ABOUT ME
My name is Rin Ray (they/them). I’m a researcher-artist and a math postdoc at Münster, working where homotopy theory meets arithmetic.
Before the math and alongside: measurement and inference where the observable is missing, degraded, or not addressed to the observer — in biology, neural signals, robotics and physics.
Here for… the math ↓ the science ↓ the art ↓
THE PATH
I am currently a postdoc at Uni-Münster in the Arithmetic and Homotopy Theory Working Group led by Thomas Nikolaus and Christopher Deninger.
Before that, I graduated from George Mason University at 16 with a B.S. in Computational Physics, and accepted the Thiel Fellowship in 2014 to develop medical technology and study mathematics under my mentor, Edward Frenkel.
I came to mathematics from the laboratory, and never really left: computational physics, robotics, machine learning on biological signals, and medical devices since 2011. That work is in The Lab, below.
Interview — WIRED, on Medical Tech
I graduated with my Master’s degree from UChicago working with Peter May and Kazuya Kato (加藤 和也), and with my PhD from Northwestern working with Paul Goerss. Yifeng Liu advised me too, without the title.
THE PROOFS
In the beautiful mountains of mathematics, the same shadow often falls into two valleys that do not talk to each other. My work is to find the peak, name the light, and see what else it casts. The arithmetic of number fields and function fields, with its zeta functions, automorphic spectra and special values, lives at chromatic height one. My program is to find what it is a shadow of at every height.
- Scattering (function fields): an inner product of Eisenstein series and an index on the Langlands dual side are two shadows of one object; Kazhdan and Okounkov proved they agree. I look for the object behind both, a trace of Frobenius on a Morita context, and ask what it casts at every height.
- Echoes (number fields): special values of zeta functions and homotopy groups count the same things, and the prime 691 appears both in exotic spheres and in the K-theory of the integers. I look for the objects behind these coincidences at every chromatic height, starting with a zeta function whose functional equation is a duality.
- Symmetry you can touch (where the two meet): a symmetry coming from a division algebra over \(\mathbb Q_p\) and the symmetry of a curve in characteristic \(p\) are two shadows of one action. I prove the curve is forced by the group alone, and find the action inside it.
Arrows point from what supplies to what needs it. Click a box to jump to it.
What the labels on papers mean
- published in a journal
- submitted under review at a journal
- preprint public, not yet submitted
- manuscript full draft written, not yet public
- in preparation main theorems proved, being written up
- conjecture, checked computations done, no proof yet
- early stage started, nothing to share yet
- note a short write-up, not a paper
Scattering current
The spectral measure of Eisenstein series is an index on the other side of the Langlands mirror.
needs ← an L-function at height h, from Echoes needs ← the height-h theories, from Symmetry
One project, three papers
The Unramified Eisenstein Spectrum as a Categorical Trace
manuscript
solo
On a curve over a finite field, the inner product of two Eisenstein series is an index on the Langlands dual side, by a theorem of Kazhdan and Okounkov. In that formula a whole moduli of bundles is replaced by \(2g\) numbers and a sign: the curve enters only through a square root of its zeta function, and the answer forgets which square root. I show the measure is the Euler characteristic of an explicit sheaf, the Koszul complex of the tangent bundle of the Steinberg stack twisted by the chosen half of the curve’s cohomology. The measure is canonical; the sheaf computing it is not. Categorically, I conjecture the formula is one reading of a single object, a trace of Frobenius on a Morita context, with the square root as the orientation the category leaves out.
L-genera and Localizations in K-theory
early stage
in conversation with Daniel Berwick-Evans and Natalia Pacheco-Tallaj
The Eisenstein formula is canonical, but writing it down takes a choice of square root that the answer forgets. The Witten genus has the same problem one height up: its class is canonical, but a cocycle for it takes a choice of rational string structure. We ask whether these are one phenomenon, and whether the forgetting is an anomaly that cancels.
Habiro Characteristic Classes of L-function Genera
early stage
in conversation with Ferdinand Wagner
In the Eisenstein formula \(q\) does two jobs: on one side it is the size of a finite field, on the other a formal variable. We let \(q\) vary and ask whether the formula’s two faces, function fields in K-theory and number fields in cohomology, are specializations of one Habiro characteristic class. The Habiro ring sees \(q=1\) directly; the function field value \(q=p^n\) is not a root of unity, and that is the whole difficulty.
Echoes current
The same numbers, computed twice: once by homotopy theory, once by arithmetic.
gives → Scattering its height-h L-function
The \(\lambda\)-twisted \(T(h+1)\)-local Zeta Function
in preparation
joint with Gabe Angelini-Knoll
A zeta function for chromatic ring spectra, built from \(T(h+1)\)-local syntomic cohomology. A chromatic Tate twist lets it see the height: for \(BP\langle h\rangle\) it has \(2^{h+1}\) zeros and poles, one for each exterior class, and its pole at \(T=1\) is redshift. Its functional equation holds with no correction factor, and it is the higher arithmetic duality of Devalapurkar, Hahn and Rognes.
L-functions and the Relative Image of J in Moore Spectra
conjecture, checked
Adams showed that the denominators of \(\zeta(1-n)\) count the image of J. For the mod \(p^k\) Moore spectrum the naive analogue is false once \(k\ge2\). I conjecture that its homotopy is glued from the two relative pieces of the extension \(F_k/\mathbb Q\), the cofibers of the unit and of the transfer, so that \(|\pi_{2n}(L_{K(1)}\mathbb S/p^k)|\) is the gcd of the denominators of \(\zeta_{\mathbb Q}(1-n)\) and of \((\zeta_{F_k}/\zeta_{\mathbb Q})(1-n)\). This recovers Salch’s theorem at \(k=1\).
checked numerically for \(p = 3, 5, 7\) and \(k \le 3\) from generalized Bernoulli numbers
Toward Categorifying Symplectic L-functions and Reidemeister Torsion
in preparation
joint with A. Petrov and M. Volpe
A curve over a finite field behaves like a 3-manifold. Abdurrahman and Venkatesh went past the analogy: they proved that the central value of a symplectic L-function, mod squares, is a Reidemeister torsion, the invariant topologists use to tell 3-manifolds apart. The natural route to categorifying it, through L-theory, provably fails: their invariant is exactly the part of the symplectic Chern class that L-theory cannot see. So we work in hermitian K-theory instead, where it appears as a secondary invariant, left over when the L-theory class vanishes for a canonical reason.
