## Research talks

- Another quantifier-elimination result in arithmetic under negated induction. Online Models of PA (MOPA) seminar, City University of New York, USA, 31 June, 2022. Logic Seminar, National University of Singapore, 17 August, 2022. Online Warsaw Mathematical Logic Seminar, Poland, 14 June, 2023.
- I presented another quantifier-elimination result in arithmetic under negated induction. This gives new information about pigeonhole principles and expansions to second-order models. This work is joint with David Belanger, CT Chong, Wei Li, and Yue Yang.
- The slides are available here in pdf.

- Arithmetic under negated induction. Online Models and Sets seminar, University of Leeds, UK, 27 January, 2021. Online Logic Seminar, National University of Singapore, 3 February, 2021. Online Logic Seminar, Institute of Mathematics of the Czech Academy of Sciences, 13 December, 2021. Computability Theory and Applications (CTA) Online Seminar, 15 December, 2021.
- Arithmetic generally does not admit any non-trivial quantifier elimination. I talked about one exception, where the negation of an induction axiom is included in the theory. Here the Weak König Lemma from reverse mathematics arises as a model completion.
- The work presented is joint with Marta Fiori-Carones, Leszek Aleksander Kołodziejczyk and Keita Yokoyama.
- There were some inaccuracies in the slides I used. A corrected version of the slides is available here in pdf.

- Properties preserved in cofinal extensions. Online Models of PA (MOPA) seminar, City University of New York, USA, 22 July, 2020. Online Warsaw Mathematical Logic Seminar, Poland, 19 May, 2021.
- I discussed how much induction cofinal extensions of models of arithmetic preserve.
- The slides are available here in pdf.

- When the Weak Pigeonhole Principle is weaker. Online Warsaw Mathematical Logic Seminar, Poland, 24 June, 2020.
- I introduced the Σ
_{1}Weak Pigeonhole Principle and related it with other familiar theories in arithmetic. The work reported comes from a preprint written with David Belanger, CT Chong, Wei Wang, and Yue Yang.

- I introduced the Σ
- End extensions and subsystems of second-order arithmetic. Talk at the Japan Advanced Institute of Science and Technology, Nomi, Japan, 12 April, 2019. Logic seminar, Department of Mathematics, National University of Singapore, 13 March, 2019.
- I attempted to justify the importance of the Big Five in reverse mathematics mathematically in terms of the characteristics of their models. The work presented is joint with Stephen G. Simpson (Vanderbilt).

- Extracting proof bounds from forcing-style arguments. Talk at model theory session, Chinese Mathematical Logic Conference, Qiannan Normal University for Nationalities, Duyun, China, 13 May, 2018.
- Using my joint work with Leszek Kołodziejczyk (University of Warsaw) and Keita Yokoyama (Japan Advanced Institute of Science and Technology), I demonstrated how to extract proof bounds from forcing-style arguments. The general framework involved was originally developed by Avigad.
- The slides are available here in pdf.

- Induction and collection in arithmetic. Logic seminar, Department of Mathematics, National University of Singapore, 7 March, 2018.
- The classic Friedman–Paris conservation result tells us that the Σ
_{n+1}collection scheme and the Σ_{n}induction scheme prove the same Π_{n+2}sentences. Clote, Hájek, and Paris asked whether one of these schemes is more efficient than the other in proving such sentences. I presented a partial answer to their question via a syntactic approach to forcing developed by Avigad. This research is joint with Leszek Kołodziejczyk (University of Warsaw) and Keita Yokoyama (Japan Advanced Institute of Science and Technology).

- The classic Friedman–Paris conservation result tells us that the Σ
- Remarks on Wilkie and Paris’s notion of fullness. Mathematical logic seminar, University of Warsaw, Poland, 15 November, 2017.
- We discussed the connections between the notion of fullness and the End-Extension Question. I also presented a few new results from my recent paper on this topic.

- Satisfaction classes on restricted formula classes. Contributed talk, Warsaw Workshop on Formal Truth Theories, University of Warsaw, Poland, 29 September, 2017.
- I showed how to characterize some popular notions of partial recursive saturation in terms of the existence of satisfaction classes. I also mentioned an application in end-extension constructions.

