Building on a talk delivered with Anita Wager at NCTM, I will use the notion of a zeroth law in physics -- a principle that comes before all others -- to argue that joy is foundational to both the content and practice standards in mathematics. Attending to joy provides a powerful signpost for making decisions about both research and teaching. I will unpack the concept of joy by examining a variety of classroom interactions, demonstrating that joy comes not only from engaging in mathematics in conjunction with other pleasurable activities, such as talk and play, but also from the work of mathematics itself.

Title: Tree Cover Number and Maximum Semidefinite Nullity of Graphs Part 2

Date: 11/28/2017

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

Speaker: Rachel Domagalski, Michigan State University

In this talk, we continue our discussion of the tree cover number of a graph and its relationship to maximum semidefinite nullity. We find the tree cover number of line graphs, shadow graphs, corona of two graphs, cartesian product of two graphs, and a few more. In these cases, we verify the conjecture: the tree cover number of a graph is less than or equal to the maximum semidefinite nullity of its associated complex Hermitian matrix.

Title: Spacetime resonance and asymptotic behavior for wave equations

Date: 11/29/2017

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

Speaker: Yu Deng, Courant Institute of Mathematical Sciences, NYU

The method of spacetime resonance was developed by Germain-Masmoudi-Shatah to study small data long time problems for nonlinear dispersive equations. In this talk we will consider the wave equation in 3+1 dimensions. Global existence for this equation was proved by Lindblad and Alinhac. By applying the method of spacetime resonance, we are able to obtain the precise asymptotics for small solutions, which turns out to exhibit a subtle nonlinear behavior. Our work also justifies the ideas behind a conjecture of Lindblad and Rodnianski. This is joint work with F. Pusateri.

Title: Entropy stable high order discontinuous Galerkin methods for hyperbolic conservation laws

Date: 11/30/2017

Time: 3:05 PM - 3:55 PM

Place: C304 Wells Hall

Speaker: Chi-Wang Shu, Brown University

It is well known that semi-discrete high order discontinuous Galerkin (DG) methods satisfy cell entropy inequalities for the square entropy for both scalar conservation laws and symmetric hyperbolic systems, in any space dimension and for any triangulations. However, this property holds only for the square entropy and the integrations in the DG methods must be exact. It is significantly more difficult to design DG methods to satisfy entropy inequalities for a non-square convex entropy, and / or when the integration is approximated by a numerical quadrature. In this talk, we report on our recent development of a unified framework for designing high order DG methods which will satisfy entropy inequalities for any given single convex entropy, through suitable numerical quadrature which is specific to this given entropy. Our framework applies from one-dimensional scalar cases all the way to multi-dimensional systems of conservation laws. For the one-dimensional case, our numerical quadrature is based on the methodology established in the literature, with the main ingredients being summation-by-parts (SBP) operators derived from Legendre Gauss-Lobatto quadrature, the entropy stable flux within elements, and the entropy stable flux at element interfaces. We then generalize the scheme to two-dimensional triangular meshes by constructing SBP operators on triangles based on a special quadrature rule. A local discontinuous Galerkin (LDG) type treatment is also incorporated to achieve the generalization to convection-diffusion equations. Numerical experiments will be reported to validate the accuracy and shock capturing efficacy of these entropy stable DG methods.This is a joint work with Tianheng Chen.

Title: Elliptic curves, Euler systems, and Iwasawa theory

Date: 11/30/2017

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

Speaker: Francesc Castella, Princeton University

Some of the most fascinating pieces of mathematics, such as Dirichlet’s class number formula and the Birch and Swinnerton-Dyer conjecture, build a bridge between the distant worlds of arithmetic and analysis. Euler systems and Iwasawa theory provide an intermediate step between the two, and both have been at the source of much of the progress to date on the BSD conjecture and its many generalizations. In this talk, I will expand on some of these ideas, including a brief overview of some of the recent developments in the area.