One of the biggest open problems in mathematical physics has been the problem of formulating a complete and consistent theory of quantum gravity. Some of the core technical and epistemological difficulties come from the fact that General Relativity is fundamentally a geometric theory and, as such, it ought to be invariant under change of coordinates by the arbitrary element of the diffeomorphism group Diff(M) of the ambient manifold M.
We discuss several recent results for Navier-Stokes equation in EMAC form, including longer time accuracy through an improved Gronwall estimate, a lower bound on error for forms that do not conserve momentum and angular momentum, exact local discrete balances of momentum and angular momentum with EMAC, and lastly the importance of consistency in ROMs for NSE using EMAC.
This talk begins by discussing the role of PDE-constrained optimization in the development of digital twins. In particular, applications to identify weaknesses in structures and aneurysms are considered. Next, we analyze a data-driven optimization problem constrained by Darcy’s law to design a permeability that achieves uniform flow properties despite having nonuniform geometries. We establish well-posedness of the problem, as well as differentiability, which enables the use of rapidly converging, derivative-based optimization methods.
Arithmetic statistics is an area devoted to counting a wide range of objects of algebraic interest, such as polynomials, fields, and elliptic curves. Fueled by the interplay of analysis and number theory, we'll count polynomials and number fields, which though basic objects of study in number theory, are quite difficult to actually count. How often does a random polynomial fail to have full Galois group? How many number fields are there? We will address both of these questions today.
We focus on the history and the mathematics of the apportionment problem for the US House of Representatives. An apportionment is a function from {1, 2, ..., s} (where s is the number of states) to the positive integers, A(i), so that the sum of the A(i) is H, the house size. Today s=51 (including the District of Columbia) and H=435, although s and H have had many values since 1790. Based on census data, one can compute the fair share of representatives for state i, call it f_i, which might turn out to be 4.69435.
This final talk in the series shall discuss the problem of geodesic deviation for the sub-Riemannian metric on the three-sphere, and how it relates to the problem of null geodesic deviation on the non-commutative gravitomagnetic monopole.