- Characterizing strong theories using end extensions. Arithmetic seminar, University of Warsaw, Poland, 29 March and 5 April, 2017.
- It is well known that end extensions of models can be used to characterize the strengths of theories in first-order arithmetic. Many of these characterizations carry over easily to the context of second-order arithmetic. However, this does not give much information about theories stronger than Peano arithmetic, e.g., ATR
_{0}and Π^{1}_{1}-CA_{0}, because there is apparently no natural theory in first-order arithmetic whose strength is at this level. I showed in these talks that such theories can indeed be used to construct interesting end extensions. This work is joint with Stephen G. Simpson.

- It is well known that end extensions of models can be used to characterize the strengths of theories in first-order arithmetic. Many of these characterizations carry over easily to the context of second-order arithmetic. However, this does not give much information about theories stronger than Peano arithmetic, e.g., ATR
- Subsets coded in end extensions. Arithmetic seminar, University of Warsaw, Poland, 11, 18, and 25 January, 2017.
- This series of three talks was about two subtly different forcing constructions based on the Arithmetized Completeness Theorem for building end extensions of models of arithmetic. I demonstrated in detail the power of these constructions in controlling the family of sets coded in the end extension.

- ACT forcing. Research seminar, Kurt Gödel Research Center for Mathematical Logic, Vienna, Austria, 24 November, 2016.
- ACT stands for ‘Arithmetized Completeness Theorem’. The usual proof of Gödel’s Completeness Theorem for first-order logic is evidently a forcing-style construction. In many applications, such a construction can easily be transformed into a (complicated perhaps, but natural) recursive construction. I talked about one example for which this is not the case in the model theory of arithmetic.

- Model theory of arithmetic and inductive theorem proving. Talk at the Third Workshop on Automated Inductive Theorem-Proving (WAIT), TU Wien, Vienna, Austria, 18 November, 2016.
- I talked about some pieces of information related to inductive theorem proving that Stefan Hetzl (Vienna) and I extracted from the logical study of arithmetic.

- A new construction of models of the Weak König Lemma. Contributed talk, Logic Colloquium, University of Leeds, UK, 2 August, 2016.
- I presented a construction of models of the Weak König Lemma based on a version of the Arithmetized Completeness Theorem and some ideas from forcing. This work is joint with Ali Enayat (Gothenburg).

- Upgrading the Arithmetized Completeness Theorem. Invited talk, Journées sur les Arithmétiques Faibles 35, Universidade de Lisboa, Lisbon, Portugal, 7 June, 2016.
- I showed how to combine the Arithmetized Completeness Theorem with forcing to re-prove two theorems in the Simpson–Tanaka–Yamazaki paper (Ann. Pure Appl. Logic 118, pp. 87–114). This research is joint with Ali Enayat (Gothenburg).
- The slides are available here in pdf.

- Revisiting some conservation results on Weak König’s Lemma. Midlands Logic Seminar, University of Birmingham, UK, 28 April, 2016.
- I presented a new proof of the Simpson–Tanaka–Yamazaki conservation result between RCA
_{0}and WKL_{0}using the Arithmetized Completeness Theorem. This research is joint with Ali Enayat (Gothenburg).

- I presented a new proof of the Simpson–Tanaka–Yamazaki conservation result between RCA
- Models of Weak König’s Lemma. Invited talk, Computability Theory and Foundations of Mathematics conference, Tokyo Institute of Technology, Japan, 10 September, 2015.
- Tanaka and his collaborators made significant contributions to the understanding of nonstandard models of Weak König’s Lemma (WKL). On the one hand, he introduced self-embeddings to second-order arithmetic, and explained why WKL is relevant in such constructions. On the other hand, research about Tanaka’s conjecture on the conservativity of WKL led to the discovery of a novel technique, due jointly to Simpson, Tanaka and Yamazaki, for producing very similar yet very different models of WKL. In this talk, I surveyed these results, and reported on some ongoing work in refining them.
- The slides are available here in pdf.

- The Arithmetized Completeness Theorem. Talk at the Sets and Computations program at the National University of Singapore, 17 April, 2015.
- This talk was a gentle introduction to the Arithmetized Completeness Theorem. I presented a number of classical applications, and discussed generalizations of these by Ali Enayat (Gothenburg) and me to models with only bounded collection and exponentiation.
- The slides are available here in pdf.

- Variations of the Arithmetized Completeness Theorem. Logic seminar, Institute of Mathematics of the Academy of Sciences, Prague, Czech Republic, 16 February, 2015.
- I presented some non-standard applications of the Arithmetized Completeness Theorem about expansions and ω-submodels that satisfy the Weak König Lemma. I discussed, in addition, the problems one has to face when, instead of Σ
_{1}-induction, the ground model only has Σ_{1}-collection with exponentiation. This research is joint with Ali Enayat (Gothenburg).

