Latest paper solves the simplest TDA inverse problem of all:
How many different barcodes have the same Betti curve?
Turns out that the Kostant Partition Function measures this exactly and one can view every barcode as a juggling sequence! π€Ή
arxiv.org/abs/2602.09011
Starter Pack: Max Planck scientists on Bluesky
Many Max Planck scientists have started sharing their #research findings on #BlueSky. Follow their posts and join the conversation! π go.bsky.app/BYcBy6R #StarterPack
I am sorry to say that AIs absolutely are going to replace regular office workers.
It's an artistic representation of a circulatory system walking through a kitchen, as drawn by Dave Gibbons and written about by Alan Moore. It happens a lot 'round here.
Happy There Is A Circulatory System Walking Through The Kitchen Day to all who celebrate.
Sadly, winning an election has always been more important to the Dems. We are terrible governors and this is why the Rs have been so frighteningly good at breaking things. I donβt think the Dems could do the same if they tried.
No health care, no deal.
arxiv.org/abs/2511.01272
/Design and Fabrication of Origami-Inspired Knitted Fabrics for Soft Robotics/
Sehui Jeong, Magaly C. Aviles, Athena X. Naylor, Cynthia Sung, Allison M. Okamura
Why can't we just say 0/0=1?
I showed one way why this would break math. What's another way?
A Sharon Begley byline, almost 5 years after her death.
Upon hearing the news James Watson had died, a STAT reporter said in our Slack, "I wish I could read what Sharon would have written."
Incredible news: Sharon in fact did pre-write a Watson obit. And it is masterful and excoriating.
π§ͺπ§¬π§«
π
RIP Sayan Mukherjee. You were a fantastic mentor, collaborator, and friend.
youtu.be/XF8B5afS1DI?...
![From p.1 of https://math.jhu.edu/~eriehl/context.pdf:
In 1941, Saunders Mac Lane gave a lecture at the University of Michigan in which
he computed for a prime p that Ext(Z[
1
p
]/Z, Z) Zp, the group of p-adic integers, where
Z[
1
p
]/Z is the PrΓΌfer p-group. When he explained this result to Samuel Eilenberg, who had
missed the lecture, Eilenberg recognized the calculation as the homology of the 3-sphere
complement of the p-adic solenoid, a space formed as the infinite intersection of a sequence
of solid tori, each wound around p times inside the preceding torus. In teasing apart this
connection, the pair of them discovered what is now known as the universal coefficient
theorem in algebraic topology, which relates the homology Hβ and cohomology groups H
β
associated to a space X via a group extension [ML05]:
(1.0.1) 0 β Ext(Hnβ1(X),G) β H
n
(X,G) β Hom(Hn(X),G) β 0 .
To obtain a more general form of the universal coefficient theorem, Eilenberg and Mac
Lane needed to show that certain isomorphisms of abelian groups expressed by this group
extension extend to spaces constructed via direct or inverse limits. And indeed this is the
case, precisely because the homomorphisms in the diagram (1.0.1) are natural with respect
to continuous maps between topological spaces.](https://cdn.bsky.app/img/feed_thumbnail/plain/did:plc:ysaiuavm2f7d4th42c4valhw/bafkreidx34fck7zb2vcc6roymvmpf5pjxf3dplzd6idmys7z5m4c5eyq7m)
From p.1 of https://math.jhu.edu/~eriehl/context.pdf: In 1941, Saunders Mac Lane gave a lecture at the University of Michigan in which he computed for a prime p that Ext(Z[ 1 p ]/Z, Z) Zp, the group of p-adic integers, where Z[ 1 p ]/Z is the PrΓΌfer p-group. When he explained this result to Samuel Eilenberg, who had missed the lecture, Eilenberg recognized the calculation as the homology of the 3-sphere complement of the p-adic solenoid, a space formed as the infinite intersection of a sequence of solid tori, each wound around p times inside the preceding torus. In teasing apart this connection, the pair of them discovered what is now known as the universal coefficient theorem in algebraic topology, which relates the homology Hβ and cohomology groups H β associated to a space X via a group extension [ML05]: (1.0.1) 0 β Ext(Hnβ1(X),G) β H n (X,G) β Hom(Hn(X),G) β 0 . To obtain a more general form of the universal coefficient theorem, Eilenberg and Mac Lane needed to show that certain isomorphisms of abelian groups expressed by this group extension extend to spaces constructed via direct or inverse limits. And indeed this is the case, precisely because the homomorphisms in the diagram (1.0.1) are natural with respect to continuous maps between topological spaces.
