A selfie I took in the desert

Welcome to my website! I'm Zoltan, a mathematical logician and educator based in Sydney, Australia. I am currently an Adjunct Lecturer at the University of New South Wales. My academic work spans multiple areas within mathematics, including proof theory, category theory, finite algebraic structures, combinatorics, and even nonstandard analysis. I also enjoy working with interactive theorem proving software, and advocate for its integration into mathematical research and education.

Read about:

Nonstandard Analysis

Nonstandard analysis allows us to take infinite limits of finite structures, and to approximate infinite structures via finite ones

Internal Set Theory

My PhD thesis (available here) explores two novel algebraic applications of Nelson's Internal Set Theory, a syntactic framework for nonstandard analysis.

In my thesis, I propose an explicitly topological formalism of structural approximation of groups, generalizing previous work by Gordon and Zilber. Using this new formalism, one can show that every profinite group admits a finite approximation in the sense of Zilber, and prove various results about LEF groups. My main result states that well-behaved actions of the approximating group on a compact manifold give rise to similarly well-behaved actions of periodic subgroups of the approximated group on the same manifold. The theorem generalizes earlier results on discrete circle actions, and gives partial non-approximability results for SO(3).

Formulating and proving results about continuous functions in a so-called "pure" fragment of Internal Set Theory allows us to extract various computational bounds from them syntactically. I showed that sheaves over topological spaces also admit such a "pure" presentation, equivalent to the usual definition over standard objects.

By introducing a non-standard (and very much non-constructive) extension of Martin-Löf Type Theory with a hierarchy of universes for external propositions along with an external standardness predicate, I was able to formally verify my results using the Agda proof assistant.

Degree of Satisfiability

Degree of primality diagram for the number 45

The idea

Which non-Abelian group is closest to being Abelian? Which composite number is farthest from being prime? Such questions can be formulated and answered using the theory of Degree of Satisfiability (and result in striking visual arrangements that feature heavily on this website). Degree of satisfiability has connections to property testing and black-box algebraic structures.

The degree of satisfiability of a logical formula φ(x,y) is the probability that φ(a,b) holds for uniformly randomly chosen values a,b. An interesting phenomenon occurs: for many formulas, the degree of satisfiability is either exactly 1, or else much less than 1. While infinite algebraic structures don't usually come equipped with a notion of uniform distribution, degree of satisfiability nonetheless generalizes well both to virtually nilpotent infinite groups, and to LEF structures.

Heyting algebras

Heyting algebras: in our article, Degree of satisfiability in Heyting algebras, Ben Bumpus and I characterized all the 1-variable equations in Heyting algebras according to their degree of satisfiability. In particular, we showed that the equation p∨¬p=1 has satisfiability gap 1/3: if a Heyting algebra is not Boolean, then at least 1/3 of its elements fail to satisfy the law of excluded middle! As a corollary, we also get that all infinite T0-separable topological spaces, except for the discrete ones, have more non-closed open sets than clopen sets. Recently, Charles Evans obtained similar results for BCK-algebras as well.


In Degree of satisfiability of some special equations, I showed that the equations xyy=yyx, xyyy=yyyx and yxy=x (as well as an infinite family of others obtained from these) have satisfiability gaps. I also introduced a simple criterion to identify which equations hold in a finite index subgroup of an infinite group precisely if they have positive degree of satisfiability.


Degree of primality diagram for the number 45

The idea

Which one of these three point clouds can we get by sampling the unit cube using a pseudo-random sequence? Surprisingly, the answer is any one of them, depending on the sequence we use! Above, I used three different pseudo-random number generators to sample points from the cube, and got three drastically different results.

As a topological manifold, the unit cube has one connected component, and no holes. However, the sample obtained using the first pseudo-random number generator above (the infamous RANDU) seemingly consists of 15 or so separate connected components. In the second sample (XorShift64, scaled), there is only one component, but the sample still looks like the union of multiple planes, enclosing many holes.

Persistent Homology

Using persistent homology, one can rigorously define the Betti numbers of point samples, and investigate their relationship to the Betti numbers of the sampled manifold. In my honours thesis (University of Stirling, 2016; available on request) I classified some pseudorandom sequences based on their ability to reflect the topological properties of the manifolds they sample, and developed an algorithm that can test for failures of uniformity by calculating persistent homology groups of the samples. This can be seen as an alternative to the spectral test for the assessment of the quality of uniformity of d-dimensional points that remains stable under small perturbations of the generator. The thesis also contains an alternative proof of the Cohen-Steiner, Edelsbrunner, Harer result on the stability of persistence diagrams.

