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AlphaGeometry:

A Step Toward Automated Math Reasoning

Hoang Huy Nguyen

ISyE Georgia Tech & former Student Researcher at Google DeepMind

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Overview of my work

Stochastic Control

Markov chain mixing

[TAC, under review]

Optimal Transport

[JMLR 2024], [AAAI 2024]

AI reasoning

AI for Sciences

AlphaGeometry 2

[JMLR 2025]

[Nature Geoscience 2025]

Foundations

Applied Probability

AI & Deep Learning

Transfer Learning

Stochastic Networks

[SIGMETRICS 2025]

About myself, my research spans from developing foundational frameworks to solving applied probability problems and investigating AI applications.
In particular, I develop frameworks to analyze the mixing time of infinite state space Markov chains, such as using Lyapunov drift or the curvature of Markov chains. I then use these frameworks to obtain the finite-time behavior of practical service systems, such as data centers or ride-hailing systems such as Uber or Lyft.

Note: Talk about finite + uncountable vs countable

Additionally, I also study Stochastic Control through the lens of Stochastic Approximation, and I develop efficient computational methods to solve large-scale Optimal Transport problems, which have many machine learning applications such as Transfer Learning.
Finally, with the recent advances in AI, the ultimate goal is to achieve AGI. One of the final frontiers to achieve AGI is to achieve the ability to perform deep reasoning. To this end, I had the opportunity to work with Google DeepMind to advance AI’s capability in math olympiad problem solving, thereby winning the gold medal at IMO 2025. I also transfer some of these insights to other areas, such as Earth Science.�For the purpose of today’s talk, I will focus on the work AlphaGeometry 2 and how it was able to beat gold medallists in solving IMO geometry problems/able to achieve a gold medallist’s performance at IMO.�
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In addition, I would love share a little bit about my other works as well. Besides my work on AI for Math, I also work extensively on Infinite state space Markov chain mixing with applications to Queuing Theory. Moreover, I also work on Optimization algorithms under the stochastic viewpoint, with applications to large scale Optimal Transport problems. I would love to chat further on these directions.
My research spans from theoretical foundations of Markov chains to Applied Probability and using AI & Deep Learning to tackle real world challenge.

In particular, I develop frameworks to analyze the mixing time of infinite state space Markov chains, such as Lyapunov drift or Markov chain curvature. I then use these machinery to analyze practical service systems such as data centers or ride hailing systems.

New AI paradigm: requires high-quality, structured data

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Mathematics and AI

First analog computer (300 BC)

Computation

Four color Theorem (1976)

Search/Enumeration

Kepler Conjecture (1998)

Verification

What would it take to achieve end-to-end automated reasoning?

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International Mathematical Olympiad (1959 – Present)

Topics

Algebra
Combinatorics
Number Theory
Geometry

One of the most prestigious pre-collegiate math competitions
Elegant, yet difficult problems, require only high school knowledge
AI-IMO challenge: Build the first AI to get gold at IMO

What would it take to build an AI to solve IMO?

Topics

Algebra
Combinatorics
Number Theory
Geometry

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Challenges

Hallucinations

Data scarcity

?

1+1=3

Answer hallucination

Citation hallucination

Lacks high quality, structured data!

Next token predictor

Reward outcome

Mechanism

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Challenges

Hallucinations

Data scarcity

?

1+1=3

Answer hallucination

Citation hallucination

Lacks high quality, structured data!

Next token predictor

Reward outcome

Mechanism

A consistent solver is needed!

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Idea #1: AlphaGeometry is a neuro-symbolic solver

7

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Symbolic Engine

Not use:

Advanced Theorems
Coordinates
Transformations

The symbolic engine translates our problem into a specific language, which encodes the geometric premises using a very small vocabulary. This language was first proposed by Chou, Gao, Zhang
The symbolic engine then uses these premises to make deductions, which forms the deductive aspect of the engine
In geometry, we also need to establish algebraic relations between sidelengths and angles, and so we have an algebraic component as well. We cast these angles and sidelengths into a system of equations and then use linear algebra techniques to solve them.
With that, we combine deductive and algebraic reasoning to obtain DDAR, and as you can see, it only uses simple stuff
Simple Gaussian elimination-based algorithm
Only uses simple Geometry techniques such as angle chasing, length chasing, ratio chasing
Use simple theorems => long proofs but fast and scalable�
Despite various claims, this component alone is very weak and cannot solve very simple theorems

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Idea #1: AlphaGeometry is a neuro-symbolic solver

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Idea #1: AlphaGeometry is a neuro-symbolic solver

Magic Construction

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Idea #1: AlphaGeometry is a neuro-symbolic solver

How to train the language model to make constructions?

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Idea #2: Synthetic Data Generation at Scale

Knowledge-building process similar to humans.

Synthetic data is artificially generated data to emulate the knowledge building process.

For example, imagine Professor Sajad wants to train his PhD students to do research.

The lack of training data for maths, especially those at the Olympiad level, motivated us to generate everything synthetically.

This actually emulate the knowledge-building process similar to humans. For example, if my advisor wants me to do research in applied probability, he will give me lots of simple exercises in stochastic process so that I can notice patterns and gain intuition about the area, which I then use to solve harder research problems that could be out of the training data distribution.

Humans can learn geometry using a pen and paper, examining diagrams and using existing knowledge to uncover new, more sophisticated geometric properties and relationships. Our synthetic data generation approach emulates this knowledge-building process at scale, allowing us to train AlphaGeometry from scratch, without any human demonstrations.

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Overview of the data generation process

Solves data scarcity w/o human intervention!

(to find the minimal problem)

Masking

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The Results

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All-time IMO Geometry results (2000-2024)

Faster symbolic engine

Knowledge sharing

Better search algorithm

Autoformalization

AG 1

AG 2

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Takeaway

Hallucinations

Data scarcity

?

1+1=3

Symbolic engines

Language models

Synthetic data generation

LLM for Math Research

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Thank you for listening!

Stochastic Control

Markov chain mixing

[TAC, under review]

Optimal Transport

[JMLR 2024], [AAAI 2024]

AI reasoning

AI for Sciences

AlphaGeometry 2

[JMLR 2025]

[Nature Geoscience 2025]

Foundations

Applied Probability

AI & Deep Learning

Transfer Learning

Stochastic Networks

[SIGMETRICS 2025]

In addition, I would love share a little bit about my other works as well. Besides my work on AI for Math, I also work extensively on Infinite state space Markov chain mixing with applications to Queuing Theory. Moreover, I also work on Optimization algorithms under the stochastic viewpoint, with applications to large scale Optimal Transport problems. I would love to chat further on these directions.
My research spans from theoretical foundations of Markov chains to Applied Probability and using AI & Deep Learning to tackle real world challenge.

In particular, I develop frameworks to analyze the mixing time of infinite state space Markov chains, such as Lyapunov drift or Markov chain curvature. I then use these machinery to analyze practical service systems such as data centers or ride hailing systems.

New AI paradigm: requires high-quality, structured data