All Bernoulli Numbers in Homotopy Theory Are Shifts
early stage
joint with Andres Mejia and Noah Riggenbach
The prime 691, the numerator of \(B_{12}\), appears twice: in the exotic 23-spheres, through Kervaire–Milnor, and in \(K_{22}(\mathbb Z)=\mathbb Z/691\), through Quillen–Lichtenbaum. Both are \(\zeta(-11)\). The classical route between them is rational; we propose an integral one, through Blumberg–Mandell’s K-theoretic Tate–Poitou duality, so that an exotic sphere is detected by the K-theory of the integers.
Symmetry you can touch
Curves, Jacobians and periods built toward the group action Lubin–Tate space hides.
gives → Scattering the height-h theories
The Lubin–Tate action, in four papers
The Morava stabilizer group acts on Lubin–Tate space. That action drives the \(K(h)\)-local sphere, yet it is explicit almost nowhere. For its maximal finite subgroup \(G'\), the group alone determines a unique curve, and the action is the symmetry of that curve.
- Part II (my thesis, 2023): at height \(p-1\), the deformations of the curve are Lubin–Tate space. This solved a 40-year-old problem, and from the curve I compute \(\pi_*E_{p-1}\) and its cohomology.
- Part III (2026): at every height \(p^{k-1}(p-1)\), the moduli are surgeries. Take the curve’s quotient by the centre of \(G'\), cut out the formal disk around each of its \(p^k\) branch points, and glue each one back in along an automorphism of the punctured disk. I prove the stack of these surgeries has exactly the Lubin–Tate tangent.
- Chromatic splitting (2026, joint): with the two-tower method of Part I we prove the coheight-one weak chromatic splitting conjecture at every height at once, for generic primes. Before, it was known only at heights 1 and 2.
- Kervaire: at \(p=3\), height 6, this is the action behind the detection theorem in the last open odd-primary case, with Eva Belmont.
- Part I is the toolkit: two ways to pin down a group acting on a formal moduli stack. Geometric modeling is what Parts II and III carry out; the two-tower method is what the splitting paper runs on. The tools are standard, but here they are set down in the stronger form the Lubin–Tate action needs: equivariantly, in families, and with the grading kept.
Modeling Group Actions on Stacks (Especially the Lubin–Tate Action)
Part IJuly 2025submitted: Expositiones Mathematicae
The groundwork for the thesis: before you can compute with a symmetry you cannot see directly, you need a way to make it visible.
Abstract
Suppose we are given a profinite group \(G\) acting on a formal moduli stack \(\mathcal{M}\), and we want to understand the group action, and compute cohomology related to this group action. How can we do it? This prolegomenon conveys two methods of pinning down such an action: geometric modeling and the two-tower method. We highlight their use on a specific action – the automorphisms of a formal group acting on its deformation space, called the Lubin–Tate action.
We provide the toolkit that these two methods run on. The pieces of the kit are standard in their own subjects, but the Lubin–Tate action requires us to sew them together and ask more of them than their usual statements give, and several are set down here in that stronger form: equivariantly, in families, and with the grading kept. The tools are put to work in the two sequels, which build geometric models at height \(p-1\) and at the heights \(p^{k-1}(p-1)\) above it; the two-tower material is a companion to the proof of the coheight one weak chromatic splitting conjecture.
Background for two projects. It surveys two ways of pinning down the Lubin–Tate action: geometric modeling, which Parts II and III carry out, and the known two-tower method, which the chromatic splitting paper uses.
v2, Oct 2026: expanded into three parts, with the shared background from Part II moved here
Tiny and Unique: The Lubin–Tate Action at Height \(p-1\) from the Minimal Curve with Its Symmetries
Part IISept 2025submitted: AG&T
A symmetry hidden inside homotopy theory turns out to be the symmetry of an actual curve, one you can write down and draw.
Abstract
Let \(p\) be odd and \(G' = \mathbb{Z}/p \rtimes \mathbb{Z}/(p-1)^2\) the maximal finite subgroup of the Morava stabilizer group at height \(p-1\). Inverse Galois theory produces from \(G'\) alone a curve \(X\), the unique curve of minimal genus with \(\operatorname{Aut}(X) \simeq G'\); its ramification, its field of definition and its equation are consequences of the group, not choices. We prove a \(G'\)-equivariant equivalence between the deformations of \(X\) and Lubin–Tate space, so that the Lubin–Tate action of \(G'\) is the action of \(\operatorname{Aut}(X)\) on deformations of the curve. The proof is a coordinate-free Kodaira–Spencer argument reducing to a single character count. The action becomes explicit: \(G'\) acts through \(\mathbb{F}_p \rtimes \mathbb{F}_p^\times\) shifting and scaling \(p+1\) points on \(\mathbb{P}^1\). From this we compute \(H^*(G', π_* E_{p-1})\) and its Tate cohomology. One identity, \(π^{p-1} = -p\), runs through every section.
v3, Oct 2026: retitled and revised together with Part I
Only the Gauge Grows: Moduli Stacks of G-Curves in Homotopy Theory at \(h = p^{k-1}(p-1)\)
Part IIImanuscript
At every height \(h = p^{k-1}(p-1)\), the moduli are surgeries on \(Y_k\), the quotient of the curve by the centre of \(G'\), at the free orbit \(B_0\) of its \(p^k\) branch points over \(0\in\mathbb P^1\). A surgery is a deformation of \(Y_k\) obtained by regluing it from \(Y_k\smallsetminus B_0\) and the formal disks at the points of \(B_0\), along automorphisms of the punctured disks. The pole order of that automorphism filters the surgeries; at \(k=1\) a surgery of pole order zero just moves a branch point. I build the smooth stack of these surgeries and prove that its tangent space and the Lubin–Tate tangent are the same representation: functions on the branch points, modulo a gauge. The moduli stay on the branch points; only the gauge grows.
