Abstract: While much of quantum cryptography requires communication of quantum states, recent research has proved major theoretical advantages in cryptography using quantum algorithms and classical communication only. This is an active and mathematically rich field which incorporates tools from lattice-based cryptography. The goal of this seminar series is to get a broad view of this area (including ongoing research at QuICS) with the hope of inspiring new ideas and collaboration. We welcome talks on related research directions, including proofs of quantumness and post-quantum cryptography. The seminar homepage is at https://qcrypto.umiacs.io .
Abstract: High dimensional distributions, especially those with heavy tails, are notoriously difficult for off-the-shelf MCMC samplers: the combination of unbounded state spaces, diminishing gradient information, and local moves, results in empirically observed "stickiness" and poor theoretical mixing properties -- lack geometric ergodicity. In this talk, we introduce a new class of MCMC samplers that map the original high-dimensional problem in Euclidean space onto a sphere and remedy these notorious mixing problems. In particular, we develop random-walk Metropolis type algorithms as well as versions of the Bouncy Particle Sampler that are uniformly ergodic for a large class of light and heavy-tailed distributions and also empirically exhibit rapid convergence in high dimensions. In the best scenario, the proposed samplers can enjoy the ``blessings of dimensionality'' that the mixing time decreases with dimension.
(join work with Krzysztof Łatuszyński and Gareth O. Roberts)
Abstract: Many machine learning algorithms have been developed to elicit information from large datasets, however, techniques for quantifying the uncertainty in the estimates and for conducting statistical inference are not as well developed. We discuss resampling inferential techniques for these tasks, and present a unified way of studying the theoretical properties of these techniques using a framework involving random resampling weights. Depending on distributional properties of these random weights, the framework we propose can be used for consistent inference under very general conditions involving dependent data, high or infinite dimensional parameters and estimators obtained as approximate solutions to optimization problems, or alternatively for very precise and accurate inference in several other problems. Thus, our framework can balance the desirable robustness and efficiency goals of statistical inference. We further extend the framework for computational efficiency in big data applications, and prove that the extended framework is both uniformly consistent as well as computationally efficient under mild conditions. Illustrative examples from climate sciences, biomedical applications and ethical data sciences are discussed.
Abstract: Magic angles are a hot topic in condensed matter physics: when two sheets of graphene are twisted by those angles the resulting material is superconducting. I will present a very simple operator whose spectral properties are thought to determine which angles are magical. It comes from a 2019 PR Letter by Tarnopolsky--Kruchkov--Vishwanath. The mathematics behind this is an elementary blend of representation theory (of the Heisenberg group in characteristic three), Jacobi theta functions and spectral instability of non-self-adjoint operators (involving Hörmander's bracket condition in a very simple setting). Recent mathematical progress also includes the proof of existence of generalized magic angles and computer assisted proofs of existence of real ones (Luskin--Watson, 2021). The results will be illustrated by colourful numerics which suggest many open problems (joint work with S Becker, M Embree, J Wittsten in 2020 and S Becker, T Humbert and M Hitrik in 2022).
Abstract: A monovariant is a quantity which is either non-increasing or non-decreasing, such as the number of primes up to $x$, but not the number of factors of $n$. Many challenging problems can be solved by associating the right monovariant to it; unfortunately it is often challenging to find the right quantity to study. After describing classic problems such as the Zombie Apocalypse and Conway's Soldiers we turn to recent applications. Zeckendorf proved every positive integer can be written as a sum of non-adjacent Fibonacci numbers (1, 2, 3, 5, 8, ...); using mono-variants we can show no decomposition as a sum of Fibonacci numbers has fewer summands, and discuss generalizations to other sequences. These are key ingredients in analyzing a game involving Fibonacci numbers, where we can prove Player 2 has a winning strategy but it is not known what it is. This work only requires elementary mathematics and is joint with many students (I will mention research opportunities, such as the Polymath Jr REU, for students this summer); there is a $500 reward for a constructive proof of Player 2's winning strategy!
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