Structured Decompositions

An abstract setting for treewidth


One can define the treewidth of an undirected graph in several equivalent ways: e.g. as the size of the largest set of vertices that occur in all tree decompositions of the graph, or as the maximum order of connected subgraphs that touch each other, i.e. brambles.

Spined categories

In our article, Spined categories: generalizing tree-width beyond graphs Ben Bumpus and I obtained a new characterization of treewidth as a functor preserving proxy pushout structure on a category of unordered graphs. We obtained a whole bunch of categories which admit tree-width-like invariants, allowing us to recover hypergraph tree-width and complemented tree-width using the same consrtuction.

Structured decompositions

Graphs of bounded treewidth often come up when examining the parametrized complexity of graph algorithms. The idea is to find graph invariants k so that the complexity of an otherwise intractable algorithm depends only polynomially on the size of the input graph, but may depend exponentially on the value of the invariant invariant k. Many families of graph algorithms that have been studied in detail display fixed-parameter tractability, i.e. can run in polynomial time on graphs of fixed bounded treewidth. In joint work with Ben Bumpus and Jade Master, we introduce the notion of structured decompositions, which provide a categorial setting for the study of fixed-parameter tractability, including the ability to define generic algorithms, and yields generalizations of many combinatorial invariants such as layered tree-width and graph decomposition width. The resulting decompositions also have striking similarity to graphs of groups familiar from Bass-Serre theory.

Proof Theory

A proof net, and mustard watches

Internal Type Theory

Nonstandard analysis is a rich formalism for analysis that allows us to work directly with infinite and infinitesimal ideal objects. Nelson's Internal Set Theory extends ZFC Set Theory with a new predicate "is standard", and three new axiom schemata governing it, which allows one to provide a syntactic treatment of nonstandard analysis, and to write down an explicit proof-theoretic translation showing the conservativity of nonstandard analysis over ordinary set theory. In my PhD thesis (available here) I introduced Internal Type Theory, an extension of Martin-Löf Type Theory largely analogous to Nelson's extension of ZFC. This allowed me to verify the main group-theoretic results of my PhD thesis using Agda, an interactive proof assistant based on Martin-Löf Type Theory.

Apartness relations

Apartness relations are irreflexive, symmetric and cotransitive binary relations. They act dually to equivalence relations: instead of saying when two things are the same, they state when two things differ. Apartness relations have fundamental importance in constructive algebra: unlike in classical mathematics, equivalence relations cannot take their place. In my article, Apartness relations between propositions, I classify all apartness relations definable in propositional logics extending intuitionistic logic, and show that one cannot construct non-trivial apartness relations between propositions without invoking the weak law of excluded middle, nor tight apartness relations between propositions without invoking excluded middle. This answers a question of Rijke on the correct notion of apartness between propositions. Using parametricity techniques, I also show that Martin-Löf Type Theory cannot construct non-trivial apartness relations between propositions.

A unary proof system

The usual Hilbert system for classical propositional logic uses three axioms and one rule of inference, modus ponens. However, the modus ponens rule is binary (requires two premises). In a Math StackExchange answer, I construct a proof system that consists only of unary rules. The idea is to use conjunction to emulate three stacks, and have each inference rule operate by taking arguments from one of the stacks.

News and Media

Diagram in a spined category

04 Jan 2024

My article with Ben Bumpus, Degree of satisfiability in Heyting algebras, will appear soon in the Journal of Symbolic Logic! In it, we show that the Law of Excluded Middle either holds for all elements, or fails for at least 33% of all elements in any Heyting algebra. We get combinatorial results about topological spaces as corollaries: e.g. in an infinite T0 space, the number of clopens cannot exceed the number of non-closed opens.

06 Oct 2023

My latest article, Proof-theoretic methods in quantifier-free definability, has appeared on the arXiv! In it, I introduce a proof-theoretic approach to studying the quantifier-free definability of second-order intuitionistic connectives. The method proves that Taranovsky's "realizability disjunction" connective does not admit a quantifier-free definition, and can be used to obtain new results and more nuanced information about Kreisel's and Połacik's families of unary connectives. Accompanying Agda code.

23 Jul 2023

Our work on the verification of the seL4 Core Platform will be presented at the ACM Asia-Pacific Workshop on Systems 2023, Seoul, Republic of Korea (joint work w/ Mathieu Paturel, Isitha Subasinghe and Gernot Heiser).

26 Jun 2023

I've given an invited talk on degree of satisfiability in Heyting algebras at Constructive Mathematics: Foundation and Practice 2023, Nis, Serbia. Slides.

01 Jun 2023

I am leaving full-time work as Lecturer in Formal Methods at UNSW! Stay tuned for news about my future affiliation here.

14 Nov 2022

My seL4 Summit 2022 talk on the formal verification of the Core Platform is now available on YouTube.

