Associate Professor of Math & Statistics at UAlbany-SUNY
Albany, New York, USA
Joined November 2009
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Justin Curry retweeted
New Anthropic research: How AI agents shape experts' judgment calls
When experts delegate work to an agent, do they still make the key decisions? We tested this in a randomized trial with 50 professional data analysts.
Justin Curry retweeted
Mathematics begins with mere collections of objects, yet builds towering castles of abstract symmetry.
We start with a simple Set equipped with a binary operation, stepping up into Monoids and Groups where every element finds its inverse.
As we introduce a second operation, these structures blossom into Rings and Fields, capturing the familiar arithmetic of numbers.
Justin Curry retweeted
now with lean
github.com/anthropics/formal…
Justin Curry retweeted
Today I was trying to give a good, geometric proof of the Loomis-Whitney inequality. It states that for a measurable subset E in R^d, vol(E)^d-1 is bounded by the product of areas of the projections of E to the d coordinate hyperplanes.
A very simple corollary of this inequality is the Euclidean isoperimetric inequality with non-sharp constants in every dimension, because area of the projection can be compared upto a factor of 2 with the area of the boundary of E.
Justin Curry retweeted
This playlist on is meaning created, given, or revealed by Billy Clark is an engaging watch on the different approaches through which we can make sense of our lives: youtube.com/watch?v=Y4hphh9L…
Justin Curry retweeted
"The Linear Algebra Behind Google" is a short paper that explains the mathematics behind Google's original PageRank algorithm.
The paper covers eigenvalues and eigenvectors, stochastic matrices, Markov chains, and iterative methods, using numerical examples and graphs to illustrate how PageRank works.
Beyond the algorithm itself and the original paper by Larry Page and Sergey Brin, this is an interesting read for anyone curious about how linear algebra applies to one of the most influential algorithms in the history of the web.
sites.cs.ucsb.edu/~gilbert/c…
Justin Curry retweeted
This AI acknowledgement takes the win. From this paper: arxiv.org/pdf/2610.27918.
Justin Curry retweeted
I’d like to recommend this piece by Sartre. I guess we all think we write because we want to know that we mean something in this world. That we spotted something that had eluded us and gave it expression. Sartre removes that from us. We uncover things about our world we didn’t++
Justin Curry retweeted
Just spotted this. Many thanks to Proofs & Prompts for the shoutout in their newsfeed.
Justin Curry retweeted
Conley Index🤝 Flow Categories
Landfall Geometry
arxiv.org/abs/2610.30993
Joint with Frances Kirwan!
Landfall Geometry
Frances Kirwan, Vidit Nanda
arxiv.org/abs/2610.30993 [𝚖𝚊𝚝𝚑.𝙰𝚃 𝚖𝚊𝚝𝚑.𝙳𝚂]
ALT We study the negative gradient flow of analytic maps f:X→mathbbR defined on compact analytic Riemannian manifolds without imposing any nondegeneracy or transversality assumptions. We construct tight isolating blocks around the connected components of the critical set Crit(f) and record only bounded-time contacts made after an orbit exits one block and before it makes landfall within the interior of another. These contacts assemble into a strict topological category \cL – called the landfall category – whose classifying space B\cL is shown to be simple homotopy equivalent to X naturally over all down-sets of the flow partial order. The filtered homotopy type is independent of the isolating blocks. Passing to a finite subanalytic triangulation yields a flow-induced acyclic partial matching, and hence, an explicit finite chain complex generated by the non-exit simplices of the blocks which computes the homology of X. The finite model also computes the Morse–Conley spectral sequence over a
Justin Curry retweeted
Pope Leo: “We live in a performance-driven culture that constantly compares and ranks people, sometimes ruthlessly” and a “misguided understanding of meritocracy ultimately fails to respect people.” vaticannews.va/en/pope/news/…
Justin Curry retweeted
A new NPR investigation has found a troubling problem in AI-generated mathematics.
Cambridge mathematician Anders Hansen identified a discrepancy between an AI-generated proof and its Lean formalization.
A constant had changed from 4 to 5, making the formally verified statement weaker than the original claim.
The Lean verification itself was not necessarily incorrect. The concern is that it may have verified something different from what the original mathematical argument asserted.
The report also features Daniel Litt, who identified an error invalidating a separate AI-generated paper, and Dana Moshkovitz, who discusses a proposed solution to the Unique Games Conjecture.
The investigation highlights a distinction that is becoming increasingly important: verifying a formal proof is not enough unless the formal statement faithfully represents the intended theorem.
Source: NPR, 21 October 2026 (US), 22 October 2026 (IST).
Justin Curry retweeted
Act only according to that maxim whereby you can at the same time will that it should become ... a global section of the law sheaf.
events.blackthorn.io/en/2Eo1…
h/t Dave Anderson
Justin Curry retweeted
In 1852, Ludwig Schläfli proved that above four dimensions only three regular shapes can exist. The paper sat unpublished for 34 years - too far ahead of its time. Two dimensions have infinitely many regular shapes. Three have five. Four have six. Then the universe hard-codes exactly three forever - and every LLM embedding space and every quantum chip operates in the dimensions where this permanent pattern lives.