- I presented some non-standard applications of the Arithmetized Completeness Theorem about expansions and ω-submodels that satisfy the Weak König Lemma. I discussed, in addition, the problems one has to face when, instead of Σ
- Σ
_{1}-Collection and the Completeness Theorem. Talk at the Joint Prague–Vienna Logic and Set Theory Meeting, Institute of Mathematics of the Academy of Sciences, Prague, Czech Republic, 29 September, 2014.- The formalized version of Gödel’s Completeness Theorem is one of the most powerful tools in constructing models of arithmetic. By passing to second-order arithmetic, Ali Enayat (Gothenburg) and I made this tool readily available to many models of Σ
_{1}-collection. Some nice applications were presented in the talk.

- The formalized version of Gödel’s Completeness Theorem is one of the most powerful tools in constructing models of arithmetic. By passing to second-order arithmetic, Ali Enayat (Gothenburg) and I made this tool readily available to many models of Σ
- Going beyond Peano arithmetic? Talk at the IMS–JSPS Joint Workshop in Mathematical Logic and Foundations of Mathematics, National University of Singapore, 3 September, 2014.
- I discussed some recent attempts of mine in finding model-theoretic properties of cuts whose strengths are above Peano arithmetic.
- The slides are available here in pdf.

- An application of the Arithmetized Completeness Theorem to second-order arithmetic. Contributed talk, Logic Colloquium, TU Wien, Vienna, Austria, 15 July, 2014.
- I presented a simple proof of Harrington’s theorem, that every model of IΣ
_{1}expands to a model of WKL_{0}, using Cornaros and Dimitracopoulos’s variant of the Arithmetized Completeness Theorem (Arch. Math. Logic 39, pp. 459–463). - The slides are available here in pdf.

- I presented a simple proof of Harrington’s theorem, that every model of IΣ
- Understanding BΣ
_{1}+exp via WKL_{0}^{∗}. Contributed talk, Journées sur les Arithmétiques Faibles 33, University of Gothenburg, Sweden, 18 June, 2014.- The theory of Σ
_{1}-collection (BΣ_{1}) is the subject of many important open questions in arithmetic. These questions are usually more tractable when the axiom exp asserting the totality of exponentiation is added to the theory. Nevertheless, the arguments involved remain mostly rather technical. We show that all these technicalities can be wrapped up into one black box, namely, the Simpson–Smith conservation theorem between BΣ_{1}+exp and WKL_{0}^{∗}(Ann. Pure Appl. Logic 31, pp. 289–306), modulo which other arguments become conceptually much more transparent. This research is joint with Ali Enayat (Gothenburg). - The slides are available here in pdf.
**Note**: a gap was later discovered in my argument for showing what is referred to as the Dimitracopoulos–Paschalis Theorem in the talk. So far, I do not see how this gap can be repaired.

- The theory of Σ
- Internal and external counting in nonstandard models of arithmetic. Logic seminar, Department of Philosophy, Linguistics and Theory of Science, University of Gothenburg, Sweden, 21 February, 2014.
- The interplay between internal and external counting is one of the most beautiful aspects of the model theory of nonstandard arithmetic. I presented one classical example of this about the
*regularity scheme*, and mentioned some relevant ongoing work about cuts.

- The interplay between internal and external counting is one of the most beautiful aspects of the model theory of nonstandard arithmetic. I presented one classical example of this about the
- Some model theory of Peano arithmetic. Research seminar, Kurt Gödel Research Center for Mathematical Logic, Vienna, Austria, 31 October, 2013.
- I presented a personal view on Peano arithmetic and generic cuts.

- The various faces of generic cuts. Logic seminar, Department of Mathematics, National University of Singapore, 25 September, 2013.
- I discussed the robustness of genericity in the context of cuts of models of arithmetic.

- A model-theoretic view of reverse mathematics. Contributed talk, Thirteenth Asian Logic Conference, Sun Yat-sen University, Guangzhou, China, 19 September, 2013.
- I presented an attempt to explain the robustness of the Big Five systems in reverse mathematics by model-theoretic means.

- The generic choice of a cut. Invited talk, 32ème Journées sur les Arithmétiques Faibles, University of Athens, Greece, 25 June, 2013.
- Pick a cut in a nonstandard model of arithmetic ‘at random’. What are we ‘likely’ to get? In the talk, I presented a natural notion of ‘likelihood’ based on indicators, with respect to which this question admits a nice answer.