How far along: proved: the curve, the stack, the matching representation, the derivative of the period map, and the Cartier chain at \(k=2\) for every \(p\). Conjectured: that Lubin–Tate space is the horizontal surgeries modulo the kernel of the period map, checked by machine at \((p,k)=(3,2),(5,2)\).
Abstract
Fix an odd prime \(p\) and an integer \(k\ge1\). The Morava stabilizer group \(\mathbb{G}_h\) has finite subgroups of order divisible by \(p\) only at heights \(h=p^{k-1}(p-1)m\), and at \(m=1\) a maximal one, maximal in the sense of not being properly contained in a larger finite subgroup, is \(G'=\mathbb{Z}/p^k\rtimes\mathbb{Z}/(p-1)^2\). Set \(h=p^{k-1}(p-1)\); then \(G'\) acts on the Lubin–Tate deformation space \(\mathrm{Def}(H_h)\) of a formal group of height \(h\). Our first theorem identifies the tangent space of \(\mathrm{Def}(H_h)\) at its closed point as a representation of \(G'\): it is \((\mathfrak{p}_L/p\mathcal{O}_L)^\vee\) for \(L=\mathbb{Q}_p(\zeta_{p^k})\) and \(\mathfrak{p}_L=(\zeta_{p^k}-1)\); equivalently, it is the space of functions \(W_k(\mathbb{F}_p)\to\overline{\mathbb{F}}_p\) on the Witt vectors of length \(k\), modulo those of difference degree at most \(p^{k-1}\), with \(\mathbb{Z}/p^k\) acting by translating \(W_k(\mathbb{F}_p)\) and \(\mathbb{Z}/(p-1)^2\) by Teichmüller scaling.
We then realize this representation geometrically. There is a unique curve \(X_k\) of smallest genus carrying a faithful \(G'\)-action. Writing \(Y_k\) for its quotient by the centre of \(G'\), the subgroup \(\mathbb{Z}/p^k\) acts on \(Y_k\) with \(Y_k/(\mathbb{Z}/p^k)\cong\mathbb{P}^1\), and the fibre \(B_0\) over \(0\) is a free orbit of \(p^k\) points, canonically identified with \(W_k(\mathbb{F}_p)\). A surgery is a deformation of \(Y_k\) obtained by regluing it from \(Y_k\smallsetminus B_0\) and the formal disks at the points of \(B_0\) along automorphisms of the punctured disks; the pole orders of those automorphisms filter the surgeries. We prove that the surgeries of pole order exactly \(p^{k-1}-1\) form a smooth formal stack \(\mathfrak{S}_k\) over \(W(\overline{\mathbb{F}}_p)\) of dimension \(h-1\), and that its tangent space is the representation above. At \(k=1\) a surgery of pole order \(0\) moves a branch point, and \(\mathfrak{S}_1\) is the configuration space of Part II.
Along a family of surgeries the crystalline cohomology of \(Y_k\) is constant as an isocrystal, and only its lattice moves. Projecting that lattice to the slope-\(1/h\) summand and reading its Hodge line defines a period map \(\Phi\) to \(\mathrm{Def}(H_h)\), on those surgeries for which the Hodge line stays in the summand. We compute the derivative of \(\Phi\) as an explicit residue, and conjecture that \(\Phi\) identifies \(\mathrm{Def}(H_h)\) with the stack of such surgeries modulo the kernel of \(d\Phi\). For \(k=2\) we locate the representation inside the curve: on the isotypic part of \(H^0(X_2,\Omega^1)\) for the centre, the Cartier operator has exactly one Jordan block of length at least \(p\), of length \(p^2-2\); dually, the image of \(\bar F^{\,p-1}\) on \(H^1(X_2,\mathcal{O})\) is, after one explicit character twist, the representation above, for every odd \(p\).
On the Chromatic Splitting Conjecture in Coheight 1
Sept 2026preprint
joint with Tobias Barthel, Lucas Mann, Andy Senger, Tomer Schlank, Jared Weinstein, and Xinyu Zhou
Homotopy theory breaks the sphere into layers, one for each height, and Hopkins predicted how neighbouring layers fit back together. We confirm that prediction for neighbouring layers at every height, using geometry rather than brute-force computation.
Every height at once. We prove the coheight-one form of chromatic splitting at every height \(h\), for all primes with \(p-1\nmid h-1\). Before this it was known only at heights 1 and 2.
Abstract
This article confirms predictions made by Hopkins’ chromatic splitting conjecture in stable homotopy theory. A consequence of our main theorem is that a certain map of \(K(h-1)\)-local \(E_\infty\)-rings \(\mathbb{S}^{B\mathbb{Z}_p}_{K(h-1)} \to L_{K(h-1)}\mathbb{S}_{K(h)}\) admits a retract, so long as \(p-1\) does not divide \(h-1\). If \(h=2\) and \(p\) is odd, we show this map is an isomorphism.
We use methods from the theory of \(p\)-divisible groups and \(p\)-adic geometry. Along the way we prove an entirely new kind of “two-tower isomorphism”, in analogy with Faltings’ isomorphism between the Lubin–Tate and Drinfeld towers, which parametrizes \(p\)-divisible groups over characteristic \(p\) perfectoid spaces with fixed generic fiber in terms of negative Banach–Colmez spaces. Using the two-tower isomorphism, we gain control over the continuous cohomology of the Morava stabilizer group \(\mathbb{G}_h\) with coefficients in \(\pi_* L_{K(h-1)} E_h\).