So Mac Lane and Eilenberg invented category theory so they could prove the universal coefficient theorem, which they discovered because Saunders showed an algebra computation to Sammy and Sammy went "Huh. That's how I compute the complement of the p-adic solenoid inside a 3-sphere."
I just shared a new article, "The State of Reasoning Models", where I am exploring 12 new research articles on improving the reasoning capabilities of LLMs (all published after the release of DeepSeek R1): magazine.sebastianraschka.com/p/state-of-l...
Happy reading!
LLMs may be good for something new. Some of them are good at βfast thinkingβ that sets up βslow thinkingβ by combinatorial optimization. This flavor of βcoherenceβ has deep roots in cognitive science. We made an algorithmic benchmark: o1, Sonnet, and QwQ do very wellβapparently superhuman, even.
Maybe a hot take, but what about the following advice to the next gen:
Don't get an AI degree; the curriculum will be outdated before you graduate. Instead, study math, stats, or physics as your foundation, and stay current with AI through code-focused books, blogs, and papers.
βThe essence of tyranny is the denial of complexity.β
β Jacob Burckhardt
.
.
.
.
.
Made with #python #mlx #matplotlib
#particlelenia #alife
Sweet! Letβs catch up while weβre there
A mathematical visualization of polynomial roots creating a soft, organic shape against an off-white background. The image has a delicate, watercolor-like quality with translucent layers in pale blue and dusty rose. The structure has four-fold symmetry with rounded lobes at each corner containing subtle circular patterns. The center features a teardrop-shaped void. The entire composition is rendered with a grainy, textured effect that enhances the watercolor appearance. The overall effect is minimalist and ethereal, resembling a sophisticated botanical illustration.
Polynomial roots mist.
#MathArt #Mathematics
Made with #python #matplotlib #numpy #sympy
*hugs* See you at JMM?
The final Calvin and Hobbes cartoon. The entire page is mostly white as they're in the winter snow. The pair is walking with a sledge, nicely duffled in in scarves, hats, gloves. Calvin delightedly exclaims 'wow, it really snowed las night! Isn't it wonderful?'. Hobbes follows up with 'Everything familiar has disappeared! The world looks brand-new!' -- A new year... a fresh, clean start!' adds Calvin. As they look over the landscape, Hobbes observes 'it's like having a big white sheet of paper to draw on!' -- 'A day full of possibilities!' Calvin affirms. They get ready on their sledge and Calvin tells his friend 'it's a magical world, Hobbes, ol' buddy... let's go exploring!', and the pair sledges off into the wintery landscape.
31 December 1995. Still the perfect goodbye.
Yessssβ¦K_0 = Euler Calculus!
Today at Grading for Growth: A guest post from two Dutch faculty on how they're using alternative grading in probability and statistics.
gradingforgrowth.com...
Random autumn snapshots
my younger brother is at the point of his undergraduate math degree (taking galois theory) where he no longer believes in the reality of the irrationals. they grow up so fast!
Are you interested in Root responses to climate change? Read our latest synthesis article about tropical root responses to global changes led by Daniella Yaffar and Laynara Lugli @laylugli.bsky.social on Global Change Biology. #fineroots #tropicalroots onlinelibrary.wiley.com/doi/10.1111/...
Iβm liking The Alignment Problem so farβ¦
brianchristian.org/the-alignmen...
A complex mathematical visualization showing a black square rotated 45 degrees, with symmetrical, flowing colored curves emanating from each corner. The curves create an elegant abstract pattern resembling a four-petaled flower or cross. Each corner features intricate swirling patterns in multiple colors including red, blue, and yellow lines that interweave and curl inward. The equation for this parametric visualization is shown at the top of the image. The artwork is displayed on a black background, creating strong contrast with the colored curves. The overall effect is both mathematically precise and artistically beautiful, demonstrating the intersection of mathematics and visual art.
Roots of parametric polynomials.
Made with #python, #matplotlib, #numpy and #sympy.