- End-extensions of models of second-order arithmetic. Invited talk, 32ème Journées sur les Arithmétiques Faibles, University of Athens, Greece, 24 June, 2013.
- End-extensions play a central role in the theory of first-order arithmetic. In the context of second-order arithmetic, they have apparently not received much attention. During the talk, I surveyed some known results around this topic, and compared them with the analogous statements in first-order arithmetic.

- Interstices and existentially closed cuts in nonstandard models of arithmetic. Seminar talk at the Institute of Mathematics, Polish Academy of Sciences, Warsaw, Poland, 14 June, 2013.
- An
*interstice*in a nonstandard model of Peano arithmetic (PA) is a maximal convex subset of the model that does not contain any definable element. The study of interstices was initiated by the papers of Kaye–Kossak–Kotlarski and of Bamber–Kotlarski in the investigation of the structure of the automorphism group of a countable recursively saturated model of PA.*Existentially closed models*are generalizations of algebraically closed fields in model theory. It is perhaps unexpected that such models have connections with interstices. The two notions turn out to meet at the theory of*cuts*, or initial segments, of nonstandard models of PA.

- An
- Reverse mathematics and model theory. Logic and Analysis seminar, Department of Mathematics, Ghent University, Belgium, 17 May, 2013.
- I proposed a model-theoretic version of the reverse mathematics program. I presented some known theorems and new results in this direction.

- Generic cuts in models of arithmetic, reprise. Logic seminar, Department of Mathematics, Pennsylvania State University, USA, 26 March, 2013.
- This was a reprise of the talk I gave at the Eleventh Asian Logic Conference in Singapore. I described in much greater detail what generic cuts look like, and how one can picture them.

- Generalizing the notion of interstices. Models of PA (MOPA) seminar, City University of New York, USA, 14 March, 2013.
- I presented a generalization of the notion of interstices that originated from the study of generic cuts.

- Understanding genericity for cuts. Seminar talk at the CUNY Logic Workshop, City University of New York, USA, 1 March, 2013.
- I talked about what
*genericity*means amongst the great variety of cuts in nonstandard models of arithmetic. This notion of genericity came from a version of model theoretic forcing devised by Richard Kaye in his 2008 paper.

- I talked about what
- Where closure under Turing jumps can replace elementarity between structures. Contributed talk, Computability Theory and Foundations of Mathematics Workshop, Tokyo Institute of Technology, Japan, 20 February, 2013.
- Vladimir Kanovei observed that if
`M`is a proper elementary extension of the natural numbers, then the external Scott algebra of`M`contains a real from the Turing degree**0**^{(ω)}. This talk was about some variants of this observation from my joint work with Richard Kaye (University of Birmingham, UK) and Roman Kossak (City University of New York, USA). Amongst other things, we showed that Kanovei’s elementarity condition can be replaced by`M`being a nonstandard model of Peano arithmetic whose standard system is closed under the Turing jump operation. - The slides are available here in pdf.

- Vladimir Kanovei observed that if
- Countable numbers in a model of arithmetic: a survey. Talk at the JSPS–FWO Joint Research Seminar on Mathematical Logic, Tokyo Institute of Technology, Japan, 15 February, 2013.
- A
*countable number*in a model`M`of Peano arithmetic is a number that has countably many predecessors in`M`. This talk included a survey on the known results about these countable numbers as an initial segment of the model`M`. It was aimed at graduate students in mathematical logic.

- A
- What are generic cuts? Talk at the JSPS–FWO Workshop on Mathematical Logic, Ghent University, Belgium, 6 December, 2012.
- I argued that the genericity of cuts is a robust notion by presenting four new characterizations of it.

- Revisiting cuts and indicators. Talk at the Workshop on Logic and Analysis, Ghent University, Belgium, 16 November, 2012.
- I talked about under what functions ω-limit cuts are closed. I asked whether there is some relationship between these cuts and phase transitions of independence results.

- Cuts closed under a specified family of functions. Talk at the 2012 International Workshop of Logic, Sun Yat-sen University, Guangzhou, China, 23 September, 2012.
- Suppose we are interested in, for example, the primitive recursive functions. Given a nonstandard model of Peano arithmetic, it is natural to look for
*cuts*that are closed under all such functions. This, in itself, is an easy exercise. However, if it is additionally required that the cut is not closed under any function other than the primitive recursive ones, then we need to get more involved. In this talk, I explained why the general problem relates to*existentially closed models*. I also pointed out some applications to the study of generic cuts. - The slides are available here in pdf.