To study the continuous action of \(\mathbb{G}_h\) on \(L_{K(h-1)} E_h\), it was necessary to develop the theory of solid Lubin–Tate spectra. Adapting Lurie’s constructions, we attach a solid Lubin–Tate spectrum to \(p\)-divisible groups over a broad class of solid rings in characteristic \(p\).
The background for this paper is in Modeling Group Actions on Stacks, which surveys the two-tower method.
Towards the p = 3 Kervaire Invariant Problem: The \(E_2\)-page for the Homotopy Fixed Points Spectral Sequence Computing \(\pi_*(E_6^{hC_9})\)
July 2025submitted
joint with Eva Belmont
The Kervaire invariant problem, one of the classical questions about high-dimensional manifolds, has one case left: the prime 3. This paper sets up the computation that decides it.
Used in a claimed resolution. In September 2026 OpenAI posted a preprint claiming to resolve the Kervaire invariant problem at the prime 3, using our results and the method Eva and I proposed in this paper.
Abstract
Hill, Hopkins, and Ravenel suggest that the last remaining Kervaire invariant problem, the case of \(p=3\), can be solved by computing the homotopy fixed points spectral sequence for \(π_* E_6^{hC_9}\). We prove a detection theorem for this case and use a conjectural form of the \(C_9\)-action on \(E_6\) to compute the \(E_2\) page of this spectral sequence away from homological degree zero.
The Eigenvalues of Frobenius of Artin–Schreier–Witt Curves are Gauss Sums
note
The Frobenius eigenvalues of Artin–Schreier–Witt curves are Gauss sums, and reading off their slopes gives new families of Newton strata in the Torelli locus.
This is what certifies the curves above as models: the Newton polygon of a curve, the slopes of its Frobenius eigenvalues, is what splits a height-\(h\) formal summand off its Jacobian.
Earlier
Before the three threads: parallel processes, polarizations, period maps.
Toward Directed Collapsibility
2020published
joint with R. Belton, R. Brooks, S. Ebli, L. Fajstrup, B. T. Fasy, N. Sanderson, E. Vidaurre
A space can look trivial and still hide many essentially different ways for parallel programs to run. We give a way to simplify such spaces without losing any of them.
Abstract
In the directed setting, the spaces of directed paths between fixed initial and terminal points are the defining feature for distinguishing different directed spaces. The simplest case is when the space of directed paths is homotopy equivalent to that of a single path; we call this the trivial space of directed paths. Directed spaces that are topologically trivial may have non-trivial spaces of directed paths, which means that information is lost when the direction of these topological spaces is ignored. We define a notion of directed collapsibility in the setting of a directed Euclidean cubical complex using the spaces of directed paths of the underlying directed topological space relative to an initial or a final vertex. In addition, we give sufficient conditions for a directed Euclidean cubical complex to have a contractible or a connected space of directed paths from a fixed initial vertex. We also give sufficient conditions for the path space between two vertices in a Euclidean cubical complex to be disconnected. Our results have applications to speeding up the verification process of concurrent programming and to understanding partial executions in concurrent programs.
Advances in Mathematical Sciences, vol. 21, pp. 255–271
Automorphisms of Abelian Varieties and Principal Polarizations
2022published
joint with D. Lee
One geometric object can often be polarized in several genuinely different ways. We wrote a computer search that counts them on some famous curves.
Abstract
The Narasimhan-Nori conjecture asks for a closed formula for the number of non-isomorphic principal polarizations of any given abelian variety. In this paper, we introduce a new algorithm that gives a lower bound on the number of non-isomorphic principal polarizations on any given abelian variety. We show, for example, that the Jacobian of the genus four underlying curve of Schoen's I-WP minimal surface has at least 9 non-isomorphic principal polarizations. We also explore the Jacobians of Klein's quartic, Fermat's quartic, Bring's curve, and more.
Rendiconti del Circolo Matematico di Palermo, Series 2, vol. 71, pp. 483–494 · circulated in preprint as Automorphisms of the Jacobian
A Global Crystalline Period Map
2018preprint
joint with M. Neaton and A. Pieper
A tool that turns a curved moduli problem into linear algebra only worked one point at a time. This version works on the whole space at once.
Abstract
The crystalline period map is a tool for linearizing \(p\)-divisible groups. It has been applied to study the Langlands correspondences, and has possible applications to the homotopy groups of spheres. The original construction of the period map is inherently local. We present an alternative construction, giving a map on the entire moduli stack of \(p\)-divisbile groups, up to isogeny, which specializes to the original local construction.
EXPOSITORY
Zeta Functions and THH
Using Automorphism Groups of Curves to Control the Slopes of their Jacobians
K-theoretic Tate–Poitou Duality Seminarcoming soon
A 4-page summary of my graduate work for a general audience
Geometry for Prime Addicts
The Hecke Orbit Conjecture and Homotopy Theory
An Overview of the Classic Theory of p-Divisible Groups
Fiber Bundles of Formal Disks
A Complete Proof of the Polynomial Ham Sandwich Theorem
My two Master’s theses
TEACHING
stub An updated teaching statement is on its way; this is the current one.
I came into academia from outside, so I go out of my way to present mathematics in ways that are accessible to everyone. For me it is a visual, interactive, intuitive and artistic experience, and above all a narrative one. I keep learning materials for students of all levels on my resources page. My teaching statement:
THE LAB
I build things, and I believe almost anything will answer you if you are willing to learn how it speaks. What I have spent my time building, mostly, is instruments for overhearing. A blinded robot, feeling for its own hand. Rats, telling each other how they are feeling. A cat, feeling for his colon. A brain, through the scar it grows around anything that listens. Half a page of Braille, about the missing half. Arithmetic, about superconductors nobody had made yet. A drone and a wheelchair, microwaved argon and a plate of food, a face, septic blood — and others. There are always others.
