In previous blogs, I described conversations with Andrew Bauman, an up-and-coming mathematician who is an undergrad at UALR and who spends spare time tinkering with problems in linear algebra and quantum mechanics. Today, in one of our gym-facilitated meetings, Andrew brought three distinguished and enjoyable intellects besides himself. This led to a wide-ranging conversation that I will attempt to recap here for posterity. We will start with a sidebar since the evening was full of them, some recursive, some eight layers deep.
Before tonight’s impromptu, mostly math and quantum computing discussion, I checked in with our pool lifeguard, a pole vaulter (not the back flipping fellow above, but certainly capable). A review of video footage of his record pole vault revealed excellent form, with one specific moment that could benefit from an improvement that consisted of pressing to a handstand and walking on one’s hands. This body position is attained for less than a second during the pole vault but is critical to obtaining greater heights. He showed me footage of his hand standing and walking, which was quite good. It gave me ideas for a follow-up exercise involving an inverted shoulder shrug, an inverted pole grip change/grip walk, and an inverted kip-up that could benefit him further. I’m writing this here as part of a stream-of-consciousness recap so I don’t forget it in the twists and turns of what follows.
Weight, there's more:
I strapped a 5 lb weight on each hip for my nightly mile walk/run tonight. However, due to the chance encounter with Andrew (and company), I did not make it to my walk, but I sported the weights like a pair of revolvers from the Wild West. Anyone seeing our extended conversation would have to wonder, “Why a weight belt for talking about math?”. Answer, “Heavy Topic”!
Andrew
paused his ping-pong game to introduce me to his ping-pong partner, Dr. Sudan
Xing, a mathematician and professor at UALR who specializes in geometric
projection and embedding problems.
A peek at Google
Scholar introduces us to the central theme of Dr. Xing's work, which is focused on
the Orlicz-Brunn-Minkowski theory and related Minkowski problems in convex
geometry. This theory is an extension of the classical Brunn-Minkowski theory,
which deals with the relationship between the volumes of convex bodies and
their Minkowski sums.
Now I’m a dolt, so I wanted to know what a simple Minkowski sum looked like, so
I asked ChatGPT-4 to write me some code and produced the figure below. It was
almost a 1-shot job; more recent work I have done has taken nearly 40 shots/redos to
get right. The code is here.
The mathematics involved in her work primarily comes from convex geometry, functional analysis, and measure theory. Key concepts include:
- Convex bodies and functions
- Minkowski addition and Orlicz addition
- L_p norms and Orlicz norms
- Surface area measures and the Minkowski problem
- Brunn-Minkowski and isoperimetric inequalities
- Log-concave functions and measures
Dr. Xing's work contributes to developing a more general theory of convex bodies and related geometric inequalities, with potential applications in mathematics and beyond.
Next,
we met her bioinformatics colleague, Ju Ni. We discussed how exciting it was to
live in the time of AI/ML and the OMIM database, where we can know the genes
involved in almost any affliction of human beings. We discussed the importance of visualizing gene and biochemical pathways for specific conditions. We discussed the particular example
of Thyroid Cancer, one of the only truly “curable” cancers, since it can be treated
with Iodine-131, which is preferentially taken up by the thyroid, thus
neutralizing the cancer, but alas, the thyroid as well, necessitating lifelong
medication afterward.
We briefly referenced a certain Calculus
book, a Facebook math site,
and my AddSubMulDivia
five-book series “that nobody reads.” We had a good laugh about how pathetic it
is to care so much about things no one else does.
Dr. Xing and I found out we shared appreciation for 2010 Fields medal laureate Cedric Villani, a mathematician and politician who is friends with the president of France. Quoting from my favorite LLM:
Villani revolutionized mathematical physics with contributions to optimal transport theory, kinetic theory, partial differential equations, and the study of Ricci curvature in metric spaces. His work, notable for bridging pure mathematics with applied physics, includes groundbreaking analyses of the Boltzmann equation and its convergence to the Landau equation, shedding light on gas behaviors in varied regimes. Villani's innovative use of optimal transport for exploring metric spaces with Ricci curvature bounds has influenced areas ranging from plasma physics to network analysis.
We talked about:
the advent of AI/ML is enabling the revisiting of unsolved math problems
Andrew mentioned he was working on a problem that starts with drawing a bisecting
line on a piece of paper, and proving that the halfspaces generated by the line
are distinct.
Dr. Xing responded with H+/H- halfspaces, then visualized the problem in 3D
with her 'ping-pong paddle' analogy. She discussed the 'floating problem' – a
sphere trimmed into equal-volume sections by tangent planes. This process seems
to give the sphere unique properties, but we switched topics too quickly.
We touched on projective geometry and how representation (explicit, implicit, parametric, iterated, chaotic) influences what we can understand about a problem. I mentioned the challenges of intersecting closed tensor product surfaces made with B-splines and wondered if Dr. Xing's dualized projection approach could help.
I mentioned my curiosity about a set of planar ray tracing problems as a family of reachability problems that are quite interesting. These live under the heading “Visibility Polygon” and the Art Gallery Problem.
At this time, Greg, a friend of Andrew’s, had arrived. He has a Master’s degree
and was focusing on math education. Dr. Xing and Ju Ni had to go, so Greg,
Andrew, and I started drilling down on several topics.
The topic space exploded before we settled down and had a heart-to-heart on
quantum computing and Bell’s Inequality.
Topic Explosion (Free Association Gone Wild)
- inner products and their connection to standard deviation and variance
- Hilbert spaces, norms
- distance metrics: Euclidean, Manhattan, Minkowski
- closure loss on the inclusion of zero
- maintaining state on chained binary operations of AddSubMulDiv-ia to enable reversibility and prevent the loss of structure of the path of a calculation that would otherwise be non-unique if results were discarded at intermediate states. Undoability.
- skew and kurtosis being higher moments in statistics
- discrete and continuous distributions
- moments in structural mechanics and moments of inertia.I demonstrated how a phone tossed in space will land without a change in rotation in two of its three axes but not the third. The Veritasium link below explains it better.
- Spaghetti Sort as an analogy to quantum computing simultaneous equation solving using entangled particles.
- The mystery of entanglement is the same as having tossed a coin that landed heads and automatically knowing that the other particle’s spin is “tails”.
- The Bloch sphere
- Bell’s inequality
- Alice and Bob’s experiment: A stream of entangled particles
- Quantum and Classical Interpretations of the experiment
- The interchangeability of streaming and fixed interval experiments
- Interval arithmetic
- The register operation of comparing A and B’s results,
- XNOR was the decision operator correctly identified by Greg
- Marvin Minsky’s XOR catastrophe started the AI winter, requiring two neurons.
Links
- Veritasium: Pilot wave theory
- Veritasium: Two-way vs. one-way speed of light
- Veritasium: The strange physics of rotating bodies
- Van TSP Blog Entry
- Van Sorting Blog Entry
- Van Animating Arc Length Integration Blog Entry