- Suppose we are interested in, for example, the primitive recursive functions. Given a nonstandard model of Peano arithmetic, it is natural to look for
- Axiom schema for a model of arithmetic with a cut. Talk at the Model Theory and Proof Theory of Arithmetic conference, Będlewo, Poland, 25 July, 2012.
- I presented some ongoing joint work with Richard Kaye, Birmingham and Roman Kossak, City University of New York on the model theory of pairs
`(M,I)`, where`I`is a cut of a model`M`of Peano arithmetic. More specifically, we isolated some combinatorial principles that one needs to build special elementary extensions of such pairs. The strength of these principles were studied and compared with the usual theories in reverse mathematics. - The slides are available here in pdf.

- I presented some ongoing joint work with Richard Kaye, Birmingham and Roman Kossak, City University of New York on the model theory of pairs
- The model theory of generic cuts. Contributed talk, Logic Colloquium, University of Manchester, UK, 17 July, 2012.
- I am interested in pairs of the form
`(M,I)`, where`M`is a nonstandard model of Peano arithmetic and`I`is a cut of`M`. Cuts have been extensively studied since the 1970s, mainly because of their relationship with independence results such as the Paris–Harrington theorem. However, surprisingly little about such pairs`(M,I)`exists in the literature. This talk was about my attempt with Richard Kaye, Birmingham to fill this gap along the tracks of Robinson-style model theory. Arithmetic usually does not fit well into this theory, but it turns out that the generic cuts, a new family of cuts recently discovered by Richard Kaye, fit in rather nicely. Amongst other results, we showed that pairs`(M,I)`with`I`generic are*existentially closed*. - The slides are available here in pdf.

- I am interested in pairs of the form
- Model theory, arithmetic and cuts. Logic seminar, Department of Mathematics, Ghent University, Belgium, 20 April, 2012.
- I discussed how generic cuts fit into the framework of Robinson-style model theory. The research reported is joint with Richard Kaye, Birmingham.

- Generic properties of cuts in a nonstandard model of arithmetic. Talk for the logic workshop at the Department of Mathematics, Ghent University, Belgium, 16 March, 2012.
- This research is joint with Richard Kaye, Birmingham. We used a class of Banach–Mazur games to define a notion of ‘almost all cuts’, and hence a filter of
*generic properties*, in a given a nonstandard model of Peano arithmetic. In the talk, some cases in which a law of excluded middle holds, and others in which it fails, were presented.

- This research is joint with Richard Kaye, Birmingham. We used a class of Banach–Mazur games to define a notion of ‘almost all cuts’, and hence a filter of
- Omitting Types Theorem and reverse mathematics. Talk at the Workshop on Proof Theory and Computability Theory, Sendai, Japan, 21 February, 2011.
- Omitting certain types in a purely model-theoretic context sometimes leads to the rediscovery of combinatorial principles that are of interest in reverse mathematics. I demonstrated this with a selection of examples that originated from the work of Scott, Gaifman, Paris, Kirby, and others.

- Some model theory of generic initial segments. Logic seminar, Mathematical Institute, Tohoku University, Sendai, Japan, 18 February, 2011.
- This was ongoing joint work with Richard Kaye. We study structures of the form
`(M,I)`, where`I`is an initial segment of a nonstandard model of arithmetic`M`. Such structures play an important role in the recent work of Keisler and Yokoyama in nonstandard analysis and reverse mathematics. I presented a few model-theoretic properties of`(M,I)`where`I`is a generic initial segment of`M`, including some results on quantifier elimination, existential closure, and extensions.

- This was ongoing joint work with Richard Kaye. We study structures of the form
- Generalizations of the Law of Excluded Middle. Talk at the Philosophy Department, Sun Yat-Sen University, Guangzhou, China, 31 December, 2010.
- I showed how several combinatorial principles can be viewed as generalizations of the Law of Excluded Middle via the model theory of arithmetic.

- Model-theoretic forcing in arithmetic. Logic seminar, Department of Mathematics, National University of Singapore, 10 November, 2010.
- It is well-known that the regularity scheme is intimately related to end-extensions. In the same sense, it was shown that the existence of cohesive sets is connected to conservative extensions.

- Building nonstandard universes using Ramsey-type principles. Logic seminar, Department of Pure Mathematics and Computer Algebra, Ghent University, Belgium, 10 December, 2009.
- Suppose we want to apply nonstandard methods to prove theorems about the natural numbers. We need to build nice nonstandard universes that contain the natural numbers as well as some ideal numbers which we can utilize. Gaifman, Phillips, and others observed that this can be done using Ramsey’s Theorem. In this talk, I briefly explained their construction, and reported some recent work to strengthen it to make better nonstandard universes. All these constructions work for general models of arithmetic.