Most of this came before the mathematics found me — or before I found it. Two threads have remained in my hands: a cluster of diseases that keep showing up together and keep being treated apart, and a treatment that works and was never written down.
I build things, and I believe almost anything will answer you if you are willing to learn how it speaks. For over a decade I have studied the three kinds of silence: what is absent, what is fading, and what is spoken in another room. Three threads remain in my hands: a cluster of diseases that keep being treated apart, a treatment that was never written down, and a cheap drug for pain that nobody will test.
A decade of work on measurement and inference where the observable is missing, degraded, or not addressed to the observer: motor intention from a thinning population of cortical units, affective state from ultrasonic vocalization, proprioception without vision, lexical structure from partial parallelism, superconducting transition temperature from electronic structure.
Living things
Hidden states, read from the traces a system emits.
Biome of Overstimulation: Sensory Sensitivity as a Unifying Mechanism2024 – ongoingcurrent
Sensory processing as an organizing principle · gut microbiome · autoimmunity · chronic pain · glutamate excitability · joint with Luca Estinto
The people I interviewed about wheelchairs kept telling me about pain, and pain was the thing nobody had an answer for. Years later the same shape kept appearing: autoimmune conditions, gut trouble, chronic pain, sensory sensitivity, and certain neuropsychiatric diagnoses turning up together in the same people and being treated by five different specialists who never speak to each other.
Restarting the nerves that move a colon2026current
Feline idiopathic megacolon · enteric motility and cholinergic transmission · neostigmine · owner documentation of an unpublished treatment
A cat of mine was dying because his colon was paralysed, and the two options put in front of me were surgery and euthanasia. He was too heavy and too unwell to survive the anesthetic, so in practice that was one option. There is a third, and it is not in the literature: it circulates as folklore between practitioners, never reaches a journal, and so whether it is offered to your animal depends almost entirely on who you happen to be standing in front of that day.
The six-cent drug nobody will test2026current
Memantine · NMDA receptor antagonism · central sensitization · opioid tolerance and hyperalgesia · drug repurposing · trial design
People who use mobility aids told me they were choosing every day between being stoned past the point of finishing a sentence and being lucid and in agony. I wanted to know whether anything avoids that choice.
Learning the ultrasonic language of ratslate 2013 – mid 2014
Computational bioacoustics · unsupervised clustering of rodent ultrasonic vocalizations · quasi-real-time monitoring in the animal’s own cage · preclinical safety pharmacology · affective-state readout · 3Rs refinement
A mouse cannot report a symptom, but it is talking constantly in a register we cannot hear. Two things follow from listening properly. You get some understanding of what these animals are actually saying to each other. And you get far more out of each experiment while doing far less to the animals in it: a microphone over a cage takes readings continuously, at night, in the dark, from an animal nobody has touched, picked up, restrained or moved to a testing room — which is both a kinder life and a cleaner measurement, since a frightened mouse is not the animal you meant to study. More information out of every cohort, less disturbance per animal, and a clearer picture of what a compound does before it goes anywhere near a person.
A multi-company validation in 2025 put three compounds that had passed conventional safety pharmacology under continuous non-invasive monitoring in their own cages and found signals anyway, some persisting for days after dosing. Quietly watching animals who are left alone turns out to see things that handling them on a schedule does not.
The signal under the scarSummer 2014
Intracortical brain–computer interfaces · decoder convergence analysis · foreign body response and chronic signal loss · optical recording
Neuroprosthetics have two major issues. At the tissue end, putting an electrode into a brain is an injury: microglia arrive, astrocytes proliferate, and a glial scar closes around it, pushing the neurons you wanted away from the thing that is listening — so the signal thins and gets noisier across the months in which a person is coming to depend on it. At the other end, any single neuron tells you almost nothing; the intention to move is legible only across a whole population of them. I came at this from the software side and assumed both were software problems. My first thought was that a decoder which recalibrated fast enough could just track the signal as it drifted — that convergence was the thing to fix. I worked on that for a while before accepting you cannot infer your way out of an electrode with fewer and fewer neurons left to hear.
Most of the answer has been materials. The stiffness mismatch is absurd — silicon around 180 GPa against a brain of a few kPa — so the field went soft: ultraflexible polymer threads, mesh electronics injected through a syringe, carbon fibers seven microns across, substrates stiff enough to go in and then soften once wet. In mice it works beautifully. One open-mesh design tracked the same neurons for thirteen months, most of the animals’ adult lives. Coatings do the rest: PEDOT to drop impedance, neural adhesion proteins, dexamethasone eluted to quiet the response. Then the uncomfortable part. Across fourteen BrainGate participants and up to seven and a half years, the arrays held spiking on about a third of their electrodes and declined only about seven percent — and when arrays are taken out and examined, the leading failure is the silicon eroding and the metal coming away, not the tissue closing in. Softer probes may be the right fix for the wrong failure. “Scar-free” was a mouse result in 2017 and has been quoted a long way past its evidence. Meanwhile the thing these arrays turned out to decode best was not a limb but speech, which in 2014 I would not have guessed at all.
A laboratory on a keyring2011 – 2014
Point-of-consumption allergen detection · laser absorption spectroscopy · G12 antibody colorimetric assay · immunoassay cost and sampling limits
For someone with celiac disease or a severe food allergy, every meal they did not cook themselves is an act of trust in a stranger, and the cost of that trust being misplaced runs from a ruined week to a funeral. The idea was to hand the judgment back to the person eating, in something small enough to live on a keyring.
