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
Part 1 and Part 2
Part 3: Examples
Part 3: Examples
Wednesday, September 25, 2024
Thursday, September 19, 2024
Wednesday, September 4, 2024
Autumn 2024
 |
| From https://people.math.harvard.edu/~knill/pedagogy/pde/index.html |
5 Sep and 12 Sep
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.
Wednesday, August 28, 2024
Monday, April 29, 2024
Wednesday, April 10, 2024
Spring 2024
11 Apr 2024
18 Apr 2024
25 Apr 2024
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.
Thursday, April 4, 2024
Thursday, March 21, 2024
Spring 2024
21 Mar 2024
K. V. Shajesh
[video]
Punit recommended these two papers: by Berg and Marshall, and by Wilberforce, and a video.
Another related topic: Arnold tongue.
Wednesday, February 28, 2024
Spring 2024
 |
| From https://www.youtube.com/watch?v=spUNpyF58BY |
29 Feb and 7 Mar, 2024
Mohammad Sayeh
On Fourier and fractional Fourier transforms, and more.
Video of Part 1
Video of Part 2 (no sound...)
Tuesday, February 20, 2024
Spring 2024
Mathew Gluck
Introduction to Principal Component Analysis
Abstract: Principal component analysis (PCA) is a data dimensionality reduction technique whose goal is to throw away the least essential components of a data set. It is commonly used by data science practitioners whose data belongs to a vector space of large dimensions. It is well-known that the eigenpairs of a suitable covariance matrix “know” which components of data are most important and which are least important. However, in many expositions on PCA, the reason behind this well-known fact is rarely explained. I will offer an explanation and I will show a concrete application of PCA in image data.
Wednesday, October 11, 2023
Autumn 2023
Part 1: 12 Oct 2023 video
Part 2: 19 Oct 2023 video
Part 3: 26 Oct 2023 video
Part 4: 2 Nov 2023 Notes (4 pages)
Part 7: 30 Nov 2023 video
Philip Feinsilver
Symmetric functions: traces, determinants, the full story.
See references
Philip shares his notes here:
The links are to pdf files. Added visualizations, namely Young diagrams. It should help clarify the definitions. Suggestions welcome - - -
From JK: This will be an opportunity to learn this topic in a novel and unique way. The presentation is planned for a few meetings. Do not miss them! You will be among the first few on the planet to get a new creative insight into the subject. Matrices in use!
Wednesday, October 4, 2023
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