Autumn 2025

30 October

Jackson Lewis

Hyperbolic Angular Momentum from the ground up

Abstract:  We present the construction of “hyperbolic angular momentum” as given by Schwinger (1951), starting with the basic one-dimensional oscillator of classical mechanics. Then we introduce harmonic oscillators in quantum mechanics, angular momentum in classical and quantum mechanics, and the Jordan-Schwinger Map. Some applications will be discussed, possibly including constructions of spin networks and discrete spacetime operators in loop quantum gravity.  

From "Math for Poets..."

16, 23  Oct 2025

Mathew Gluck

The Method of Moving Spheres

Abstract. The method of moving spheres is a powerful and versatile method for analyzing partial differential equations with conformal symmetry. At the core of this method is the amazing fact that one can classify all suitably nice functions f  defined on Euclidean space for which both of the following properties hold:

1.  For every point x there is a sphere centered at x about which f  has inversion symmetry, and

2.  for every direction e, there is a hyperplane with normal direction e about which f  has reflection symmetry.

I will give some examples of functions for which both properties hold, and I will discuss the historical development of the classification of all such functions. Finally, I will overview the method of moving spheres and provide some applications of the method in the analysis of conformally covariant partial differential equations.

Autumn 2025

3 and 10  October

Viska Arbogast

Numbers as Equations

Autumn 2025

 

From Wikipedia

11, 18, 25  Sep 2025

Jerzy Kocik

Coxeter groups, and an unconventional view on the unimodular group (with pictures)

Part 1:  Coxeter groups (the idea)

Part 2:  Modular group as a Coxeter group

Part 3:  "Supermodular" group and an unexpected application 

For part 1, you may consult a nicely written text by J. Baez:  Coxeter and Dynkin diagrams. 

From Nature Reviews Physics 1,437-449 (2019)

5 Dec, Thursday, 1pm

Ines Cavero-Pelaez
University of Zaragoza, Spain

About the geometrical phase

Video

Autumn 2024

21 Nov 2024

Stacey Staples

Spectral Properties of the Zeon Combinatorial Laplacian

Video

Autumn 2024

7 and 14 Nov, Thursday

Jerzy Kocik

A certain expansion (evolvement) of ${\rm SL}(2,\mathbb Z)$ and its role as the symmetry group of the Apollonian "time crystal."

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