In March 2014 I found GlutenTox, TellSpec and 6SensorLabs already at it — the good kind of disappointment. 6SensorLabs became Nima, which shipped in 2017, missed gluten at the legal threshold more than a fifth of the time, died in 2020, and came back in 2026 claiming 10 ppm, by its own measurement. My wall got routed around rather than climbed: the expensive part was the antibody, and the field swapped it for aptamers, synthetic DNA that folds around a target for a fraction of the cost. One 2024 design reads gliadin off gold nanoparticles with a phone camera. The part underneath is still unsolved: a negative on one bite does not clear the plate, and sampling error swamps measurement error.
Polyglass — reading a pulse off a face, and why I would not build it nowHackMIT 2013
Remote photoplethysmography from video · contactless physiological sensing on a head-mounted display · consent and surveillance ethics
The idea was to help autistic people read social communication: if the signals everyone else is supposedly picking up on are real, a machine ought to be able to surface them.
I would not build this now. It takes a reading off someone’s body without their knowledge and gives them no way to refuse — the person being measured is the one person in the room with no say in it. The need was real; meeting it this way moves the cost onto somebody who never agreed to carry it.
Bodies in motion
A body reading the traces of its own movement.
Fixing what is wrong with wheelchairs2013 – 2015
Modular retrofit robotics for powered wheelchairs · automated pressure redistribution · powered seat elevation · assisted transfer · rough-terrain drive · user-led requirements · gaze-controlled assistive mobility · mentorship
My grandfather lost his mobility, so I volunteered at a spinal cord injury rehabilitation unit and asked the people there: What would improve your quality of life? Three answers kept coming back: getting between bed and chair, being at eye level in a conversation, and ground rougher than the floor the chair was built for. Separately I went looking at what actually kills people with spinal cord injuries. Mostly infection, and an open pressure sore is one of the ways infection gets in. They are not the biggest killer — that is pneumonia; they are the biggest killer mechanical engineering can do anything about.
- five robotic pressure-injury relief mechanisms, shifting the occupant’s load without them having to do anything — depending on someone to remember it, which many cannot, fails on exactly the days they are unwell
- a transfer mechanism, for getting between bed and chair
- a seat that rises, for talking at eye level
- a drive and tires for ragged ground
I expected to hear about getting around. What I kept hearing about was pain —
constant, and largely unaddressed by anyone they had seen.
It was not a thing I could answer with mechanical engineering. What I could do was
make sure the mechanisms were not locked up: I wrote up all the mechanical engineering
and soft robotics I’d invented as a nonprovisional patent application,
Robotic Mobility Assistive Wheelchairs, and then decided not to file it, and
they went out under a Creative Commons license instead. A patent would have meant the
people I built them for waiting on somebody else to license it first.
Then I went to work on understanding and treating chronic pain, which I am still doing.
Both halves have since arrived, about a decade on. Medicare began covering powered seat elevation in 2023 — on the grounds of transfers and reach, not of meeting people at eye level, which is what the users I spoke to actually said. And automated pressure redistribution that asks nothing of the occupant is now a pilot study. I should say plainly that all of this, mine included, measures interface pressure, which is a proxy; nobody has shown it prevents an injury.
When the cameras die, the hands take overSummer 2012
Reinforcement learning · proprioception-only contact-rich manipulation · blind peg-in-hole insertion · two-arm load equalization · Willow Garage PR2 · human–robot interaction
Blindness in robots. A machine that stops dead the moment its camera fails is a machine you cannot trust around people, and sensors fail constantly.
What is recoverable once a sense is gone?
The Braille work, the following spring. Here, a robot reaching for a hole it cannot see and finding it by touch; there, a language meant to be read by touch, worked out from fragments by a machine that cannot see either. Both are asking what is recoverable when the channel everyone assumes you have is missing.
Drawing the map while flying through itFall 2012
Simultaneous localization and mapping · monocular visual SLAM with parallax bootstrapping · loop closure · scale from sonar and inertial fusion · motion planning on a partial map
To know where you are, you need a map. To build a map, you need to know where you are. So how does anything ever get off the ground?
I would not take this on now. Navigating terrain nobody has surveyed is not a neutral capability — there is very little daylight between solving it for a toy quadcopter and solving it for something armed. I was treating it as a puzzle about maps and motion, not thinking of the consequences.
The homesickness robot2011
Mimicking animal motion and companionship · robot person following · gait-based re-identification · statically stable hexapedal locomotion
My first robotics project, and the motive was homesickness — I missed my dog at university, so I built something that would trail after me.
The way out was not more legs, even though I personally find that cuter. Boston Dynamics went the other way with BigDog — four legs, which cannot fall back on geometry at all and so have to catch themselves, force-controlled limbs choosing where to put a foot in the next fraction of a second. That is the machine everybody remembers getting kicked sideways on ice and skittering back upright, and it is the approach that generalized. Quadrupeds now learn the recovery in simulation and cross ground I could never have got a hexapod over. Following a person around has a name now too — robot person following — with benchmarks, and re-identification that keeps learning your target as they change.
What if, rather than planning motion across a floor, you add flight to the problem? Three dimensions, no ground to stand on, and no option to stop and think.
Matter
Behaviour inferred from the traces that structure leaves.
Calculating a material into existence2012 – 2013
Electronic-structure calculation · augmented plane wave method · superconducting transition temperature of intermetallic compounds · computational materials screening
A superconductor carries current with no resistance: nothing lost, nothing turned to heat, and magnets far stronger than their power draw. Every one we have needs liquid helium, so the chase is for one that works warmer — and you cannot run that chase by building candidates one at a time.
The approach paid off, in other people’s hands. H3S was calculated to superconduct near 200 K and then measured at 203 K; LaH10 was predicted in 2017 and found at 250 K two years later. Materials predicted on a computer before anyone made them, which is the outcome the whole approach was betting on. They need megabar pressures, so nobody is wiring a city with them yet.
