UCSC Colloquium $\cdot$ 11/5/2026

AlphaCentaur

Nic Brody $\cdot$ UCSC

Part 1

AlphaCentaur

An environment for human-AI math collaboration

Starting point

What is intelligence?

“Intelligence measures an agent's ability to achieve goals in a wide range of environments.”

Legg-Hutter, 2007

Hey! ... I'm an agent!

And I want to achieve goals!

Why I care

What are these goals?

• Learn and understand beautiful things
  
• Prove some cool theorems

For a while I tried "learning a bunch of math," thinking that would increase my "intelligence", ie, my ability to achieve these goals.

It seems clear that this won’t be what proves this theorem…

How else can I try to achieve this goal?

“Intelligence measures an agent's ability to achieve goals in a wide range of environments.”

Legg-Hutter, 2007

Maybe I can at least create a nice environment for my AI and I to work in.

Language models are cool, but sometimes language isn't the right interface!

Introducing: AlphaCentaur!

The name

Alpha + Centaur

Alpha

• AlphaGo, AlphaZero, AlphaFold, AlphaProof, etc.

Centaur

• Human + chess computer collaboration team

Keep the human in the loop,
and build interfaces for happy human–AI interactions.

Part 1 · AlphaCentaur

What have I been doing as a mathematician?

Read

AlphaLibrary

Collaborate

AlphaAgent

Think

AlphaNote

Write

AlphaMarkdown · AlphaTeX

Compute

AlphaPy

Check

AlphaLean

Present

AlphaPresent

Part 1 · AlphaCentaur

A suite of applications for math research

AlphaLibrary

Agents hate PDFs. We (kinda) like them.

AlphaAgent

Agent instructions and harnesses.

AlphaNote

I love pen and paper.

AlphaMarkdown

Pretty good for both of us.

AlphaTeX

LaTeX source and PDF, side by side.

AlphaPy

Reactive Python notebooks.

AlphaLean

Lean, checked against Mathlib.

AlphaPresent

Most presentation editors don't allow live web elements. (Nice remotes!)

1 of 8 · Read

AlphaLibrary

Live: AlphaLibrary in this page · your edits stay in your browser

AlphaLibrary · AlphaAgent · AlphaNote · AlphaMarkdown · AlphaTeX · AlphaPy · AlphaLean · AlphaPresent

2 of 8 · Collaborate

AlphaAgent

Lives inside AlphaCentaur: switch windows to show it live

AlphaLibrary · AlphaAgent · AlphaNote · AlphaMarkdown · AlphaTeX · AlphaPy · AlphaLean · AlphaPresent

2 of 8 · Collaborate

Two ways to plug in an AI

API key

Paste in an API key: Claude, OpenAI,
Gemini, or a local model.

Chat with an agent in the sidebar

MCP

Or, you can use Claude Code or Codex on your desktop, and simply point it to AlphaCentaur! Your AlphaCentaur app has specialized instructions Claude will read.

Either way, you can always see what the AI changed.

AlphaLibrary · AlphaAgent · AlphaNote · AlphaMarkdown · AlphaTeX · AlphaPy · AlphaLean · AlphaPresent

3 of 8 · Think

AlphaNote

Live: AlphaNote in this page · your edits stay in your browser

AlphaLibrary · AlphaAgent · AlphaNote · AlphaMarkdown · AlphaTeX · AlphaPy · AlphaLean · AlphaPresent

4 of 8 · Write

AlphaMarkdown

Live: AlphaMarkdown in this page · your edits stay in your browser

AlphaLibrary · AlphaAgent · AlphaNote · AlphaMarkdown · AlphaTeX · AlphaPy · AlphaLean · AlphaPresent

5 of 8 · Write up

AlphaTeX

Live: AlphaTeX in this page · the PDF was compiled ahead of time; click-sync works both ways

AlphaLibrary · AlphaAgent · AlphaNote · AlphaMarkdown · AlphaTeX · AlphaPy · AlphaLean · AlphaPresent

6 of 8 · Compute

AlphaPy

Shown read-only here: Markdown rendered, Python code displayed. It runs inside AlphaCentaur.

AlphaLibrary · AlphaAgent · AlphaNote · AlphaMarkdown · AlphaTeX · AlphaPy · AlphaLean · AlphaPresent

7 of 8 · Check

AlphaLean

Lives inside AlphaCentaur: switch windows to show it live

AlphaLibrary · AlphaAgent · AlphaNote · AlphaMarkdown · AlphaTeX · AlphaPy · AlphaLean · AlphaPresent

8 of 8 · Present

AlphaPresent

Live: AlphaPresent in this page

AlphaLibrary · AlphaAgent · AlphaNote · AlphaMarkdown · AlphaTeX · AlphaPy · AlphaLean · AlphaPresent

8 of 8 · Present

This slide was made in AlphaPresent

• Live web elements: your apps run on the slide.
• A phone remote.
• A deck is a folder in the project, next to the math it's about.

AlphaPresent — alphaCentaurTalk/index.html

Speaker notes: “To understand this slide, see this slide.”

