Autumn 2024

The art from here

24 and 31 Oct, Thursday

Philip Feinsilver

Convergence theorems for matrices with some negative entries

See the outline.

Video:  Part 1 and Part 2

Autumn 2024

17 Oct 2024

Jerzy Kocik

${\rm SL}(2,\mathbb Z)$  --  The modular and the Dedekind tesselations 

I suggest seeing this site as a "prerequisite:" Lieven LeBruyn: Neverendingbooks

Autumn 2024

26 Sep, 3 Oct, and 10 Oct

K.V. Shajesh

Conjugate functions: A correspondence between analytic functions on a complex plane and electrostatic configurations in two dimensions

Part 1 and Part 2
Part 3:  Examples

Autumn 2024

19 September 

Punit Kohli

Calligraphic fabric – and who should be afraid of it?

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.

VIDEO OF PART 1

VIDEO OF PART 2

AUTUMN 2024

 

29 Aug 24

Jackson Lewis

Using AI in Elementary Number Theory

Video

Spring 2024

2 May, Thursday

John McSorley

Finding unitype graphs amongst uncyclic graphs

Video

Spring 2024

11 Apr 2024
18 Apr 2024

25 Apr 2024

Mathew Gluck

The role of compactness in partial differential equations having variational structure

Video 

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.

Spring 2024

 4 April

 Gossip and conversation in math, physics, and philosophy

Spring 2024

21 Mar 2024

K. V. Shajesh

Sympathetic oscillators

[video]

Punit recommended these two papers:  by Berg and Marshall,  and by Wilberforce, and a video.

Another related topic:  Arnold tongue.

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...)

Spring 2024

22 Feb 2024

Mathew Gluck

Introduction to Principal Component Analysis

Video

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.