1/ yesterday’s thread on the genesis of figures in “Elementary Applied Topology” ended on a cliffhanger: what do all the chapter heading illustrations mean? it’s a puzzle.
get ready for a myth/math mishmash.

https://twitter.com/robertghrist/status/1698839707801497793

2/ the title page & the back cover carry an iconic stylized pomegranate, with some motion induced on what looks like half-a-dozen seeds? hmmm… that reminds me of something…

1/ in 2009, i began writing a book on topology, meant to be a short introduction to the core concepts, in the context of lots of interesting applications. every idea would be paired with one or more uses, as much outside of mathematics as possible: “applied topology”. 2/ i wanted the book to have a lot of pictures – at least one per page on average – and i wanted the pictures to be the exercises. each picture is a puzzle, and if you understand the picture, then you have solved the exercise & understand something.

=> new paper w/yuliy baryshnikov <=
"navigating the negative curvature of google maps"
/1 one thing that google maps gets right is the user interface for navigating -- it just feels right to pinch-zoom, scroll, etc.
/2

1/ this week in multivariable calculus @Penn ...
constrained optimization 2/ this builds on previous investigations into level sets of scalar fields -- a topic for which my students often struggle to grasp the notation & motivation...
a bit of animation can help!

1/ this past week in multivariable calculus @Penn
the chain rule & its consequences... 2/ my students all remember how to use the chain rule "by hand", even if writing out / explaining the principle takes more thought. we start off discussing composition and what it means for functions and rates of change...

mathematical art is increasingly important in research & AI is about to revolutionize what can be done... 🧵 i have been doing art & animation for my research & teaching for years & have been convinced that this is the future of imaginative thinking & teaching... but it's hard & has a high learning curve...

billiards are a classic example of chaotic dynamical systems: choose a table shape, fire a ball, and find out what happens, using angle of incidence = angle of reflection
1/ some table shapes give chaotic dynamics, but circles are especially simple. orbits on circular tables are periodic or nonperiodic, depending on (ir)rationality of the angles involved
2/

1/ observations on having taught math both virtually & in-person over the past month: a brief thread 2/ just finished up a 4-week "boot camp" for incoming students @Penn in the @Penn_CAP pre-freshman-program. i was responsible for teaching math to ~30 incoming engineers. this is a great program for acclimating students from a variety of backgrounds.

1/ recently, the @jm_collective and others have been pushing for mathematicians (& the @amermathsoc) to cut ties with the NSA (@NSAGov).
i would like to comment on that. 2/ i begin by affirming their right (& everyone’s right) to free speech & expression of beliefs, political or other. go for it & sign whatever petitions you want. make your arguments & let us all listen.

a quick thread on what the @penn mathematics department is doing starting this fall to improve the student experience in calculus courses...
1/ of course, this is motivated by the pandemic and the need for online instruction; however, this is not a hackjob emergency plan. these reforms are more structural and sustainable, and have been in the works for some time.
2/

what ties together machine learning, rational homotopy theory, rough paths, lie groups, topological data analysis, and how to teach undergraduate vector calculus?
a thread…
1/ this is a long, paper-laden thread that sets up a new paper with darrick lee, a ph.d. student at @Penn in applied mathematics. buckle up!
arxiv.org/abs/2007.06633
2/

a technical thread for a technical paper:
"Cellular sheaves of lattices & the Tarski Laplacian"
with @hansmriess
arxiv.org/abs/2007.04099
1/ here's the setup: cellular sheaves are data structures over a cell complex (such as a social or neural network) that attach algebraic data to nodes, edges, etc.
they are wonderful mathematical entities.
2/

i get a lot of questions about how to make math vids...
my workflow is so byzantine, it's not worth writing up.
but...for those making vids for fall semester in a hurry: a thread of advice...
/1 for quick lo-rez vids, you do not want to stand in front of a camera or be filmed at a board. nononono...
write on paper or a digital pad, and then do voiceover.
splice in visualizations if you got 'em.
that's the basic winning formula.
/2

one of the things i did this semester in my multivariable calculus class was to give the option of doing an "extra credit" project. i've never done that before, and was inclined against it. but: it went so very well. let me tell you...
1/ ground rules: you do the project because you want to learn more than what we cover in class. it counts for an unspecified amount of bonus credit if done well. no rules on format (essay? video? comic book? yay!), but you have to explain what you learned well.
2/

henry wente: geometer & teacher.
he left this world today: 20-jan-2020.
he's the reason i'm a mathematician. i was an engineering undergraduate at the university of toledo, ohio, in 1987. i had AP calculus in high school, but there was an "honors calculus 1 for engineers" course, so i signed up for that. the teacher was *so* *weird*.
his name was dr. wente.

the invariant set of the smale horseshoe is obtained by intersecting all the horizontal & vertical strips in the square under iterations of the map & its inverse. this type of set is a cantor set. it looks like dust, but is uncountably infinite. twitter is going to compress this into mush, i'm sure. there is a better quality version up on youtube:

the lorenz equations define a classic chaotic dynamical system. any orbit meanders about an attractor in a way that, though deterministic, appears impossible to predict.
1/4 being chaotic, the lorenz system exhibits SDIC = sensitive dependence on initial conditions. if you start multiple orbits very close to one another, they will eventually diverge and have independent long-term behaviors.
2/4

what is this?
it has something to do with a spinning tennis racket...
1/5 spinning a tennis racket about the longest or shortest principal axes is stable, but when you spin about the "middle" axis, there's a curious instability... 2/5

periodic phenomena (vibrations, tremors, mood swings?) often arise from a [supercritical] Hopf bifurcation, in which a spiral sink becomes a spiral source, spawning an attracting limit cycle as the parameter passes the critical value
1/4 seeing this in the full 3-d space (x-y-mu) reveals the limit cycles forming a paraboloid. this \sqrt{\mu} - growth in radius is why the bifurcation feels so sudden: the amplitude of oscillation is growing very quickly at first
2/4

a (local) bifurcation is a change in the number/type of equilibria in a parametrized dynamical system. in 1-D continuous-time, it's helpful to plot state (x) versus parameter (mu) to see the "multiverse" of potential dynamics as a function of parameter. 1/6 these local changes can be classified. most common is the "saddle-node" bifurcation, in which two equilibria of opposite stability coalesce and annihilate as you change the parameter. this one shows up all over the place. 2/6

tomorrow i am teaching coupled oscillators in my applied dynamical systems class. it all begins with a pair of independent identical pendula, just doing their thing... 1/9 if you couple them -- let each slightly influence the other -- then interesting things happen. the question is to what extent do coupled systems converge to consensus? 2/9