2 May, Thursday
John McSorley
Finding unitype graphs amongst uncyclic graphs
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
Wednesday, September 27, 2023
Sunday, September 10, 2023
Autumn 2023
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| A. Pasieka: Lorenz Attractor |
Thursday, Sept 14, 21 (2pm, Nc 356)
Mike Sullivan
Geometric Lorenz Attractor
Here are links sent by Mike:
http://galileo.math.siu.edu/
Plus, this is a series of well done videos on chaos theory. Number 7 covers some of the topics covered in my seminar talks.
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.
Wednesday, April 26, 2023
Monday, April 3, 2023
Spring 2023
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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
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.
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