Personal statements

My involvement began at the Wolfram Winter School 2025, where I worked on the insertion Lie bracket suggested by Carlos and Xerxes. I generalized it to hypergraphs, using a hyperedge as the root, and uncovered its relation to the commutator of linearized “blow-up” rewriting rules. I also explored graded versions and higher-arity constructions based on the insertion or gluing of hypergraphs, as well as the representability of Lie algebras on subsets of rules and subspaces of the hypergraph vector space. I implemented these ideas in code and contributed to the Hypergraph paclet.

In October 2025, I joined the Wolfram Institute, funded by the JTF grant. After some initial contributions to the Wolfram Function Repository, the DiscreteGeometry paclet, and the PvsNP blog project, I started to work with Stephen Wolfram on Infrageometry. Together, we set out to develop the dictionary from the point of view of minimal models of geometric phenomena on graphs.

After initial meetings with Stephen Wolfram and Nik Murzin, where I absorbed the main philosophy and vision and learned about causal graphs, branchial graphs, spatial reconstruction, and the concept of computational minimal models, I proposed a concrete pipeline for obtaining an emergent spacetime from a single-history branch of the multiway system of hypergraph rewriting. The pipeline starts with a given observer chain and foliation, together with nearby observer chains whose intersections with the foliation represent germs based at the original observer. It produces spatial slices fibered over ℤ³, with gauge degrees of freedom and coarse-graining maps. I then committed to developing a mathematically sound narrative and tools for this proposal.

In particular, I implemented the idea of observer-scale dependence in minimal models of spatial geometry and arrived at the notion of renormalizable geometry. I introduced vertex and edge densities for infrageometric multiobjects, counting how often each vertex or edge occurs among their representatives. I proposed infrageometric convergence as scaled Gromov–Hausdorff convergence of metric-measure spaces with specified coarse-graining maps, together with weak convergence of the measures associated with these densities to measures on geometric objects in the limit. This provides a framework for studying how geometric structures emerge as the graph becomes larger and the observer scale increases.

I also implemented a minimal model of fibered graphs compatible with elementary Wolfram Language structures and, together with my student collaborator Ioana Milea, began exploring its connection to lattice gauge theory, viewed as a “mesoscale theory.” Her previous work on this topic provided important inspiration for this direction.

For infracalculus, I viewed integration as aggregation along directed graphs, seen as graphs fibered over ℤ by shortest-path distance from their sources. I defined differentiation as a left inverse of the corresponding integration and classified all sensible versions of these operations.

In parallel with this work, I developed several research repositories exploring these ideas within the broader vision of a computable universe advocated by Stephen Wolfram. In particular, I started developing InfraCausality, a toolkit for exploring discrete special relativity on causal graphs, intended as part of the JTF deliverables. These repositories combine my original code with contributions from Nik Murzin, Dugan Hammock, and student collaborators, as well as code later developed with the assistance of AI tools.

I am very grateful to Stephen Wolfram, Nik Murzin, Dugan Hammock, Carlos, Xerxes Arsiwalla, Ioana Milea, and other members of the institute for sharing their perspectives, ideas, discussions, and suggestions throughout this work. I was also positively influenced and motivated by my mentees and acquaintances at the Wolfram Summer School and Winter School, which provided a great environment for exchanging ideas and sharing our enthusiasm for this novel view of science.

My involvement with the physics project came in as someone with a background in differential geometry and an interest in hypergraph rewriting independently, motivated from the Arity Science project. When I first joined the Wolfram Institute, discussions with Xerxes Arsiwalla and Jonathan Gorard led to the ideation of the infrageometry project. Terms such as infracalculus were being used at that time by Stephen Wolfram, and I thought that the physics project needed a foundation in a different kind of mathematics that uses discrete combinatorial data structures as a substance and pure rewrite rules as the computational framework to give it life.

This kickstarted a research process by which I tried to build a dictionary with all of conventional differential geometry, to which Jonathan was also contributing some ideas, where he would suggest some applications in physics while I was leading the conceptual framework and general directions. Then I worked for a few months directly with Stephen Wolfram to make those things formalized until around June 2024, when that collaboration ended.

At that point, I was starting to work on a JTF grant application that eventually became the Causality, Observers and Computation grant, which I led as project leader until October 19th, 2026, when my position at the Wolfram Foundation ended. That grant had two scientific pillars of open questions and research directions. One was the questions stemming from the Arity Science project, particularly those that had a metamathematical angle and had some overlap or conceptual resonance with the metamathematics project that Wolfram did a few years before. The other was the infrageometry project, where geometry-like structures were to be derived from pure hypergraph rewriting. The grant application was written, developed and coordinated with three other universities, with Nicolas Behr, Josh Grochow and Alex Kolpakov. Eventually it was awarded, and in October 2025 it began.

In that process, Pavel Hajek joined as postdoc in the project, and together we had several sessions where we ideated and discussed multiple directions. During this time, I also had conversations with Utkarsh Bajaj about discrete analogs of differential geometry, having worked with him in a winter school where he did a project on discrete difference operators on graphs that to this day has led to several further developments. Pavel went on to work with Stephen on infrageometry. My involvement in the infrageometry project ended, and later my position at the Wolfram Foundation ended as well.

During that time, I also developed and finalized a research project in which axiomatic relativity, as presented in the original writings of Marzke and Wheeler, and the construction of a geometric clock were going to be applied to causal graphs arising from hypergraph rewriting. I concluded some investigations with AI as well, to conclude that there was a formal set-theoretic structure that allowed for the construction of geometric clocks and signal functions in discrete causal graphs, and how they connected to the hypergraph rewriting systems that generated them.

This is the extent of my involvement so far.