- End-extending models of arithmetic using Ramsey-type principles. Talk at the 2009 Ramsey Theory in Logic, Combinatorics and Complexity Workshop, Bertinoro, Italy, 30 October, 2009.
- The role of Ramsey’s Theorem in end-extension constructions. Logic seminar, School of Mathematics, University of Leeds, UK, 14 October, 2009.
- Gaifman (and independently others) showed that Formalized Ramsey’s Theorem can be used to build elementary end-extensions of models of arithmetic. It is also known that this use of Ramsey’s Theorem is necessary. In this talk, I discussed some work in progress which attempts to generalize these to stronger Ramsey-type theorems at the level of ATR
_{0}or above. The aim of this work is to better understand the roles played by Ramsey’s Theorem and related combinatorial principles in these constructions.

- Gaifman (and independently others) showed that Formalized Ramsey’s Theorem can be used to build elementary end-extensions of models of arithmetic. It is also known that this use of Ramsey’s Theorem is necessary. In this talk, I discussed some work in progress which attempts to generalize these to stronger Ramsey-type theorems at the level of ATR
- Generic cuts in models of Peano arithmetic. Contributed talk, Logic Colloquium, Sofia University, Bulgaria, 5 August, 2009.
- Some recent progress on this topic was reported.
- The slides are available here in pdf.

- Generic cuts in models of Peano arithmetic. Contributed talk, Eleventh Asian Logic Conference, National University of Singapore, 25 June, 2009.
- This was another report of the results in my MPhil (qualifying) thesis, presented in a different way from what I did in the Manchester talk.
- The slides are available here in pdf.

- Generic cuts of models of Peano arithmetic. Contributed talk, Fourth MATHLOGAPS Training Workshop, University of Manchester, UK, 10 July, 2008.
- Some results proved in my MPhil (qualifying) thesis were reported.
- The slides are available here in pdf.

- Peano arithmetic and finite set theory. MSci presentation, University of Birmingham, UK, 6 June, 2006.
- This presentation was an introduction to nonstandard models of Peano arithmetic. I put special emphasis on describing how one can turn these models into models of finite set theory, and vice versa.

## General-audience talks and expositions

- Proof and truth in mathematics. Math Interest Talk for the Math Society, National University of Singapore, 3 September, 2021.
- We discussed what mathematical logic tells us about mathematics and logic.
- The slides are available here in pdf. More technical details of the proofs I gave can be found in my MA5219 notes on the consistency of arithmetic.

- Gödel’s Incompleteness Theorems and more. Seminar at the Department of Mathematics, Chinese University of Hong Kong, China, 6 September, 2016.
- This was an introduction to Gödel’s Incompleteness Theorems and reverse mathematics.

- Nonstandard models of Peano arithmetic. A short course aimed at a general mathematical audience. Chinese University of Hong Kong, China, August–October, 2011.
- We went through the basics of mathematical logic and of nonstandard models of arithmetic. The key results proved included Gödel’s Completeness Theorem, the MacDowell–Specker Theorem, the Tennenbaum Theorem, and the Paris–Harrington Theorem.

- Computer programming and natural number arithmetic. Guest lecture for the Enrichment Programme for Young Mathematics Talents. Chinese University of Hong Kong, China, 27 July, 2011.
- I gave an exposition of a correspondence between computer programs and natural number arithmetic, and showed that the halting problem is not solvable by a computer. Building on this, I presented Gödel’s Incompleteness Theorem and the MRDP solution to Hilbert’s Tenth Problem.
- The slides are available here in pdf.

- Nonstandard models of arithmetic. A short course aimed at graduate students. National University of Singapore, February–April, 2011.
- We followed Richard Kaye’s book Models of Peano Arithmetic. In the seven weekly lectures, we went through Chapters 11–15.

- Introduction to coalgebras. Two talks in the informal logic seminar series. University of Birmingham, UK, 12 and 19 March, 2009.
- We investigated how coalgebras help understand certain circular phenomena. Algebras and coalgebras were looked at in parallel, showing the beauty of the duality.

- Games in set theory. Talk in the informal logic seminar series. University of Birmingham, UK, 8 February, 2008.
- This talk was a gentle introduction to the Axiom of Determinacy.