Monday, March 26, 2018
03/26/2018 - 4:00pm to 5:00pm
This talk presents boundary integral equation (BIE)methods and related software for problems with moving boundaries. We will discuss two problems. In the first, we simulate the dynamics of highlyconcentrated vesicle suspensions in a Stokesian fluid. Such fluids are representative of real biological systems and are useful for studying the rheology of blood and other complex biofluids. In the second, we will discuss the problem of computing the equilibrium position of magnetic flux surfaces in magnetically confined plasmas. Such studies enable better designs for devices used for plasma confinement.We will discuss the challenges in developing BIE solvers for these problems\; such as accurate boundary discretization, efficient quadratures for singular and near-singular integration, need for reparameterization/remeshing and scalable implementation. We will present algorithms and software tools to address these issues.
03/26/2018 - 4:15pm to 5:15pm
Fix two positive integers a and b. Scott showed that a homogeneous coordinate ring of the Grassmannian Gra, a+b has the structure of a cluster algebra. This homogeneous coordinate ring can be decomposed into a direct sum of irreducible representations of GLa+b which correspond to integer multiples of the fundamental weight wa. By proving the Fock-Goncharov cluster duality conjecture for the Grassmannian using a sufficient condition found by Gross, Hacking, Keel, and Kontsevich, we obtain bases parametrized by plane partitions for these irreducible representations. As an application we use these bases to show a cyclic sieving phenomenon of plane partitions under a certain sequence of toggling operations. This is joint work with Jiuzu Hong and Linhui Shen.
03/26/2018 - 4:15pm to 5:15pm
1. We discuss Newton’s method as a dynamical system: ifp is a polynomial, then the Newton map is a rational map that very naturally ``wants to be iterated”. Among all rational maps, Newton’s method has the best understood dynamics, and these dynamical systems can be classified (in the sense of a theory developed by Bill Thurston). As a byproduct, we offer an answer to a question of Steve Smale on existence of attracting cycles of higher period (joint work with Kostiantyn Drach, Russell Lodge and Yauhen Mikulich).2. We outline theory about the complexity of Newton’s method as a root finder: unlike various other known methods,Newton as a root finder has both good theory and good implementation results in practice (partly joint work with Magnus Aspenberg, Todor Bilarev, Bela Bollobas, and Malte Lackmann). 3. We present recent experiments on finding all roots of Newton’s method for a number of large polynomials, for degrees exceeding one billion, on standard laptops with surprising ease and in short time, with observed complexity O(d log2 d) (joint with Marvin Randig, Simon Schmitt and Robin Stoll).