Spring 2024

11 Apr 2024
18 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.

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.

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 5:    9 Nov 2023  video, diagram (NEW!)
Part 6:  16 Nov 2023   video

Part 7:  30 Nov 2023   video

Philip Feinsilver  

Symmetric functions: traces, determinants, the full story.

See references

Philip shares his notes here:

Symmetric Functions Algebras

ziglilucien.github.io

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!

Autumn 2023

 

5 Oct (Thursday), 2pm

Duston Wetzel

Triply periodic and polyhedral helical weaves

Video

Autumn 2023

P.P. Rubens: Bacchus 

28 September

K.V. Shajesh

Negative mass: A runaway solution

Video

Autumn 2023

A. Pasieka: Lorenz Attractor

Thursday, Sept 14, 21  (2pm, Nc 356)

Mike Sullivan

Geometric Lorenz Attractor

Video of Day 1
Video of Day 2

Here are links sent by Mike:

http://galileo.math.siu.edu/Courses/405/F22/MATH405Section9_8.pdf

http://galileo.math.siu.edu/Courses/405/F22/Lorenz.mp4  (18 min video)

http://galileo.math.siu.edu/Courses/DiffEq/GLA.pdf

http://galileo.math.siu.edu/Courses/DiffEq/video1384802557.mp4 (2 min video)

Plus, this is a series of well done videos on chaos theory. Number 7 covers some of the topics covered in my seminar talks.

http://galileo.math.siu.edu/Courses/405/F22/ChaosVideoSeries.html

Abstract of the talk:  The Lorenz equations, developed and studied by the meteorologist Edward Lorenz in the 1960s, were one of the first examples of a chaotic system. They are a set of three nonlinear ordinary differential equations in three variables. The trajectories move toward a bounded subset of $\mathbb R^3$ and then seem to oscillate about, but never truly repeat. Mathematicians began studying these equations using both quantitative and qualitative methods. The former proved very difficult. The latter yielded results. Guckenheimer and Williams developed a geometric model with behavior similar to the Lorenz equations. Then Birman and Williams used topological methods to analyze this Geometric Lorenz Attractor. Decades later it was proved that the two systems are equivalent.

Participants should have had basic undergraduate courses in differential equations and linear algebra. We will begin with a review of systems of ordinary differential equations. This should last about 20 minutes. Then we will cover the construction of the Geometric Lorenz Attractor. That should take about 30 minutes. Next week we will cover Birman and Williams' proofs of the knot types of periodic orbits in the Geometric Lorenz Attractor.

This material was originally covered last Fall in my MATH 405 course. For now, you may wish to amuse yourself with this video.  

Spring 2023

27 April (Thursday), 2pm

Duston Wetzel

Gyroid and the triply periodic helix linkages

Video

Spring 2023

6 April (Thursday),
13 April, and 20 Apr.

Robert Owczarek   (University of New Mexico)

Proto-topological (Quantum) Field Theories of Kijowski and Sławianowski

Video of Part 1
Video of Part 2
Video of Part 3

Here are some links suggested by Robert:
1.  Seminar talk by Kijowskiego from the last year
2.  Related topic on variational principle.  

Abstract: Topological quantum field theories that are metric independent, like, e.g., Chern-Simons theory, give topological invariants of manifolds and knots. Classical field theories of Kijowski and Sławianowski are metric independent, and are defined on manifolds of a priori any dimension, and thus may give new topological invariants of manifolds.

In this talk, I will try to make a friendly introduction to these theories.   

Spring 2023

From Treewewal

23 & 30 March (Thursdays)

John McSorley

Graphs with trees of certain types, and related topics

Video of Part 1
Video of Part 2