Nothing interesting happens at only one scaleJan 2015
Multiscale modeling · directed type systems · compositional model building · topology applied to complex biological systems
A multiphysics simulation is several different physics solved over one domain at once, and it is expensive twice over: the run itself, and the human work of stitching solvers together without introducing errors. Both of those are structural costs, and the structure is already sitting there in the mathematics.
Other people were heading the same way, and one of my own co-authors on Toward Directed Collapsibility is among them: Nicole Sanderson now writes on topology and neural circuits. Persistent homology has reached the tumour microenvironment too, picking out rare T-cell states under checkpoint blockade. The immunotherapy daydream had other people in it all along.
Both halves have since been developed, independently of me. Directed type theory was rebuilt on synthetic simplicial foundations, where the objects under study are exactly cocartesian fibrations, and the proofs are going into a proof assistant. On the applied side, Decapodes composes systems of PDEs diagrammatically: you assemble a multiphysics simulation out of component models instead of editing a monolith, and the composite is a first-class object rather than a hand-merged file. They report a drastic drop in time-to-first-simulation, though they say plainly that the claim is qualitative and unmeasured, which is the honest state of it.
What shapes will a plasma hold?2012
Standing-wave mode structure of a 2.45 GHz argon discharge · conductive polyhedral cavity · deployable structures in ionospheric plasma
Space is cool, and a satellite that folds is cooler: you launch something small and it opens into something large. But a deployed satellite is a geometric mesh sitting in the ionosphere, which is a plasma, and that plasma is part of the electrical environment the structure has to work in. So the question is what a given geometry does to the fields around it.
People do this deliberately now and call them plasma metasurfaces: discharges arranged so their geometry decides which electromagnetic modes get through, and reconfigured by changing the discharge rather than the hardware. I was making a very crude one and calling it a side project.
A thinking film you could bendSpring 2012
Flexible TiO2 resistive switching · memristive device fabrication and bend-cycle characterization · in-memory computing
A memristor is a resistor that remembers where you last left it. That sounds like a small thing and it is not: a component that stores a value and computes with it in the same place means a device built from them does not have to send its data somewhere else to be processed. Put a memristor on a flexible surface and that device can be worn: potentially as a non-invasive medical device.
Years afterward, a group wired live rat neurons in Padova to silicon neurons in Zurich through memristors in Southampton, with the memristors holding the synaptic weights between them. The devices were Pt/TiOx/Pt — the same stack I had been making by hand. I had nothing to do with it, and it is still my favorite thing to have happened to a material I once held.
Fragments
Structure recovered from the traces a language leaves behind.
Reading a language from the fragments somebody already translatedSpring 2013
Unsupervised grammar and lexicon induction from partially parallel text · decipherment of incompletely understood scripts · contracted Braille as the test language · accessible signage
This started because I read Braille and kept noticing that the signs around me were wrong. The one that decided it: a door labeled electrical room in print and safety exit in Braille. Whoever installed it could not read what they were mounting, and nobody downstream of them could catch it. So I wanted translation to be something a builder could get right at the point of installing a sign, rather than something they had to take on faith.
The grammar was the problem, and the standards body agreed. When Unified English Braille replaced the older code, nine contractions were deliberately deleted — in BANA's own words, to enable accurate automatic translation and reduce the exceptions to the rules. They simplified the language rather than the model. It is still not solved: frontier language models today refuse, hallucinate, or emit malformed Braille, and a small fine-tuned model beats all of them.
Trying to keep clinical doctors up to date2013
Automated computational semantics · abstractive summarization of biomedical literature · evidence fidelity
I kept meeting working doctors who were further from the current literature than they wanted to be. Not for any want of caring: seeing patients all day and reading a stack of new papers every week are not two things one person can do. The volume is what defeats them, so the volume is what to go after.
This is the research CAMEL grew out of, pointed the other way round. There, meaning recovered from too little text; here, meaning kept intact while most of the text goes away.
Living things
Hidden states, read from the traces a system emits.
Biome of Overstimulation: Sensory Sensitivity as a Unifying Mechanism2024–current
Synthesis arguing that failure of habituation constitutes a shared upstream mechanism, drawing on connectivity imaging, glutamatergic excitability, microbiome composition and enteric regulation of immunoglobulin secretion. With L. Estinto.
Hypothesis-generating; not experimentally validated.
Neostigmine in feline idiopathic megacolon2026current
Owner-reported case study of an acetylcholinesterase inhibitor used off-label to restore colonic motility, with lactulose and anti-NGF analgesia, documented to a standard usable in consultation.
Contraindicated without prior deobstipation and exclusion of mechanical obstruction. Medical management succeeds in roughly two thirds of cases presenting under six months and under six percent thereafter.
Memantine as a candidate non-opioid analgesic in central sensitization2026current
Review of the mechanistic rationale, comparative pharmacology against ketamine and dextromethorphan, and the clinical trial record for memantine in chronic pain, with a proposed design for a publicly funded trial.
Evidence synthesis only. The pooled analysis of eleven trials is not significant; the positive signal rests on one 63-patient fibromyalgia trial.
Unsupervised classification of rodent ultrasonic vocalization for continuous preclinical monitoring2013–2014
Unsupervised classification of rodent ultrasonic vocalizations in quasi-real time, deployed as a preclinical readout for compound effects on stress and libido — the first continuous non-invasive behavioral phenotyping system at Vium, several years before MUPET and DeepSqueak made the approach standard. Two write-ups: A New Female–Female Mouse Vocalization Discovered via Unlabeled Machine Learning and On the Detection and Prevention of Aggression in Lab Mice via Quasi-Real Time Analysis.
Non-contact acquisition removes handling stress as a confound and increases information yield per cohort; later multi-company validation supports the approach. Prior unsupervised USV clustering exists (Grimsley et al., 2013).