8 of 8 · Present

This slide was made in AlphaPresent

• Live web elements: your apps run on the slide.
• A phone remote.
• A deck is a folder in the project, next to the math it's about.

AlphaPresent — alphaCentaurTalk/index.html

Speaker notes: “To understand this slide, see this slide.”

Base case: the projector ran out of pixels.

8 of 8 · Present

This slide was made in AlphaPresent

• Live web elements: your apps run on the slide.
• A phone remote.
• A deck is a folder in the project, next to the math it's about.

AlphaPresent — alphaCentaurTalk/index.html

Speaker notes: “To understand this slide, see this slide.”

Base case: the projector ran out of pixels.

Base case: the projector ran out of pixels.

Part 1 · AlphaCentaur

Why one platform?

The advantage of integrating all of these into a single platform:

context.

We are creating an environment for our AI agent.

Part 2

Presentation Calculator

What AlphaCentaur is for, in my case

Part 2 · The question

Suppose $S\subseteq {\mathrm{PGL}}_2(\overline{\mathbb{Q}})$ is a finite set of matrices.

Part 2 · The Presentation Calculator

Type in matrices, press Compute

1 · Matrices over $\overline{\mathbb{Q}}$

Type the entries; + root adds an algebraic number by its polynomial. About 70 examples, and a die for a random group.

2 · Compute

Three steps run at once in workers: Zariski closure, discreteness at every place, arithmeticity.

3 · The answer

A verdict, and a presentation $⟨\text{generators}\mid \text{relations}⟩$ at the top: here one relation.

Live: the Long–Reid group at $t=9$

Part 2 · The question

Classification Conjecture

Let $k$ be a number field and $\mathbf{G}$ an absolutely simple $k$-algebraic group. Call a subgroup $\Gamma \le \mathbf{G}(k)$

1. algebraic if it lies in a proper $\mathbb{Q}$-algebraic subgroup $\mathbf{H}(\mathbb{Q})⪇{\mathrm{Res}}_{k/\mathbb{Q}}\mathbf{G}(k)$;
2. geometric if $\Gamma \le \mathbf{G}(k_v)$ is discrete for some nontrivial valuation $v$ of $k$;
3. arithmetic if $\Gamma$ is commensurable with $\mathbf{G}(A)$ for some subring $A\le k$.

Conjecture. Every subgroup is algebraic, geometric, or arithmetic.

Part 2 · For PGL₂

First observation

• If $\Gamma$ is finitely generated, then $\Gamma \le {\mathrm{PGL}}_2(A)$ for a finitely generated subring $A\subseteq \overline{\mathbb{Q}}$.
• $A\subseteq {\mathcal{O}}_k[S^{-1}]$ for some number field $k$ and finite set of primes $S$.
• $A$ is a lattice: $A{\le }_L{\mathbb{R}}^r\times {\mathbb{C}}^s\times {\prod }_{p\in S}{k}_p$.

Conclusion

Part 2 · Step 1

Zariski closure, field and places

1 · Closure

Finite, a torus, Borel, or all of ${\mathrm{PGL}}_2$. Here: Zariski dense.

2 · Field and algebra

The invariant trace field $k$ (here cubic, discriminant $-23$) and the quaternion algebra $A$ over it.

3 · Every place

Each generator classified at each place: hyperbolic, elliptic, parabolic. $S$ is where $\Gamma$ is unbounded.

Live: the Weeks manifold

Part 2 · Three kinds of subgroup

Algebraic subgroups

For ${\mathrm{PGL}}_2$, the algebraic subgroups are

solvable or unitary.

Part 2 · Step 1, not dense

Not dense, still presented

1 · Borel closure

$\Gamma$ fixes a point of ${\mathbb{P}}^1$, so it acts by $z\mapsto \lambda z+b$.

2 · A $\mathbb{Z}[1/N]$-presentation

$⟨u,t\mid tu{t}^{-1}=u^{3/2}⟩$: multipliers $\mathbb{Z}$, translations $\mathbb{Z}[\frac{1}{6}]$.

3 · Finitely presented?

Bieri–Strebel says no. The relation alone gives $\mathrm{BS}(2,3)$; every relation is checked on the matrices.

Live: $\mathbb{Z}[\frac{1}{6}]⋊\mathbb{Z}$

Part 2 · Three kinds of subgroup

Geometric subgroups

Groups acting properly on…

A tree

virtually free

discreteness algorithm ✓

${\mathbb{H}}^2$

virtually a surface group, or free

discreteness algorithm ✓

${\mathbb{H}}^3$

virtually a 3-manifold group

discreteness algorithm? Conjecture! (False over ${\mathrm{PGL}}_2\mathbb{C}$)

For all of these spaces, finitely generated ⟹ finitely presented. (Scott core theorem)

Discrete at ${\mathbb{Q}}_p$

Discrete at $\mathbb{R}$

Unwrapping · Figure-eight group

Start with two matrices

$A=\left(\begin{array}{cc}1& 1\\ 0& 1\end{array}\right)$

$B=\left(\begin{array}{cc}1& 0\\ \omega & 1\end{array}\right)$

$\omega =\frac{-1+\sqrt{-3}}{2}$

${\Gamma }_8=⟨A,B⟩$

Unwrapping · Figure-eight group

Build the Cayley graph

Choose a basepoint $p\in {\mathbb{H}}^3$.

