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Abstract: The k-th power Weyl sum is S_k(N,a)=\sum_{n\le N} \exp(2\pi i an^k), where a is a real parameter. The classical bound takes the form O_{k,a,c}(N^c), for any c > 1-2^{1-k}, whenever a is well-approximable by rationals. This is best possible for k=2, and has not been improved for 100…

Arnold conjecture says that the number of 1-periodic orbits of a Hamiltonian diffeomorphism is greater than or equal to the dimension of the Hamiltonian Floer homology. In 1994, Hofer and Zehnder conjectured that there are infinitely many periodic orbits if the equality doesn't hold. In this…

Sampling from high-dimensional distributions is a notoriously difficult problem, especially when the distribution isn't log-concave or has multiple modes. While Markov chain Monte Carlo is a powerful approach, new tools are much needed. I will present a different class of algorithms which is…

Abstract: In Newtonian gravity, a self-gravitating gas around a massive object such as a star or a planet is modeled via Vlasov Poisson equation with an external Kepler potential. The presence of this attractive potential allows for bounded trajectories along which the gas neither falls in…

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The Kauffman bracket skein module S(M) is an invariant of a 3-manifold M, which was independently introduced by Turaev and Przytycki as a generalization of the Jones polynomial of knots. It can be defined as a quotient of the free Z[A,1/A]-module spanned by isotopy classes of links in M modulo…

Abstract: Classical algorithms are often not effective for solving nonconvex optimization problems where local minima are separated by high barriers. In this talk, we introduce optimization algorithms based on quantum dynamical systems. On the one hand, we leverage the global effect of quantum…

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