Departmental Colloquium

We first develop random batch methods for classical interacting particle systems with large number of particles. These methods use small but random batches for particle interactions, thus the computational cost is reduced from O(N^2) per time step to O(N), for a system with N particles with binary interactions. For one of the methods, we give a particle number independent error estimate under some special interactions. This method is also extended to molecular dynamics with Coulomb interactions, in the framework of Ewald summation, and to quantum Monte-Carlo methods for the N-body Schrodinger equation. In each case we will show its superior performance compared to the current state-of-the-arts methods for the corresponding problems, in the computational efficiency and parallelizability. This talk is based on joint works with Lei Li, Jian-Guo Liu, Zhenli Xu, and Xiaotao Li.

Approximations via conforming and non-conforming finite elements can be used to construct validated and computable bounds on eigenvalues for the Dirichlet Laplacian in certain domains. If these are to be used as part of a proof, care must be taken to ensure each step of the computation is validated and verifiable. In this talk we present a long-standing conjecture in spectral geometry, and its resolution using validated finite element computations. Schiffer's conjecture states that if a bounded domain Ω in R^n has any nontrivial Neumann eigenfunction which is a constant on the boundary, then Ω must be a ball. This conjecture is open. A modification of Schiffer's conjecture is: for regular polygons of at least 5 sides, we can demonstrate the existence of a Neumann eigenfunction which does not change sign on the boundary. In this talk, we provide a recent proof using finite element calculations for the regular pentagon. The strategy involves iteratively bounding eigenvalues for a sequence of polygonal subdomains of the triangle with mixed Dirichlet and Neumann boundary conditions. We use a learning algorithm to find and optimize this sequence of subdomains, and use non-conforming linear FEM to compute validated lower bounds for the lowest eigenvalue in each of these domains. The linear algebra is performed within interval arithmetic. This talk is based on the following paper, which is a joint work with Bartlomiej Siudeja and Ben Green at University of Oregon.

The minimal free resolution of a graded module encodes many geometric properties of the corresponding sheaf on projective space. However, when the ambient space is a product of projective spaces or a more general smooth projective toric variety X, minimal free resolutions over the Cox ring are too long and contain many geometrically superfluous summands. In joint work with Daniel Erman and Gregory G. Smith, we propose considering instead virtual resolutions, which more closely reflect the geometry of sheaves on X. We will survey recent results which build this case.

There is a unique line through 2 points in the plane, and a unique conic through 5. These counts generalize to a count of degree d rational curves in the plane passing through 3d-1 points. Surprisingly, the problem of determining these numbers turns out to be deep and connected to string theory, and it was not until the 1990's that Kontsevich determined them with a recursive formula. Such formulas are valid when you allow your curves to be defined with complex coefficients. For fields of characteristic not 2 or 3, we use A1-homotopy theory to show that by counting with bilinear forms, there is an invariant arithmetic count of rational plane curves. This is joint work with Jesse Kass, Marc Levine, and Jake Solomon.