The studio pays for itself; this is what it's for. A multi-project program in topology, number theory, and the foundations of inference — half headed toward a PhD, all of it where the methods I bring to clients come from. In the open, and honest about what's a finished thought versus a first one.
Why this exists
I want to help people make better decisions — and to leave the world a little less anxious than I found it, for the humans before me, the humans beside me, and the humans after me.
Most of the fear around a hard choice is uncertainty wearing a costume. The mathematics here — telling signal from noise, holding a belief at its honest strength, acting well when you can't be sure — is all in service of one stubborn conviction: it doesn't have to be this way. Things can be clearer, kinder, and better shaped. That's the work, and the reason the studio exists to fund it.
The program — the value of information
One question runs under all of it: what is information worth — to have, to act on, and not to have? Ten threads, each a face of that one question. Two are active, a couple are building, the rest are honest seeds.
IHaving it — reading the information in a thing
The method under most of the program: real structure persists, noise doesn't. Read a human thing by its shape — perception, a melody, a rhyme — with persistent homology, or read a covariance matrix by its spectrum with random matrices. Either way, keep only what's real and let the rest go.
Information is only worth what it changes about what you'd do. Hold a belief at its honest strength, then act — the value is exactly the gap between deciding after you know and deciding blind. This is the bridge from the research to the studio.
P2Early
QBism as Decision Theory
Quantum Bayesianism as an implementable inference engine — beliefs an agent acts on, not just a philosophical stance.
Julia · LaTeXthe bridge to the studio
IIINot having it — the hidden, and the relational
Two faces of absence: the primes' hidden discipline — the arithmetic order that underwrites what can be concealed — and information that isn't in a thing at all but between agents. A knot-record holds nothing until someone reads it.
P6Building
Shadows of F₁
The Riemann Hypothesis saga, worked computationally — the field with one element as a shadow over the integers, with Ramanujan as companion.
Where the threads start to rhyme — kept private until each connection is vetted — and the platform you're reading them on.
P7Active
The Shadow of G
The synthesis thread — where lyric topology (P5), the number-theory work, and the process-relational substrate start rhyming with each other. Active and note-heavy, and deliberately kept private until each connection is vetted. The standing question it keeps asking: how can anyone be certain this early? Hints, not claims — yet.
Obsidian · Manimsynthesis · vetting in private
P8Active
The Platform
This site, and the studio it grew into — research notes and project pages alongside honest, useful client work. You're reading it.
Next.js · Obsidianpublic-facing
The immortal problems
These questions outlive everyone who works on them. I don't expect to solve them — I orient by them, and they pull the program forward. They're also tangled in each other, and in where mathematics and AI are heading right now.
BSDBirch & Swinnerton-Dyer
How many independent rational points an elliptic curve carries — its deepest arithmetic — is supposed to be readable off how its L-function behaves at the center. Counting meets calculus.
rankE(Q)=ords=1L(E,s)
Sibling to RH: both ask what the zeros of an L-function know about arithmetic — RH about the primes, BSD about a curve's points. It's the live edge of my P6 number-theory thread, and the natural target of the P10 L-function wedge, where a family's random-matrix symmetry type is conjectured to predict exactly this central vanishing.
Every nontrivial zero of the Riemann zeta function sits on one vertical line. If it holds, the primes are as evenly spread as they could possibly be — randomness with a hidden discipline.
ζ(ρ)=0⟹Re(ρ)=21(nontrivial ρ)
The prototype every other L-function imitates. Its zeros don't just sit on a line — their spacings follow GUE random-matrix statistics (the Montgomery–Dyson link), which is the precise bridge my P10 spectral work walks between number theory, high-dimensional data, and physics. Where “signal vs. noise” and “prime vs. structure” turn out to be the same question.
A vast web of dictionaries translating between number theory and harmonic analysis — symmetry on one side, spectra on the other. The roof under which RH and BSD are special pages.
Galois representations⟷automorphic forms
It contains the other two as cases: RH and BSD are entries in its dictionary between symmetry (Galois) and spectra (automorphic forms). That same symmetry-first instinct now drives geometric deep learning in AI — structure a model can't afford to ignore — and it's the horizon my modular-forms reading and the P10 spectral work both point toward.
On a smooth complex projective variety, which topological holes are actually cut out by polynomial equations? Hodge says: exactly the ones of type (p,p). When a hole isn't just there, but algebraic.
Hk,k(X)∩H2k(X,Q)=⟨[Z]:Z⊆Xalgebraic⟩Q
The one where all three lenses meet at once: topology (the cohomology my TDA work reads), geometry (the manifold ladder), and algebra (polynomial cycles) fuse into a single question — which topological invariants are algebraic. It is the deep form of “which structure is real signal,” one floor up, and it borders the motivic world where the L-functions and Langlands live.
This program is where one belief gets tested at full scale — formulation over solution, from how I think. The method is the same one I bring to students: higher-order encoding and mind-mapping (in the spirit of Justin Sung), then retrieval, spacing, and teaching. The full walkthrough — with a live mind-map that builds itself — lives in How I teach →.
Mathematics & the wider neighborhood
The territory the program sits inside — home turf, and the adjacent currents that quietly feed it. Open it when you want the texture.
Home mathematics · other currentsexpand ▾
Mathematics I'm living in
Elliptic curves and modular forms in one hand, persistent homology in the other, with QBism as the bridge. The rest is the wider neighborhood I read in. More about me →
Number theory
Elliptic curvesModular formsq-analogAnalytic number theoryp-adic analysisRamanujan's notebooksRiemann Zeta FunctionField with one element (F₁)
Bayesian decision theoryde Finetti exchangeabilityQBismMathematical statisticsRobust estimationInformation theory
Artificial intelligence
Foundation models & LLMsAgentic systems & tool useReinforcement learning (RLHF)Probabilistic & Bayesian MLGeometric & topological deep learningCausal inferenceMechanistic interpretabilityAI alignment & safetyOptimizationNeurosymbolic reasoning
Analysis
Real & complex analysisInfinite seriesSpecial functionsSpectral methods
Foundations & philosophy
Analytic idealismPhenomenologyPhilosophy of probabilityInfinite Ethics
Other currents
Adjacent curiosities — several quietly feed the program more than the “serious” work does.
Human temporality
How lived time bends — duration, anticipation, the specious present. The phenomenology under P1, and quietly the most interesting question I know.
Poetry
Read closely and written badly on purpose — the raw material for P5.
Chess
Pattern, tempo, and sitting with a hard position without flinching.
Neuroscience
How the brain builds the perceptual space the topology is trying to measure.
Some of this turns into writing when it's ready to be read. If a thread here overlaps something you're working on, I'd genuinely like to hear about it — research@backporch.studio.