I've been making lecture notes for my analytical dynamics course and uploading them to my site. They're hand-written, but can be read standalone. aghostinthefigures.com/the-notion-o... I wrote 100 pages and haven't even gotten to dynamics yet. Why on Earth am I doing this? (1/3) #physics #dynamics

Adventures in teaching graduate-level dynamics: I have done the equivalent of asking my students to "fetch the blinker fluid"—asking for a coordinate-free representation of the "direction cosine tensor". Truly diabolical.

Obligatory conference post: I’ll be at #JMM2026 presenting some of my upcoming work on predicting the existence of qualitatively periodic orbits in dynamically stabilized systems. (It’s my first invited talk at a conference; wish me luck!)

Curious (and a bit baffled) to see all the hype surrounding the Navier-Stokes equation in the context of the Millenium Prize these days; from my perspective, the Millenium Prize N-S problem is about the least interesting parts of what we don’t know regarding Navier-Stokes!

A preview of a context-less, interesting plot coming out of my research. It has applications, I promise!

Honored to have been featured as today’s mathematician in Lathisms’ annual calendar for Hispanic Heritage Month! (Also delighted to see some familiar faces in this year’s cohort!)

I've uploaded my lectures for my summer heat transfer course on YouTube, free to watch. They are largely unedited and a tad sloppy; but I made a commitment years ago to the free distribution of knowledge, and I have to see it through.

One of my favorites parts of teaching heat transfer: Explaining that radiation is not /technically/ a form of heat transfer! (A thread.) #physics

It’s been interesting to reflect on how steeply my feelings on sharing things—educational and otherwise—on social media have changed over the last year. For example; I’m making a series of lecture videos for my summer heat transfer course. The original plan was to share them freely. Now—not sure.

Another semester of teaching at UB concluded. Eternally grateful to have the ability to teach—it is not an easy time right now to be a professor, but it is a far harder time to be a student. They’ve done amazing work in the face of great adversity and fear. What a privilege it is to help them.

More fun with computational fluid mechanics: Here's a 2-D simulation of a layer of air (red) below a layer of water (blue) in a closed container. As air is lighter, buoyant forces cause the interface to be unstable, leading to the complex flow shown here as the air eventually rises to the top.

More fun with CFD for my course; here's a simulation of air bubbles being generated in a container of water with a rapidly rotating cylinder inside of it. As time goes by, the cylinder rotation begins to affect the flow in the whole tank, and droplets start to drift to the right!

More fun with computational fluid dynamics; what does flow past an airfoil look like? #FluidDynamics A thread:

A perk of teaching students how to use finite element/volume methods; making cool animations! This is "vortex shedding" in 2-D flow past a cylinder; a pair of symmetric counter-rotating vortices develop in the back, which then destabilize to form a long, oscillatory, transient wake. #fluidmechanics

For a lot of people, the key hallmark of chaotic behavior is sensitivity to initial conditions—I disagree! To me, the key hallmark of chaos is complexity. Everything follows from that; roughly speaking, cramming “many” dynamical behaviors into a “small” set of states is what causes sensitivity!

It's interesting to think that (IMO) the axioms of topology generate structure that is right in the Goldilocks zone for mathematical exploration. There are far fewer "interesting" results about the structure of measurable spaces, because their structure is "too" fine to vary much between spaces!

Getting back in the habit of making math/physics animations in my spare time. Here's an old favorite of mine; the formation of a homoclinic tangle! I'm hoping to upload the code for a few of these on GitHub, once I feel it's polished enough for public viewing. Hopefully won't take too long!

Tried a fun exercise with my finite elements students a few weeks ago: Do the empirical equations we use for finding the stress concentration factor in the bar below match the results we'd get from doing finite element analysis?

Happy, but more importantly proud, to have received this fellowship. I am also taking the time to reflect on the fact that many of the programs that got me to where I am today—a job where I have the power to teach and help a new generation of engineers help others—are being dismantled as we speak.

Been writing a handbook on finite element methods for a bit now; unsure why. Hope to make it, and a few algorithms within it, free and publicly accessible once I'm done. Tough to feel as if this sort of thing matters in the new Age. Logic fights volition and electrochemistry. Let's see what wins.

I obtained a copy of Physics of Continuous Matter in December—was incredibly impressed by what I consider to be a singular work in continuum physics—and thought I would send a “thank you” e-mail to the author, Benny Lautrup, after the semester had started. I just learned he passed away on the 3rd.

Replayed Portal 2 for the holidays; pleasantly surprised to find some decorative fluid mechanics on the whiteboards! (Not sure if that Reynolds transport theorem statement is correct, though.)

In today's lecture in my nonlinear dynamics and chaos course, I tried to give intuition for where chaos comes from. It's caused by repeated stretching and folding of phase space.

What does flow past a sphere look like when viscous friction dominates versus when it's neglected? Take a look for yourself! #physics The flow below is creeping flow, where frictional effects dominate. Particles that get close to the sphere slow down drastically due to the no-slip condition.