Apply $A,B$ and their inverses to grow the orbit.

Join $gp$ to $gAp$ and $gBp$.

Unwrapping · Figure-eight group

See the tiling

Translate the domain by group elements.

The copies tile hyperbolic space.

Each tile corresponds to a point of the orbit.

Live: Poincaré’s Tiling view

Unwrapping · Figure-eight group

From orbit to domain

For each orbit point $gp$, draw the perpendicular bisector of $p$ and $gp$.

Intersect the half-spaces containing $p$.

The result is the Dirichlet domain.

Unwrapping · Figure-eight group

Check the polyhedron

Pair the faces by group elements.

Check the edge cycles and angle conditions.

Poincaré’s theorem: the pairings give a discrete group and a presentation.

Live check: 12 faces; exact relations, numerical geometry.

Unwrapping · Add a generator

What if we add a translation?

$C=\left(\begin{array}{cc}1& t\\ 0& 1\end{array}\right)$

Take $t=1+\pi /100$.

$A^{-m}{C}^n=\left(\begin{array}{cc}1& nt-m\\ 0& 1\end{array}\right)$

$nt-m\to 0$: the group is indiscrete.

Unwrapping · A related group

A second copy, shifted along the cusp

Now shift a second figure-eight group along its cusp.

The next picture unwraps a related double, using a translation by $\pi$ and a different generator convention.

Part 2 · Beyond PGL₂

What changes in general

More general $\mathbf{G}$

We expect discrete groups to be built by amalgamating simpler pieces.

More general $k$

2-dimensional fields!

Part 2 · Three kinds of subgroup

Arithmetic subgroups

The conjecture says: if $\Gamma$ is neither algebraic nor geometric, it is commensurable with $\mathbf{G}(A)$.

Example: PGL₂ ℤ[½]

• Not discrete in ${\mathrm{PGL}}_2\mathbb{R}$ ($\mathbb{Z}[\frac{1}{2}]$ is dense in $\mathbb{R}$).
• Not discrete in ${\mathrm{PGL}}_2{\mathbb{Q}}_2$ (it contains ${\mathrm{PGL}}_2\mathbb{Z}$, inside a compact group).
• But a lattice in ${\mathrm{PGL}}_2\mathbb{R}\times {\mathrm{PGL}}_2{\mathbb{Q}}_2$.

Part 2 · Step 1, again

Congruence closure

1 · Prime by prime

Where is the closure smaller than ${\mathrm{PSL}}_2({\mathbb{Z}}_p)$? Here at levels $N=5\cdot 7\cdot 13$.

2 · One finite quotient

Put together, with determinant classes: $[\Lambda :\Delta ]=611,520$.

3 · Congruence subgroups

By the congruence subgroup property, $\Gamma$ is $S$-arithmetic exactly when $\Gamma =\Delta$.

Live: the Long–Reid group at $t=9$

Part 2 · Step 3

Coset enumeration and Bass–Serre

1 · A presentation of $\Lambda$

${\mathrm{PGL}}_2(\mathbb{Z}[\frac{1}{3}])$ as an iterated Bass–Serre amalgam.

2 · Todd–Coxeter

$⟨A,B,\mathrm{diag}(3,1)⟩$, Sanov's free group plus one matrix, has index 12: $S$-arithmetic.

3 · Graph of groups

The quotient on the tree $T_3$, and a presentation by Reidemeister–Schreier, each generator a word.

Live: Sanov's group with $\mathrm{diag}(3,1)$

Part 2 · At scale

The database

1 · Thousands of groups

Every 2-generator subgroup of ${\mathrm{PGL}}_2(\mathbb{Q})$ up to a height, run through the same pipeline.

2 · The hard ones

FlashBeam searches the adelic product for a covering; vertex orbits with Stallings graphs give an exact index, or thin.

3 · What's left

The undetermined groups: each one a small research problem.

Live: the Presentation Calculator database

Part 2 · The tool

Presentation Calculator

The theoretical framework for this algorithm has existed in my mind for years!

Patching together: ring algorithms, DJK, Fuchsian groups, Kapovich, Markowitz, etc.

A wider audience

A moment for mathematics

Our discipline is changing—and much of the world is looking at mathematics.

What if we invited everyone to explore the ideas for themselves?

Introducing EigenReel

Beautiful little pieces of math

A fun, curated home for mathematical content creators to bring math to the masses.

Make a piece. Share it. Let someone play with an idea.

Short visualizations · interaction · explanations

See it in action

EigenReel · live demo

Open eigenreel.com ↗ · Browse a piece, interact, and explore the explanation.

Putting it together

In AlphaCentaur, I can ask the agent to use the tools we develop.

Thank you!

Recursion depth: 7. Stack overflows: 0.