21 Nov 2024
Stacey Staples
Spectral Properties of the Zeon Combinatorial Laplacian
Jerzy Kocik
A certain expansion (evolvement) of ${\rm SL}(2,\mathbb Z)$ and its role as the symmetry group of the Apollonian "time crystal."
26 Sep, 3 Oct, and 10 Oct
K.V. ShajeshConjugate functions: A correspondence between analytic functions on a complex plane and electrostatic configurations in two dimensions
From https://people.math.harvard.edu/~knill/pedagogy/pde/index.html |
Mathew Gluck
How do researchers in PDEs think about PDEs
Abstract: The classical Poisson problem, say with homogeneous Dirichlet boundary data, seeks to determine which source functions are in the image of the Dirichlet Laplacian. In this talk I will briefly discuss the classical formulation of Poisson’s problem. I will then discuss the modern formulation (i.e., the weak formulation) of this problem in the case that the source is square integrable. Compared to the classical formulation, the modern formulation of Poisson’s problem requires a bit more up-front cost to understand. Specifically, one must (a) carefully specify both the domain and the codomain of the Laplacian and (b) completely overhaul how they understand the Laplacian. However, this extra up-front cost of understanding pays off. Indeed, the existence-uniqueness theory for the modern formulation of Poisson’s problem is much simpler than that for the classical formulation.
Students with interest in PDEs or related topics are especially encouraged to attend.
Mathew Gluck
The role of compactness in partial differential equations having variational structure
Abstract: A partial differential operator is said to be variational if it can be realized as the differential of some scalar-valued functional. For partial differential equations involving variational operators, exhibiting the existence of solutions is equivalent to exhibiting the existence of critical points of the associated functional. A key tool in exhibiting the existence of critical points is compactness. In short, compactness allows one to pass from a sequence of approximate critical points to a genuine critical point. I will illustrate this concept in multiple settings, starting with an undergraduate-level problem and ending with a variational partial differential equation that one might see in a graduate-level course. Finally, I will discuss a variational problem where there is no apriori compactness. For this problem, I will emphasize how the failure of compactness can be overcome.
K. V. Shajesh
[video]
Punit recommended these two papers: by Berg and Marshall, and by Wilberforce, and a video.
Another related topic: Arnold tongue.