Long nights and low lights at UB as the semester winds down and the first few snowfalls come in. I’m so lucky I get to do what I do.

There's no simpler way to show that boundary value problems are much trickier than initial value problems than by analyzing the simple harmonic oscillator. The following boundary value problem has no solution! #math

Lectured a bit today on one of my favorite prototype aircraft; the Plymouth A-A-2004. Notice the fact that, technically speaking, it has no wings! #physics

Have you ever wondered why you can’t get to the 7th floor of tall buildings? The answer, surprisingly, is fluid mechanics! #physics

Random thought while teaching fluid mechanics: it’s crazy to consider that we could’ve just as easily developed a theory of hydraulics based on /forces/ and not power, and had it work just as well, but didn’t. Wouldn’t have made a huge difference practically, but it’s a huge conceptual difference!

Hard to think it’s already been a year. The end of an era for me—written right while I was switching jobs/career goals, at my most desperate, a last throe in the dark for a grand goal I had since I started graduate school. Finishing it was a long shot; I’m so glad I got to.

Meaningless (but neat) Sunday math fact: You can define the derivative of a function as an integral of that function! #math (You can also show that Taylor series are just a special kind of generalized Fourier series, but that requires a bit more effort.)

Time to hit the ground running on my New Year’s resolution. Looking forward to properly learning about modeling physics on warped spacetimes!

I’ve gotten a few old math books over the years, some owned by notable mathematicians—but it never felt right to just leave them in a corner to look at once a year. Better to pass them on to those who can use them! This one’s going to a dear friend who can read Russian far better than I can.

A few dynamicists have claimed visiting the tropics helped them do some of their best work—being from here, I’d like to think I have home field advantage. Happy holidays!

Revisiting some favorite Cornell spaces during my last days in Ithaca—this is the Johnson Museum of Art (well, its lobby). When it had more seating capacity, I used to hold special office hours here—students studied, took a break to see art, and got back to studying. Good memories.

Setting up the home office with my editor, Ferguson. (He likes to sit on my keyboard sometimes.)

Here's the first of a few threads I plan on making about the contents of Qualitative Dynamics and Chaos: Why bother describing things that change over time using sets of infinite symbol sequences? (1/8)

Took a small break from writing on the computer to write in my favorite medium—letters, particularly those for friends in Puerto Rico. (I need to find better seal wax!)

So I deleted the 1st post 😜 There are some people out there that might find ChaosBook.org helpful, so here is a very very short introduction to it: 1/7

Week 1 patches to Qualitative Dynamics and Chaos are now live! Added 2 exercises, mostly just fixed up typos. Nothing huge yet, but I'm staying alert; thanks to all of you who've been letting me know of any mistakes you catch! Now, time to finish my Baldur's Gate 3 run. 😅

Fun math fact: Invent any "game" or rule that turns a 2-digit number into another 2-digit number. (You can add leading zeroes as needed, i.e. 8 is 08.) If the rule only depends on input number, repeat applications of the rule will always eventually generate periodic output!

An exercise from my book: If you describe a possible game of Snakes and Ladders on this board by a sequence of snakes and ladders (in the order you landed on them), can you show there are infinitely many distinct periodic and aperiodic sequences representing possible games?

As I wait to recover from my cold, here's an interesting qualitative dynamical system to pique your curiosity: en.wikipedia.org/wiki/Kapreka... Using my book's techniques, you can show that any input number to Kaprekar's routine will eventually generate a periodic sequence of outputs!

Some cut content from the book: My illustrator Caleb’s (calebharrington.com) interpretation of a smarticle in the “fable” at the beginning of Chap. 5. I /love/ how he drew the inside; my own interpretation was something like the Astral Prism from Baldur’s Gate 3! 😅

After all this time, my free textbook on qualitative dynamics and chaos is finally done! Access it on my webpage (wp.me/P7zu73-2a8) or on Google Books/Play (bit.ly/498vI61). (1/7)

Calling all dynamicists! #dynamics Any recommendations on articles that use experimentally-generated return maps to diagnose chaos in real-world systems? Here's an example article on what I mean: doi.org/10.1016/0375...

A very early concept for the cover of the upcoming book, using Hopper's "Automat". I went through a lot of different ideas, including trying to depict Hercules fighting a fractal hydra, until I saw sense and hired an (amazing!) illustrator. Can't wait for you to see what he did!

Here's an image I cut from the book (it's now an exercise for the reader to make it themselves); a fractal attractor in the Chua dynamical system. With clever sampling techniques for these sorts of systems, you can easily argue that they exhibit chaotic behavior. You'll see how soon!

Co-hosted, with the amazing Mallory Gaspard, the second annual SIAM Open House + Applied Dynamics Seminar yesterday at Cornell. Glad to see it’s become a tradition now—it sure deserves it.

Polishing off some of the final images and examples for the upcoming book. Didn't expect my example return map to exhibit stretch-and-stack dynamics /this/ straightforwardly!