09 Oct 2022

Between 01 and 09 Oct, Ben Bumpus, Jade Master and I visited Tallinn University of Technology, where we spent a week researching the relationship between monoidal width and structured decompositions. We thank Pawel Sobocinski and Elena di Lavore for organizing and supporting this research meeting!

28 Sep 2022

I'll be giving an invited talk next year at Constructive Mathematics: Foundation and Practice, probably on the interactions between degree of satisfiability, logical anti-exceptionalism, and constructivism.

10 Sep 2022

My latest article, Apartness relations between propositions, has hit the arXiv! In it, I answer a question of E. Rijke about the correct notion of apartness for logical propositions, give a short classification of apartness relations between propositions that can occur in a Heyting algebra, and show that Martin-Löf Type Theory is not able to construct non-trivial apartness relations between propositions.

14 Jul 2022

Our new article about Structured Decompositions has appeared on the arXiv! This is joint work w/ Ben Bumpus and Jade Master, and follows up on earlier work Ben and I did on Spined Categories.

25 Oct 2021

My paper, Degree of satisfiability in Heyting algebras has hit the arXiv! (joint work w/ Ben Bumpus).

20 Oct 2021

A bunch of SMT-based tools I developed for a model-based security engineering project have been open-sourced, available from the CSIRO Data Access Portal (joint work w/ Ihor Kuz).

27 Aug 2021

Upcoming talk on combinatorics and spined categories at Category Theory 2021, 3 Sep 2021, Genoa, Italy. (Joint work, presented by Ben Bumpus).

28 Jul 2021

The recording of my Applied Category Theory 2021 talk is now available on YouTube.

20 Jun 2021

I will be giving a talk on Spined Categories at the Australian Category Seminar (Macquarie University) on 23 Jun 2021.

08 Jun 2021

I will be presenting Treewidth via Spined Categories, joint work with Ben Bumpus, at Applied Category Theory 2021 (Cambridge, UK, 12-16 July 2021).

21 May 2021

I received the CSIRO SCS Engineering and Technology Award, awarded for our high-impact work on binary verification for the 64-bit RISC-V architecture (along with my teammates Matthew Brecknell and Dr Mitchell Buckley).

06 May 2021

Click here to read the RISC-V International coverage of my work, bringing Translation Validation to RISC-V!

05 May 2021

Prof. Gernot Heiser's blog post on our seL4 verification work hit the front page of news aggregator news.ycombinator.com today!

05 May 2021

My work on Translation Validation of the seL4 kernel was featured on the HENSOLDT Cyber GmbH website. Read the full article, including my comments, in English🇦🇺 and German🇩🇪!

26 Apr 2021

I gave a talk about degree of satisfiability at the CSIRO Trustworthy Systems seminar, including announcing some results on degree of satisfiability in Heyting Algebras.

05 Apr 2021

Spined Categories: generalizing tree-width beyond graphs, joint work with Ben Bumpus, submitted to Combinatorica.

31 Mar 2021

Ben Bumpus presented our joint work on spined categories at the 37th British Colloquium for Theoretical Computer Science.

30 June 2020

I gave a talk about my type-theoretic analogue of Internal Set Theory at the Budapest Type Theory Seminar.

05 Feb 2020

My paper, Degree of satisfiability of some special equations hit the arXiv today!

01 Feb 2020

Conflict-free routing of multi-stop warehouse trucks, joint work with Sandy Brownlee and others, has appeared in the journal Optimization Letters.

23 Oct 2019

I relocated to Australia and joined the Trustworthy Systems Research Group of CSIRO!

14 Oct 2019

I successfully defended my PhD thesis, Development of Group Theory in the Language of Internal Set Theory.

07 Sep 2019

I gave an invited talk at LMS Prospects in Mathematics 2019, on applications of Nelson's proof-theoretic algorithm in topology.


Zoltan and bushcraft crew gathered around a sundial

I find the Big City and the Great Outdoors equally comfortable. On my day off, you're just as likely to meet me in the dense urban area of the Sydney CBD as in the coastal bushlands of New South Wales.

Zoltan and friends going up Kinder Scout

If you live in the UK, you may know me from Manchester Orators, a chartered club for the promotion of public speaking, presentation and leadership skills. I was a member between 2017 and 2019, and I hope to return one day.

Meal ingredients mise en place

At home, you'll find me in the kitchen. Fusion cuisine, simple ingredients, mise en place. Sometimes, something good comes out of it.

Salmon ceviche and other favorite dishes


If you have any questions, or if you wish to hire me as a consultant, use the email address below. You can also find me on the Mathematics Stack Exchange and on Semantic Scholar.

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