conference, Mitali Bafna defines the spectral notion of HDX, proves the trickle-down theorem, and defines random walks on HDX.
In this presentation from this fall’s joint boot camp for the programs on Spectral Theory Beyond Graphs and on Pseudorandomness & High-Dimensional Expansion, William Hoza presents an introduction to the “L vs. BPL” problem, which asks whether randomness is ever necessary for space-efficient computation.
Eugenia Sapir’s mini-course provides an introduction to hyperbolic surfaces, including their basic topology and (hyperbolic) geometry, several random models for surfaces (some of which are directly defined using random graphs), and the ways in which many results about random surfaces parallel those for random graphs.
Motivated by the problem of the small-scale sign distribution of Laplace eigenfunctions, we introduce a strong notion of sign-balance for (eigen)functions, and prove that random eigenfunctions are sign-balanced above a precisely determined scale with almost full probability. The scale is proven to be optimal up to a logarithmic power of the energy. Our results include the important case of random spherical harmonics, as well as more general band-limited random waves on smooth Riemannian manifolds. Extending the notion of balance to arbitrary levels, we determine the precise optimum scale above which random eigenfunctions are volume-balanced with respect to non-zero levels. Beyond their intrinsic interest, our results serve as a model for a natural conjecture on the optimal scale at which deterministic Laplace eigenfunctions are sign-balanced. This talk is based on a joint work with S. Muirhead.
Can we improve a random sample by modifying just a tiny fraction of its points? This fascinating question brings together statistics (which motivates it), probability theory (which gives us the tools), and discrepancy theory (where it takes a deterministic form). I’ll describe a series of joint works with Gleb Smirnov that led us on an interesting detour into information theory. Our central idea was recently used by AI to solve the long-standing Komlós conjecture.
The study of diffusions on Lie groups has a rich history dating back to 1950s when they were first introduced by Itô. In this talk, I will focus on a family of multiplicative Brownian motions B^N(t) on the General Linear Group of degree N, i.e. at each time t, B^N(t) is an N by N random matrix. In 1997, Philippe Biane conjectured that as N —> infinity the empirical spectral measure of B^N(t) converges to the Brown measure of a free multiplicative Brownian motion at time t, which is a diffusion on a certain infinite-dimensional space. With Nick Cook, Todd Kemp and Félix Parraud we confirm this conjecture.