The reason five platonic solids exist and not six is pure angle arithmetic. Stack regular polygons around a shared corner - sum below 360 degrees and you get a valid 3D corner, hit exactly 360 and you get a flat tiling, exceed it and the surface warps like a potato chip. Triangles give three valid solids, squares one, pentagons one. In four dimensions the same logic produces six polytopes - including the 600-cell with 600 tetrahedral faces and 1152 symmetries. Above four dimensions only three families survive: the simplex, the hypercube, and the cross-polytope. The hypercube tiles n-dimensional space the way cubes tile 3D - the same structure indexing positions across every transformer attention grid today.
25 minutes. Bookmark & watch today. The sequence 5, 6, 3, 3, 3, 3 and the geometry behind every dimension your model operates in.
Justin Curry retweeted
Over the past year, the Discovery team at @GoogleDeepMind, in a project led by student researcher @dsantosmarco, has made new discoveries in one of the most beautiful open problems at the edge of computation: Finding the 6th Busy Beaver (BB) number.
While most AI math systems solve human-generated problems, like theorems, this work tackles a different aspect of mathematical creativity: Discovering hidden problems.
For years, the BB Community has pursued one of theoretical computer science’s most captivating collaborative quests: finding the simplest programs that run the longest before halting. Because determining whether these programs ever stop is uncomputable, no universal algorithm can solve it, meaning new mathematical techniques need to be found and invented to make progress.
In this way, the BB challenge serves as an open ended domain for mathematical creativity and, in particular, a generator of hard math problems called Cryptids. These are tiny programs whose halting behavior is governed by and equivalent to deep, unsolved math problems in number theory, similar to the Collatz or Goldbach conjectures. Historically, the BB community had recognized ~20 Cryptids in BB(6), and the best problems were selected for the Busy Math Olympiad (BMO).
Building on their work, Marco built a pipeline powered by Gemini 4 Argon that:
1️⃣ Simulates raw Turing machine traces to compress recurring patterns.
2️⃣ Conjectures mathematical transition rules & runs synthetic experiments to test edge cases.
3️⃣ Proves the equivalence formally in Lean 4.
Using this pipeline, Marco and the team discovered 20 new potential Cryptids and verified 33 more machines previously studied in the BB community. All the proofs were open sourced and shared in the community discord channel and we hope that they will help the community to make progress towards resolving BB(6). We also contributed related halting/non-halting proofs directly to the community GitHub repository.
Finding these Cryptids shows hidden links between simple programs and number theory, and helps mathematicians map the limits of computation. I'm also excited about Gemini 4 Argon's ability to help researchers both solve hard problems and accelerate scientific discoveries, of which Busy Beaver is just one example.
Work led by our student researcher @dsantosmarco with paper contributions from @vivek_veeriah, @Dahoas1 , Andrew Bolt, @zuza777 , Satinder Singh, and myself with additional contributions from Hussain Masoom, @maxencefaldor , @MoritzFirsching and Petar Sirkovic.
Read more about it in our paper: arxiv.org/abs/2610.22603
Justin Curry retweeted
In 1950, Norbert Wiener explored automation’s promise and risks. Today, at Cyborg Psychology, we study how AI shapes our cognition, relationships, and sense of self and how to design it to support human agency and flourishing.
Justin Curry retweeted
R package for information-geometric centroids of probability distributions:
pregnault.github.io/igcentro…
integration with package flexclust
- Symmetrized KL/Jeffreys centroid
- Jensen-Shannon centroid
- Jeffreys-Fisher-Rao Center and the Gauss-Bregman Inductive Center
- Etc.
Justin Curry retweeted
Beautiful talk by Grant Sanderson @3blue1brown about a fun question in geometry, the problem-solving utility of thinking topologically, and the buildings of Frank Gehry. Oh yes, and also about "The Role of Exposition in the Future of Mathematics" youtu.be/PtCPLLuMK1o?is=C-UI…
Justin Curry retweeted
I’m very opinionated. So it’s nearly impossible for me to hear someone’s arguments and *not* find something I disagree with. (This includes reading things I myself wrote years - or weeks - ago.) And yet the amazing Grant Sanderson @3blue1brown has put out a long, detailed video on the future of mathematics, in which I cannot find a single flaw. I’ll be sending this to every PhD student and postdoc I know, and strongly recommend everyone watch it.
youtube.com/watch?v=LKrwhjOI…
(Of course I tried to say many of the same things in my conversation with Thomas Koberda. But Grant’s version, unsurprisingly, does an infinitely better job at communicating those ideas!)
Justin Curry retweeted
Grant Sanderson (@3blue1brown) predicts that in 3 years, there will be an end-to-end AI generated channel better than his! 🤯
He admits to feeling resigned about it:
"There's a part of me thinks, 'hey, even if it only lasts a couple glorious years, like, to fully embrace that.'"