Decoder convergence and optical recording for intracortical motor prostheses2014
Convergence analysis of intracortical decoders under signal drift — how fast a Kalman-style filter settles onto a usable mapping from neural activity to intended movement, how far it degrades as glial scarring thins the neural population, and how much adaptive recalibration recovers. Then optical recording via fluorescent calcium indicators as an alternative to penetrating electrodes.
Foreign body response degrades yield; subsequent human longitudinal data indicates abiotic electrode failure may dominate. Optical methods remain depth- and delivery-limited.
Portable detection of gluten and common allergenic proteins2011–2014
Two approaches: a laser and avalanche-photodiode optical assay, abandoned for insufficient SNR against protein-rich matrices; then a G12 antibody colorimetric assay in a toothpick or strip format.
Limiting factor was antibody cost per assay, not sensitivity. Sampling variance in heterogeneous food exceeds measurement variance, which bounds the device class; aptamer assays have since addressed the cost term.
Remote photoplethysmography on a head-mounted display2013
Recovery of cardiac pulse and its variation from video alone, without contact, intended as a social-signal aid. HackMIT, with K. Talwar and S. Hewett.
Withdrawn on ethical grounds: the measured party cannot detect or refuse the measurement.
Bodies in motion
A body reading the traces of its own movement.
Modular robotic retrofits for powered wheelchairs2013–2015
Four assemblies designed to fit chairs already configured to their users: automated pressure redistribution, powered seat elevation, assisted bed-to-chair transfer, rough-terrain drive. Requirements elicited from wheelchair users with spinal cord injury; subsequent mentorship of gaze-controlled mobility assistance for ALS and SCI.
Written as a nonprovisional patent application and deliberately not filed; released under Creative Commons. Interface pressure is a surrogate endpoint.
Graceful degradation under sensor failure: contact-rich manipulation on proprioception alone2012
A robot that stops dead the moment its camera fails cannot be trusted around people, and sensors fail constantly. With vision and all external feedback disabled, a Willow Garage PR2 acquired a shape-matched insertion task via reinforcement learning from commanded joint positions and gripper sensing alone. Also two-arm load equalization for objects with off-center mass. HCI internship, George Washington University.
Demonstrates graceful degradation at the extreme: total loss of exteroception, not a partial fault. The reinforcement learning was small-scale policy search.
Monocular visual SLAM with motion planning on a micro aerial vehicle2012
PTAM-class monocular SLAM, processed off-board, with metric scale recovered by fusing sonar altimetry and inertial measurement, coupled to frontier-based planning on a partially built map. Parrot ARDrone.
Bounded by repetitive texture, illumination change and airframe disturbance. Dual-use: unsurveyed autonomous navigation transfers directly to armed platforms.
Gait-based re-identification for robot person following2011
Hexapedal platform; target enrollment by proximity, association by gait rather than appearance, steering on bearing. Alternating tripod gait for statically stable traversal.
Static stability is a guarantee over an idealized contact model and does not survive real terrain.
Matter
Behaviour inferred from the traces that structure leaves.
Electronic-structure prediction of superconducting transition temperature2012–2013
Augmented plane wave band structure feeding an electron-phonon coupling estimate and a McMillan/Allen–Dynes transition temperature for intermetallic compounds. Advised by D. Papaconstantopoulos, George Mason University. Undergraduate thesis.
Subsequently validated at scale by the predicted-then-synthesized hydride superconductors, at megabar pressures.
Dependent and directed type systems for multiphysics models2015
Proposal that multiphysics models be expressed in a dependent type language. Classical fields are sections of fiber bundles; dependent types are semantically fibrations; the correspondence permits model structure, including direction of coupling, to be carried in the type and exploited by the compiler. Visiting researcher, Santa Fe Institute.
Motivated by run time rather than correctness alone: dependently typed elimination of runtime checks measurably reduces execution time on ordinary numerical code. Directed variants correspond to cocartesian fibrations; a related applied line of work is diagrammatic composition of PDE systems, whose reported gains remain qualitative.
Standing-wave mode structure of an RF discharge in a polyhedral cavity2012
2.45 GHz discharge in an evacuated aluminum polyhedron backfilled with argon; cavity geometry selects the supported resonant modes and the discharge organizes along the resulting standing wave. Motivated by deployable conductive structures in ionospheric plasma.
Benchtop analogue only: argon matches neither ion mass nor electron temperature, and discharge density sits orders of magnitude above ionospheric values.
Flexible TiO2 resistive-switching devices2012
Fabrication and characterization of memristive devices on compliant substrates, including switching behavior under bend cycling. Research assistantship, Mary Baldwin College.
Motivation was colocated storage and computation for body-worn sensing; now termed in-sensor and near-sensor computing.
Fragments
Structure recovered from the traces a language leaves behind.
Unsupervised grammar and lexicon induction from partially parallel text2013
Probabilistic lexicon induction over weakly parallel corpora, evaluated on contracted Grade 2 Braille, chosen as a test language with a known ground truth. Undergraduate thesis, George Mason University. Code.
This grew out of automated computational semantics research at GMU, which also included work on abstractive summarization of biomedical literature — getting a machine to read a scientific paper and return something faithful and short enough that a clinician could decide whether the full paper was worth their evening.
Generalizes to incompletely deciphered scripts. Applied motivation: transcription error in installed accessible signage.
Automated abstractive summarization of biomedical literature for clinical readers2013
Automated computational semantics directed at reducing a paper to a triage-sufficient summary. George Mason University.
Evaluation of evidence fidelity, rather than generation, remains the open problem.
THE STUDIO
Pastels, acrylic, spray-painted creature murals, tattoos, polaroids. A taste:
And beyond the gallery:
CONTACT ME
- Curiosity is welcome
- fractalcows@gmail.com
- My work email
- cray@uni-muenster.de
You will find the name Cathe(rin)e Ray on old stuff and Rin Ray on my newer works — these both refer to the same person. I prefer Rin nowadays.





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